JULIUS, HAYDEN, August 2021 PURE MATHEMATICS NONSTANDARD SOLUTIONS of LINEAR PRESERVER PROBLEMS (111 Pages) Dissertation Advisor

JULIUS, HAYDEN, August 2021 PURE MATHEMATICS

NONSTANDARD SOLUTIONS OF LINEAR PRESERVER PROBLEMS (111 pages)

Dissertation Advisor: Mikhail Chebotar

Linear preserver problems concern the characterization of linear operators on matrix spaces that leave invariant certain functions, subsets, relations, etc. We present several linear preserver problems whose solutions may be considered nonstandard since they diﬀer signiﬁcantly from classical results. In addition, we also discuss several related linear preserver problems with standard solutions to highlight the phenomena observed. NONSTANDARD SOLUTIONS OF LINEAR PRESERVER PROBLEMS

A dissertation submitted

to Kent State University

in partial fulﬁllment of the requirements

for the degree of Doctor of Philosophy

Hayden Julius

August 2021

c Copyright

Except for previously published materials Dissertation written by

Hayden Julius

B.S., Cleveland State University, 2016

M.S., Kent State University, 2019

Ph.D. Kent State University, 2021

Approved by

Mikhail Chebotar , Chair, Doctoral Dissertation Committee

Joanne Caniglia , Members, Doctoral Dissertation Committee

Feodor Dragan

Mark L. Lewis

Dmitry Ryabogin

Accepted by

Andrew M. Tonge , Chair, Department of Mathematical Sciences

Mandy Munro-Stasiuk , Interim Dean, College of Arts and Sciences TABLE OF CONTENTS

TABLE OF CONTENTS...... iv

PREFACE...... vi

ACKNOWLEDGMENTS...... ix

1 An introduction to linear preserver problems...... 1

1.1 Types of linear preserver problems...... 1

1.1.1 Subset preservers...... 1

1.1.2 Function preservers...... 5

1.1.3 Relation preservers...... 7

1.1.4 Transformation preservers...... 10

1.2 Generalizations...... 12

1.3 Nonstandard solutions of linear preserver problems...... 13

2 Maps preserving certain elements of a subset...... 17

2.1 On maps sending rank-k square-zeros to square-zeros...... 20

2.2 On maps sending rank-t idempotents to idempotents...... 24

2.3 Related problems and conjectures...... 31

3 Maps preserving nonzero Lie products...... 37

3.1 Commutativity preservers and Herstein’s Lie isomorphism problem...... 42

3.2 On maps preserving Lie products equal to e12 ...... 45

3.2.1 Some results on triangularization...... 45

iv 3.2.2 The main result...... 47

3.3 On maps preserving Lie products equal to e11 − e22 ...... 57

3.4 Related problems and conjectures...... 61

4 Maps preserving nonzero ordinary products...... 64

4.1 Maps preserving products equal to ﬁxed invertible matrices...... 65

4.2 Maps preserving products equal to a diagonalizable matrix...... 69

4.3 Related problems and conjectures...... 77

5 Maps preserving roots of nonhomogeneous polynomials...... 84

5.1 Maps preserving the square roots of an idempotent matrix...... 89

5.2 Maps preserving the square roots of a rank-one nilpotent...... 92

BIBLIOGRAPHY ...... 98

v PREFACE

The main goal of this dissertation is to illustrate the general framework of linear preserver problems

(LPPs) and introduce new results that complement, generalize, or stand in contrast to existing theorems in the literature. However, LPPs form an extremely active research topic that incorporates objects and techniques from matrix theory, operator theory, noncommutative algebra, projective and algebraic geometry, functional analysis, etc., and in what follows, we will have need to discuss many preserver problems that touch on many of these areas.

Consequently, I have decided in many places to state a known result in a simplified form and not in its full, optimal form. For instance, there are plenty of results below that hold for matrices over an arbitrary field or even on rectangular spaces of matrices, but it will be convenient to simply think of complex square matrices. Hence the theorems being represented here (more than likely) appear differently in their actual papers. There are close to 70 individual preserver problems mentioned in what follows, and thus it would be a significant task for the reader to keep track of each problem’s minor differences. I will explicitly mention the full, optimal form if relevant to the matter at hand.

I have also used uniform notation throughout.

Hence the reader can usually expect the LPPs mentioned below to be stated on complex square matrices. The Greek letter φ is always reserved for the mapping between matrix spaces being discussed (unless a speciﬁc example has been deﬁned). The hypotheses on φ might change from paragraph to paragraph and theorem to theorem. I have tried to make this as clear as possible.

A list of fixed notations that I use throughout the document without reference is provided on page viii. Now I will also list the conventions used. Each vector in a finite-dimensional vector space over a field is represented as a column vector, and matrix multiplication is on the left. Furthermore,

vi if T˜ and S˜ are two linear transformations with T and S their matrices relative to some ordered basis, then the matrix multiplication TS represents the composition T˜S˜, so that S˜ “acts ﬁrst” on the vector space.

The full algebra of n × n matrices over a ﬁeld F is denoted Mn(F ). I stipulate that whenever

Mn(F ) is written, it is inherently assumed that n ≥ 2. All rings considered here are associative but do not necessarily have a multiplicative identity. A map that ﬁxes the multiplicative identity is called a unital map. I have assumed a level of familiarity with linear algebra up to Jordan canonical forms and a basic understanding of groups, rings, and ﬁelds.

vii Frequently used notation:

C ﬁeld of complex numbers

Fp ﬁnite ﬁeld of order p

char(R) the characteristic of a ring R

Mn(R) set of all n × n matrices over a ring R (shortened to Mn if R = C)

sln(F ) set of trace-zero matrices over a ﬁeld F (shortened to sln if F = C)

dimF V dimension of a vector space V over a ﬁeld F (shortened to dim V if F = C)

(aij) a matrix with entries aij, where i indexes rows and j indexes columns

eij matrix with 1 in the (i, j)-entry and zeros elsewhere (called a matrix unit)

In n × n identity matrix, also written as e11 + e22 + ··· + enn

At transpose of a matrix A

tr(A) trace of a matrix A

det(A) determinant of a matrix A

rank(A) rank of a matrix A

σ(A) spectrum of a matrix A

A ◦ B Jordan product AB + BA

[A, B] Lie product AB − BA

C(A) centralizer of A, {B ∈ Mn(F ) | AB = BA}

FA set of F multiples of A, {cA | c ∈ F }

V1 ⊕ V2 direct sum of vector spaces

φ|V restriction of a map φ to V

bxc the largest integer n such that n ≤ x

dxe the smallest integer n such that x ≤ n

viii ACKNOWLEDGMENTS

Firstly I would like to express profound gratitude to my advisor Mikhail Chebotar. Misha’s guidance over the past few years has been incredible. He has helped me publish a nontrivial amount of papers, navigate the academic job market during a global pandemic, and advocated for me when travel was not possible. In addition to his professional support, Misha has been there for me personally. I will fondly remember all of our conversations past and present about current events, fresh ﬁsh and coﬀee, the nature of academic publishing, and entertainment; in particular, stop motion cartoons about the Cold War.

Thank you to each member of my defense committee, Mark Lewis, Dmitry Ryabogin, Feodor

Dragan, and Joanne Caniglia for their generous time spent with my monstrous dissertation. I would also like to acknowledge Misha, Mark, and Jenya Soprunova for the wonderful opportunities with the 2019 and 2020 REU programs at Kent State University.

Thank you to Victor Ginsburg, Ricardo Velasquez, Nikita Borisov, and Martha Sikora for their collaboration on problems that appear in this dissertation. I am honored to have played a (small) part in their future success as young researchers.

I am especially grateful to Artem Zvavitch and Andrew Tonge for their guidance and support throughout my time at Kent State. Thank you to Patty Housley and Virginia Wright for their constant and invaluable assistance. I would also like to acknowledge my amazing fellow graduate students, past and present, who have been there for me over the years.

Thank you to my parents, Mark and Alison, and my siblings, Maggie and Anthony, for their love and so much more. To Miranda, for everything. And to Toby, my best friend.

ix CHAPTER 1

An introduction to linear preserver problems

Linear preserver problems concern the characterization of linear operators on matrix spaces that leave invariant certain functions, subsets, relations, etc. Emerging prominently in the latter half of the 20th century, the earliest linear preserver problems arose naturally as converse problems from matrix theory. Since then, LPPs have become an active, fertile area of research with applications to many branches of mathematics, both pure and applied.

What follows is a brief discussion highlighting several signiﬁcant and classical results that moti- vate the upcoming presentation of results in later chapters. This introduction follows the expository papers [37, 52, 56].

1.1 Types of linear preserver problems

Let M denote a matrix space, by which we mean a vector space whose elements are linear operators.

1.1.1 Subset preservers

Let S be a proper subset of M. We say that the linear map φ : M → M preserves S if

A ∈ S =⇒ φ(A) ∈ S and φ : M → M preserves S in both directions if

A ∈ S ⇐⇒ φ(A) ∈ S.

A typical subset preserver problem asks for a characterization of maps φ such that S is invariant under φ. Equivalently, we write φ(S) ⊆ S if φ preserves S and φ(S) = S if φ preserves S in both

1 directions.

In the familar case of Mn := Mn(C), there are many important subsets of matrices that are well understood and of chief interest in applications. Here we record the classical results that speciﬁcally address these subset preservers. The ﬁrst result is of fundamental importance.

Theorem 1.1.1 (Marcus, Moyls [61]). If φ : Mn → Mn is a bijective linear map preserving rank- one matrices, there exist invertible matrices U ∈ Mn and V ∈ Mn such that

(1) φ(X) = UXV , or

(2) φ(X) = UXtV .

for all X ∈ Mn.

Note that X 7→ UXV and X 7→ UXtV preserves the rank of every matrix. A natural follow-up to this theorem concerns maps preserving ranks other than 1.

Theorem 1.1.2 (Beasley [2]). Fix k ∈ {1, . . . , n}. If φ : Mn → Mn is a bijective linear map preserving rank-k matrices, there exist invertible matrices U ∈ Mn and V ∈ Mn such that

(1) φ(X) = UXV , or

(2) φ(X) = UXtV .

for all X ∈ Mn.

Beasley [2] also provides numerous conditions for the description to hold without assuming bijectivity. In particular, if for every 1 ≤ j ≤ k, the image of a rank-j matrix is nonzero, the result holds.

Let GLn and Σn denote the set of invertible and singular complex square matrices, respectively.

Theorem 1.1.2 in the case n = k was proved earlier by Marcus and Purves.

2 Theorem 1.1.3 (Marcus, Purves [62]). If φ : Mn → Mn is a linear map such that φ(GLn) ⊆ GLn

(i.e., φ preserves invertibility), then there exist invertible matrices U ∈ Mn and V ∈ Mn such that

(1) φ(X) = UXV , or

(2) φ(X) = UXtV

for all X ∈ Mn.

The complementary result for singular matrices follows from a result of Dieudonn´e.

Theorem 1.1.4 (Dieudonn´e[26]). If φ : Mn → Mn is a bijective linear map such that φ(Σn) ⊆ Σn, there exist invertible matrices U ∈ Mn and V ∈ Mn such that

(1) φ(X) = UXV , or

(2) φ(X) = UXtV

for all X ∈ Mn.

Theorem 1.1.4 was actually proved under the assumption that φ was merely semilinear. Recall that a semilinear transformation is an additive mapping φ : Mn(F ) → Mn(F ) with the scalar law deﬁned with respect to a ﬁeld automorphism τ : F → F ; that is, φ(cX) = τ(c)φ(X) for

t all X ∈ Mn(F ). The theorem concludes that the mappings X 7→ Uτ(X)V or X 7→ Uτ(X) V , where τ(X) is the matrix obtained by applying τ entrywise to X, are the only bijective semilinear mappings preserving singular matrices.

Now we will discuss some subsets of matrices not distinguished by rank. A matrix A is nilpotent if there exists natural number m such that Am = 0. The smallest such m such that Am = 0 is called the nilindex of A. Recall that every eigenvalue of a nilpotent matrix is zero and the linear span of nilpotent matrices is sln, the space of n × n trace-zero matrices.

3 Theorem 1.1.5 (Botta, Pierce, Watkins [8]). If φ : sln → sln is a bijective linear map that preserves nilpotent matrices, there exists an invertible matrix U ∈ Mn and c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ sln.

We note that the proof of Theorem 1.1.5 solves the rank-one preserver problem on the space sln as a consequence. Indeed, let N be a subalgebra of Mn of maximal dimension in which every element is nilpotent (in fact, every maximal nilpotent subalgebra is conjugate to the strictly-

n−1 upper triangular matrices by a famous theorem of Gerstenhaber [33]). Notice N = CN for some rank-one nilpotent matrix N, and the correspondence between N and N is unique. Hence a bijective linear mapping preserving nilpotent matrices sends maximal nilpotent subalgebras to maximal nilpotent subalgebras, which induces a bijective linear mapping sending rank-one nilpotent matrices to rank-one nilpotent matrices. The proof in [8] then uses tools from projective geometry to obtain the description.

A matrix A is idempotent if A2 = A. It is easily seen that the linear span of idempotent matrices is Mn.

Theorem 1.1.6 (Beasley, Pullman [3]). If φ : Mn → Mn is a bijective linear map preserving idempotent matrices, there exists an invertible matrix U ∈ Mn such that

(1) φ(X) = UXU −1, or

(2) φ(X) = UXtU −1

for all X ∈ Mn.

4 Several of the above results have been considered in the inﬁnite-dimensional case (where the matrix space is B(H), the algebra of bounded linear operators on an inﬁnite-dimensional Hilbert space H) and likewise shown to have nice descriptions. For instance, the rank-one preserver problem on B(H) can be found in [68].

1.1.2 Function preservers

From a historical point of view, the “ﬁrst” linear preserver problem was the determinant preserver problem due to Frobenius [32]; that is, characterizing all linear mappings φ : Mn → Mn such that det(φ(A)) = det(A) for all A ∈ Mn. It is interesting to note that Frobenius, more commonly known for fundamental contributions to group theory, created the theory of group characters in the language of determinants of a group representation.

This inspires the following type of preserver problem. Let f be a (scalar-valued, vector-valued, set-valued) function on M. We say that φ : M → M preserves f if

f(φ(A)) = f(A), for all A ∈ M.

Most often, we are interested in (not necessarily linear) functionals into the ground ﬁelds whose values are of practical application.

Theorem 1.1.7 (Frobenius [32]). If φ : Mn → Mn is a bijective linear map preserving the determinant function, there exist invertible matrices U ∈ Mn and V ∈ Mn such that

(1) φ(X) = UXV , or

(2) φ(X) = UXtV

for all X ∈ Mn, where det(UV ) = 1.

Dieudonn´e’sresult Theorem 1.1.4 (appearing approximately 50 years after Theorem 1.1.7) of course generalizes this result. Moreover, if one takes f to be the rank function, we can similarly use

5 the rank-one preserver problem Theorem 1.1.1, but rank function preservers predate the rank-one preserver problem.

Theorem 1.1.8 (Marcus, Moyls [60]). Let φ : Mn → Mn be a linear map. The following are equivalent:

(1) φ preserves the rank function,

(2) there exist invertible matrices U, V ∈ Mn such that

(a) φ(X) = UXV , or

(b) φ(X) = UXtV

for all X ∈ Mn.

Moreover, if

(3) φ preserves the determinant function, then φ is of form (2) with det(UV ) = 1,

(4) φ preserves the spectrum function, then φ is of form (2) with UV = In.

This demonstrates the ubiquity of rank preservers. Also note that bijectivity is not assumed for rank preservers; indeed, the only matrix of rank 0 is the zero matrix.

A mapping preserving a matrix norm is typically referred to as an isometry. Descriptions of isometries are of chief interest in analytic applications. In addition, there is a good deal of interest in preservers of functions on the singular values of a matrix. For more details, see [55]. There are several other functions, both in preservers and their applications, of interest that can be found in the expository papers [56] and [52].

Analogously, the framing of a function preserver problem naturally motivates preserver problems about functions equal to speciﬁed values (though this could technically now be considered a subset

6 preserver problem). For instance, let SLn denote the set of complex n×n matrices of determinant 1.

This subset is a group in which every member is the solution to a polynomial equation det(X) = 1.

Let φ : Mn → Mn be a bijective linear map preserving SLn. Suppose A is invertible and set

−1/n −1 a = det(A) . Then aA ∈ SLn and hence 1 = det(φ(aA)) = det(aφ(A)) = det(A) det(φ(A)).

Multiplying through by det(A) we conclude that det(A) = det(φ(A)) for all A ∈ GLn. In particular,

Theorem 1.1.3 applies.

If f(x) is a polynomial in one indeterminate, one can study linear maps preserving those matrices annihilated by f; that is,

f(A) = 0 =⇒ f(φ(A)) = 0.

Problems of this type are discussed further in Chapter5. More generally, consider linear maps preserving the set of matrices annihilated by any (ﬁnite) collection of polynomials deﬁned on a matrix space. This amounts to characterizing algebraic subsets of matrices. It is interesting to note that bijective linear preservers of algebraic sets automatically preserve in both directions; see

[Lemma 1, [27]]. Preserver problems concerning algebraic groups can be found in [7, 27, 71].

1.1.3 Relation preservers

Given a relation ∼ on M, one additional type of linear preserver problem concerns the characterization of maps sending related pairs to related pairs. We say φ : M → M preserves ∼ if

A ∼ B =⇒ φ(A) ∼ φ(B), for all A, B ∈ M and φ : M → M preserves ∼ in both directions if

A ∼ B ⇐⇒ φ(A) ∼ φ(B), for all A, B ∈ M.

Here the LPP literature can become quite varied depending on the relation introduced. For instance, deﬁne a relation ∼ on Mn by A ∼ B if and only if rank(A − B) = 1. Matrices A and B satisfying this condition are called adjacent.

7 Theorem 1.1.9 (Hua [80]). If φ : Mn → Mn is a bijective linear map preserving adjacency in both directions, there exist invertible matrices U ∈ Mn and V ∈ Mn such that

(1) φ(X) = UXV , or

(2) φ(X) = UXtV ,

for all X ∈ Mn.

The relation of adjacency is just as easily introduced on the space of rectangular m×n matrices over any field. In fact, Theorem 1.1.9 as written here is an extremely simplified version of Hua’s actual result concerning surjective (not necessarily linear) mappings on rectangular matrices over a field F preserving adjacency. A mapping of the form X 7→ Uτ(X)V + R preserves adjacency, where X is a rectangular matrix, U and V are invertible matrices of appropriate sizes, R is an arbitrary rectangular matrix (of the same dimensions as X), and τ is a field automorphism of F applied entrywise to X. When the space of matrices is square, the possibility X 7→ Uτ(X)tV + R may also occur. The content of Hua’s result is that adjacency preserving surjective mappings must be one of the two forms mentioned here. The result is so important that it goes by the name

Hua’s fundamental theorem of the geometry of rectangular matrices. The text [80] is devoted to

Hua’s groundbreaking work on problems of this type. It is a useful geometric tool for exploring preserver problems without the assumption of linearity. In addition, there is a connection to rank- one preserver problems given that every rank-one matrix is adjacent to the zero matrix.

Some other well-known examples of relation preserver problems are expressed through zero products. Let ? denote a binary (not necessarily associative) operation on matrices and declare

A ∼ B if and only if A?B = 0. A map preserving this relation is said to preserve zero ?-products.

In concrete terms,

A?B = 0 =⇒ φ(A) ? φ(B) = 0.

8 The operation ? could be the ordinary product AB, the Lie product [A, B] = AB − BA, or the

Jordan product A ◦ B = AB + BA. This leads to the following collection of classical results:

Theorem 1.1.10 (Wong [84]). If φ : Mn → Mn is a bijective linear map preserving zero products,

−1 there exists an invertible matrix U ∈ Mn and c ∈ C such that φ(X) = cUXU for all X ∈ Mn.

We devote Chapter 4 to variants of the zero product preserver problem. The corresponding result for the zero Lie product is due to Watkins.

Theorem 1.1.11 (Watkins [83]). Let n ≥ 4. If φ : Mn → Mn is a bijective linear map preserving the zero Lie product (i.e., preserving commutativity), there exists an invertible matrix U ∈ Mn, scalar c ∈ C, and linear functional f : Mn → C such that

−1 (1) φ(X) = cUXU + f(X)In, or

t −1 (2) φ(X) = cUX U + f(X)In,

for all X ∈ Mn.

We devote Chapter 3 to variants of the commutativity preserver problem. Lastly we state the zero Jordan product preserver problem on matrices.

Theorem 1.1.12 (Chebotar, Ke, Lee, Zhang [23]). If φ : Mn → Mn is a bijective linear map preserving the zero Jordan product, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ Mn.

9 We note that in light of the upcoming Theorem 2.0.1, the above result is very easily obtained via matrices satisfying A2 = 0, since any mapping preserving zero Jordan products preserves square-zero elements (provided the characteristic of the ﬁeld is not 2). However, the authors in [23] considered this problem on a matrix ring Mn(R), where R is unital ring in which 2 is invertible.

In the realm of generalizations, the following result will be used in Chapter3.

Theorem 1.1.13 (Pierce, Watkins [72]). Theorem 1.1.11 also holds over an arbitrary ﬁeld F and for n ≥ 3.

Lastly note that since any (binary) relation ∼ on M is a set of ordered pairs (A, B) in the

Cartesian product M × M, it is possible to state relation preserver problems in terms of subset preserver problems in the matrix space M × M, if desired.

1.1.4 Transformation preservers

Let F : M → M be a transformation (not necessarily linear). We say that the map φ : M → M commutes with F if

φ(F (A)) = F (φ(A)), for all A ∈ M.

Preserver problems of this type are rarer than subset, function, and relation preservers. However we point out a few nice examples.

Given a matrix A ∈ Mn, one can form a new matrix C ∈ Mn entrywise by taking the determinant of the (n − 1) × (n − 1) matrix obtained by deleting the ith row and jth column of A and placing it in the (i, j)-entry of C. We call C the cofactor matrix of A. Then the adjugate of a matrix A, denoted adj A, is deﬁned as the transpose of the cofactor matrix of A.

Theorem 1.1.14 (Sinkhorn [78]). Let n ≥ 3. If φ : Mn → Mn is a bijective linear map such that

φ(adj A) = adj φ(A) for all A ∈ Mn (i.e., φ commutes with adj), there exists an invertible matrix

U ∈ Mn and scalar c ∈ C such that

10 (1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

n−1 for all X ∈ Mn, where c = c .

A description of maps preserving invertible matrices is given in Theorem 1.1.3. A description of maps commuting with the inverse function is as follows.

Theorem 1.1.15 (Essanouni, Kaidi [30]). If φ : Mn → Mn is a bijective, linear, and unital map

−1 −1 such that φ(A ) = φ(A) for all invertible A ∈ Mn, there exists an invertible matrix U ∈ Mn such that

(1) φ(X) = UXU −1, or

(2) φ(X) = UXtU −1

for all X ∈ Mn.

Lastly, we consider maps commuting with power functions.

Theorem 1.1.16 (Chan, Lim [19]). Let k ≥ 2. If φ : Mn → Mn is a linear map such that

k k φ(A ) = φ(A) for all A ∈ Mn, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

k for all X ∈ Mn, where c = c.

The same authors also obtained a similar result stated on the space of symmetric matrices.

11 1.2 Generalizations

Now that linear preserver problems have been explored, we point out a few ways in which classical preserver problems can be generalized. First, take preserver problems that hold in both directions and see if the conclusion remains valid in one direction (or vice versa). Second, some authors consider additive mappings rather than linear ones. This is advantageous when considering preserver problems in more general contexts when there is no underlying ﬁeld; say, working in associative rings. When there is an underlying ﬁeld, some authors have shown that additive maps with a prescribed invariance property must in fact be linear, reducing to previous cases. However the additivity assumptions are usually indispensable. For example, to each matrix A ∈ Mn associate

−1 an invertible matrix UA ∈ Mn. Then A 7→ UAAUA preserves every property above.

One recent direction considers preservers between matrix spaces of possibly different (finite) dimensions; typically, φ : Mn → Mr. For instance, suppose r = 2n and define a linear map

φ : Mn → M2n by A 0 A 7→ . 0 A

Certainly maps of this type preserve all sorts of properties of A but cannot be written in the form X 7→ UXV or X 7→ UXtV for invertible matrices U and V , so the solutions to such LPPs can be quite creative. Nevertheless, some fruitful preserver problems have been explored between matrix spaces of diﬀerent sizes; in particular, see [54] for some general techniques. See [48] for a generalization of Theorem 1.1.14 on matrix spaces of diﬀerent sizes. We also point out the upcoming

Theorem 4.0.2 also studies maps φ : Mn → Mr with r ≤ n preserving the zero product.

There are also results that substantially weaken the algebraic assumptions on φ; merely that it is a set map with a preserver property. See [79] for a mapping which is only assumed to be bijective (in the paper, the matrix space is B(H) with H inﬁnite-dimensional). The papers [24, 29] consider a weakened linearity assumption for the determinant preserver problem; namely, that

12 det(A + λB) = det(φ(A) + λφ(B)) for all A, B ∈ Mn and λ ∈ C. Other not-necessarily-linear results can be found in [25, 28, 76].

The matrix space itself may be generalized. In the analytic direction, preserver problems on matrix spaces such as B(H), C∗-algebras and von Neumann algebras, Banach algebras, etc., are of principal interest, including descriptions of isometries (often not-necessarily-linear), though we are primarily concerned with ﬁnite-dimensional matrix structures in all that follows. In the algebraic direction, preserver problems on Mn(F ) may be considered instead on Mn(R), where R is a ring.

For instance the idempotent preserver problem Theorem 1.1.6 was generalized by Breˇsarand Semrlˇ

1 [13] to Mn(R), where R is a commutative ring with 2 . More commonly, preserver problems can be stated on Mn(D), with D a division ring, if the preserver property does not depend too much on the commutativity of the coeﬃcients of a matrix. An example can be found in Chapter5 (Theorem

4.1.3). To go even further one can replace a matrix space with a ring R and describe additive maps φ : R → R with a prescribed invariance property. One such example is the classiﬁcation of additive commutativity preserving maps on simple rings, inspired by questions on Lie isomorphisms of simple rings. More on this in Section 3.1 speciﬁcally. In addition, several important results in mathematical physics consider preserver problems on P1(H), the set of rank-one projections on a

Hilbert space H, or the real vector space of self-adjoint operators of ﬁnite rank on a Hilbert space.

Numerous examples of preserver problems with applications in physics can be found in [64].

1.3 Nonstandard solutions of linear preserver problems

The converse statements of each of the above theorems are entirely trivial. In the language of [27],

t the set of all linear transformations of Mn of the form X 7→ UXV or X 7→ UX V is called a rigid mapping. The set of rigid mappings form a group of invertible transformations of Mn. The solutions to every LPP above are collections of rigid mappings, up to perturbation by the identity in the case of commutativity preservers.

13 Despite this, in many cases, the rigid mappings in the solution set must in fact be inner automorphisms or antiautomorphisms. Preserver problems thus can be considered as a type of classiﬁcation of matrix algebra automorphisms. On Mn especially, this demonstrates that its algebra automorphisms are determined by their local actions (on subsets, functions, relations, etc.).

Recall that an F -algebra is called central if its center is isomorphic to the ﬁeld F . The description of the automorphism group of a central simple algebra (such as the algebra of n × n matrices) is the well-known Skolem-Noether theorem.

Theorem 1.3.1 (Skolem-Noether). Every (algebra) automorphism of a central simple F -algebra is an inner automorphism.

Mn(D) (with D a division ring ﬁnite dimensional over its center F ) is a central simple algebra.

t −1 Consequently every antiautomorphism of Mn(D) is of the form X 7→ UX U .

We now present another type of mapping.

Deﬁnition 1.3.2. Let A be an F -algebra. A bijective F -linear map φ : A → A is called a Jordan automorphism if φ(A2) = φ(A)2 for all A ∈ A. If char(F ) 6= 2, every Jordan automorphism of

A satisﬁes φ(A ◦ B) = φ(A) ◦ φ(B) for all A, B ∈ A (i.e., Jordan automorphisms preserve Jordan products).

Clearly every automorphism and every antiautomorphism of an (arbitrary) algebra is a Jordan automorphism. In Mn, these are the only examples.

Theorem 1.3.3 (Jacobson, Rickart [45]). Every Jordan automorphism of Mn is either an automorphism or antiautomorphism.

One can also use Theorem 1.1.16 for a self-contained approach using preserver techniques.

Based on the exposition above it is natural to expect Jordan automorphisms to be solutions to

LPPs. Indeed, from the interpretation of A as an operator on an n-dimensional vector space V ,

14 any underlying property of A as an operator (such as its rank, idempotence/nilpotence, spectrum, minimal polynomial, determinant, trace, etc.) that is independent of a chosen basis for V will be preserved under an inner automorphism. Moreover, if the property of A as an operator is invariant under transposition, we also expect antiautomorphisms to appear.

We will say that a linear preserver problem has standard solutions if the solution set to a preserver problem can be written as families of rigid mappings of Mn (or sln). In the case of commutativity preservers we include perturbation by a scalar. The usage of the term “standard” also reﬂects the high frequency of rigid mappings as solutions to preserver problems in the literature.

On the other hand, we will say that a linear preserver problem has nonstandard solutions if the family of maps in the solution set satisﬁes any of the two heuristic criteria:

(1) Its solutions include bijective, non-rigid mappings

In Chapter 2 we examine the idempotent preserver problem (whose standard solutions are

Jordan automorphisms; Theorem 1.1.6) and obtain a generalization that allows for non-rigid

mappings. Theorem 2.3.1 demonstrates this phenomenon and has implications for similar

problems found in Section 2.3.

In addition, in Chapter 4 we encounter Theorem 4.1.3 and Theorem 4.3.6 which have non-

Jordan mappings as solutions. However we point out that these mappings are still rigid

mappings. The nonstandard behavior is how they diﬀer compared to classical results.

(2) The preserved subspace is not Mn or sln

If a linear mapping φ preserves a given collection of matrices, the action of φ is only determined

on the span of the preserved matrices. We call this span the preserved subspace of φ. For

example, the nilpotent preserver problem is stated on sln since the action of φ is only speciﬁed

on matrices of trace-zero.

15 In Chapter3 and Chapter5 we encounter preserver problems in which the preserved subspace

(a) is merely a subspace of Mn with no (ordinary, Lie, Jordan) multiplicative structure

(Theorem 3.2.6),

(b) decomposes into two disjoint subspaces with nice properties (Theorem 5.1.1), and

Despite the preserved subspace having relatively less structure than the typical Mn or sln, in

these results we still sometimes obtain a complete (multiplicative) description of φ acting on

the space. However the extension of φ up to the entire space Mn of course need not have any

particular requirements, and thus may fail to be a rigid map.

Each chapter in this dissertation contains both standard and nonstandard solutions to LPPs.

The complete results obtained herein concern diﬀerent types of matrix invariants, twists on classical problems, and represent a new and novel approach to generating LPPs. Moreover, several open questions are posed for Mn with the obvious potential for generalization to many other matrix spaces.

16 CHAPTER 2

Maps preserving certain elements of a subset

In this chapter we consider a variant of the usual subset preserver problem. First we present some speciﬁc results concerning square-zero matrices. Recall that a matrix A is called square-zero if

2 A = 0. It is also easy to check that square-zero matrices generate sln. A surprisingly useful result is the following.

Theorem 2.0.1 (Semrlˇ [75]). If φ : sln → sln is a bijective linear map preserving square-zero matrices, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ sln.

Proof sketch. Suppose A2 = 0 and consider the subspace

2 2 W (A) = hB ∈ sln | B = 0 and (A + B) = 0i.

A linear map φ preserving square-zero matrices satisﬁes φ(W (A)) ⊆ W (φ(A)). However, the condition rank(A) = 1 is equivalent to dim W (A) ≥ n(n − 2) [Lemma 5, [75]]. Since φ is bijective, we have n(n − 2) ≤ dim W (A) = dim φ(W (A)) ≤ dim W (φ(A)). Hence φ(A) is a rank-one square- zero matrix whenever A is a rank-one square-zero matrix. This reduces to the rank-one preserver problem on sln, which has standard solutions by the proof in [8].

17 A generalization of Theorem 2.0.1 to matrices over commutative rings can be found in [20]. For

A ∈ Mn, the collection of all matrices in Mn that are similar to A is called the similarity class of A (the orbit of A under conjugation). Theorem 1.1.5 concerning nilpotents and Theorem 2.0.1 concerning square-zero matrices are both special cases of the following more general statement concerning maps preserving similarity classes of nilpotent matrices [Lemma 2.5, [51]].

Proposition 2.0.2 (Li, Pierce [51]). Let T be a nonzero union of similarity classes of nilpotent matrices. If φ : sln → sln is a bijective linear map that preserves T , there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ sln.

The main result of [51] also obtains a complete description of bijective linear maps preserving unions of similarity classes of arbitrary matrices as well. Considering Jordan canonical forms and the invariance of rank under conjugation, the similarity class of a square-zero matrix A of rank k is precisely the set of all rank-k square-zero matrices. Hence one can take T to be a single similarity class of a square-zero matrix A of rank k and represent it as Rk ∩ S, where Rk is the set of rank-k matrices and S is the set of square-zero matrices. Hence a particular case of Proposition 2.0.2 is a characterization of bijective linear mappings satisfying φ(Rk ∩ S) ⊆ Rk ∩ S as scalar multiples of

Jordan automorphisms.

This motivates the following generalization of the subset preserver problem: Let R and S be two distinguished subsets of matrices and let φ : Mn → Mn be a bijective linear map such that

φ(R ∩ S) ⊆ S. (2.1)

18 There are many fruitful questions here. Must φ preserve R ∩ S? Must φ preserve S? Notice that any information about membership in R is “forgotten” by φ. In this sense, φ is preserving certain elements of the set S. This problem is most interesting when the subset S is well-understood, and the descriptions of bijective linear maps preserving S are standard. Since every map preserving S also satisfes φ(R ∩ S) ⊆ S, must a description of φ be standard?

In this chapter we explore two variants of preserver problems of type (2.1). Section 2.1 concerns the set S of all square-zero matrices while Section 2.2 concerns the set P of idempotent matrices.

In both sections, R is taken to be various unions of rank-k matrices.

Note that, at present, only one example of a problem of this type appears in our citations.

Indeed, consider Lemma 1 of [19]. It reads as follows:

Lemma 2.0.3 (Lemma 1, [19]). Let φ : Mn(F ) → Mn(F ) be a linear map, where |F | ≥ 3. If φ sends rank-one idempotents to matrices of rank at most 1, then rank(A) ≤ 1 implies that rank(φ(A)) ≤ 1.

Proof Sketch: Every rank-one matrix is either a scalar multiple of a rank-one idempotent or a rank- one nilpotent. By hypothesis, the former case is true. Suppose A is a rank-one nilpotent. There is a rank-one idempotent B such that A + λB is a scalar multiple of a rank-one idempotent for all λ ∈ F \{0}. Hence φ(A) + λφ(B) is either rank-one or zero. Hence φ(B), φ(A) + φ(B) and

φ(A) − φ(B) all have rank at most one, therefore rank(φ(A)) ≤ 2. Suppose rank(φ(A)) = 2 and suppose without loss of generality that φ(A) = diag(1, 1, 0,..., 0). Computing the ﬁrst principal

2 × 2 subdeterminants of φ(A) + φ(B) and φ(A) − φ(B) (which both equal 0), we are forced to conclude that 2 = 0, which contradicts |F | ≥ 3.

Symbolically the hypothesis is represented as φ(R1 ∩ P) ⊆ R1 ∪ {0}. If one also assumes bijectivity, we recover a problem of type (2.1). Besides this example, preserver problems like

φ(R ∩ S) ⊆ S have largely remained unexplored.

19 2.1 On maps sending rank-k square-zeros to square-zeros

The ﬁrst result is of the type φ(R ∩ S) ⊆ S, where R = R1 ∪ R2.

Theorem 2.1.1. Let n ≥ 4. If φ : sln → sln is a bijective linear map sending rank-one and rank-two square-zero matrices to square-zero matrices, there exist an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ sln.

Proof. Let S be any square-zero matrix. Write S = S1 + S2 + ··· + Sk, where the Si are pairwise

2 orthogonal (i.e., SiSj = SjSi = 0) rank-one square-zero matrices. By hypothesis, φ(Si) = 0 and

2 φ(Si + Sj) = φ(Si) ◦ φ(Sj) = 0 for all 1 ≤ i, j ≤ k. Hence

2 2 φ(S) = φ(S1 + S2 + ··· + Sk) = 0, i.e., φ preserves square-zero matrices. By Theorem 2.0.1, φ has the form as claimed.

Now suppose R = Rk is the set of all rank-k matrices, where k is ﬁxed but greater than or equal to 2.

Theorem 2.1.2. Let n ≥ 4 and k ≥ 2. If φ : sln → sln is a bijective linear map sending rank-k square-zero matrices to square-zero matrices, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ sln.

20 Proof. Let S ∈ Mn be a rank-k square-zero matrix, k ≥ 2. There is a rank-(k − 1) square-zero matrix T such that S + xT is a rank-k square-zero matrix for all x ∈ C. It suﬃces to show this in some basis. Suppose S takes the form

S = e12 + e34 + ··· + e2k−1,2k.

The rank-(k − 1) square-zero matrix

T = e14 + e36 + ··· + e2k−3,2k−2

2 satisﬁes the property rank(S + xT ) = k and (S + xT ) = 0 for all x ∈ C.

Hence φ(S + xT )2 = 0 implies that

xφ(S) ◦ φ(T ) + x2φ(T )2 = 0 for all x ∈ C. In particular, consider the system of equations obtained by taking x = 1 and x = −1 to conclude that φ(T )2 = 0. Consequently, if φ preserves the square-zero property of rank-k square- zero matrices, it also preserves the square-zero property of rank-(k − 1) matrices. This forms the base case of a downward induction argument so that rank-m square-zero matrices are mapped to square-zero matrices for all m ≤ k. Since k ≥ 2 both rank-one and rank-two square-zero matrices are sent to square-zero matrices, so φ has the form (1) or (2) by Theorem 2.1.1.

One would like to understand the preceding theorem in the case k = 1. Recall that the preserver problem concerning rank-one nilpotent matrices (which are of course square-zero) has a standard form via Theorem 1.1.5. In symbols, φ(R1 ∩ S) ⊆ R1 ∩ S implies that φ is a scalar multiple of a

Jordan automorphism. Consider also a map satisfying φ(R1 ∩ S) ⊆ R1. Since φ is map from sln to sln, the image of any rank-one square-zero matrix is once again rank-one and trace-zero. But the only rank-one trace-zero matrices are the rank-one square-zero matrices. So φ is a standard map.

21 But with the relaxed hypothesis φ(R1 ∩ S) ⊆ S, a standard description is not obvious. Indeed,

2 2 since e13, e14, and e13 + e14 are rank-one square-zero matrices, we have that φ(e13) = φ(e14) =

φ(e13) ◦ φ(e14) = 0. Of course, any two matrix units with at least one index in common satisﬁes identical conditions. Furthermore, since e13 + e14 + e23 + e24 is rank-one and square-zero, one can conclude using the above computations that

2 φ(e13 + e14 + e23 + e24) = φ(e13) ◦ φ(e24) + φ(e14) ◦ φ(e23) = 0, but there does not appear to be a straightforward way to improve this relation to conclude that

φ(e13)◦φ(e24) = 0, for example, which is a necessary condition for φ to be a Jordan automorphism.

In principle, it may be that φ(e13) ◦ φ(e24) is nonzero.

Approaching this problem seems quite challenging. If n ≥ 4, then e13, e14, e23, and e24 are strictly upper-triangular. Can one insist that their images φ(e13), φ(e14), φ(e23), and φ(e24) are also strictly upper-triangular? Does this improve anything?

Example 2.1.3. Let N4 denote the strictly upper triangular 4 × 4 matrices. Consider the linear map η : N4 → N4 deﬁned by

0 x y z 0 x + y y − z z  0 0 s t 0 0 u t − s η :   7→   . 0 0 0 u 0 0 0 s  0 0 0 0 0 0 0 0

It is straightforward to see that η is bijective and η sends rank-one square-zero matrices to square- zero matrices. Indeed, a rank-one matrix in A ∈ N4 is either a single row, single column, or the

2 × 2 oﬀ-diagonal block 0 0 y z 0 0 s t   0 0 0 0 0 0 0 0 satisfying yt − zs = 0. We check each case. The column matrix xe12 is mapped to xe12. We also

22 have 0 0 y 0 0 y y 0  0 0 s 0 0 0 0 −s   7→   . (2.2) 0 0 0 0 0 0 0 s  0 0 0 0 0 0 0 0

The column matrix ze14 + te24 + ue34 is mapped to a 2 × 2 oﬀ-diagonal block. Hence each rank-one column matrix has a square-zero image. Consider now the rank-one row matrices. The row matrix xe12 + ye13 + ze14 is mapped once again into the ﬁrst row. The row matrix se23 + te34 is mapped to the column matrix (t − s)e24 + se34. The row matrix ue34 is mapped to ue23. In every case, the image is square-zero. Lastly, we have

0 0 y z 0 y y − z z  0 0 s t 0 0 0 t − s  η :   7→   . 0 0 0 0 0 0 0 s  0 0 0 0 0 0 0 0

The image is square-zero since

0 y y − z z 2 0 0 0 y(t − s) + (y − z)s 0 0 0 yt − zs 0 0 0 t − s 0 0 0 0  0 0 0 0    =   =   = 0. 0 0 0 s  0 0 0 0  0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0

Clearly η is not a Jordan automorphism since η(e13) ◦ η(e24) = (e12 + e13) ◦ e24 6= 0.

Remark 2.1.4. This nonstandard map on N4 does not extend to sl4. Suppose now that φ : sl4 → sl4 is a bijective extension of η : N4 → N4 mapping rank-one square-zero matrices to square-zero matrices. Since φ(e12) = e12, the relation φ(e11 − e22) ◦ φ(e12) = 0 implies that   a a a a  11 12 13 14       0 −a 0 0   11  φ(e11 − e22) =   .      0 a32 a33 a34      0 a42 a43 −a33

Since φ(e11−e22)◦φ(e13) = φ(e12)◦φ(e23) = e12◦(e34−e24) = −e14 (see [Lemma 5(d), [47]] for matrix unit computations), we must have a33 = −a11, a32 = 0, and a34 = −1. Also φ(e11 − e22) ◦ φ(e14) =

φ(e12) ◦ φ(e24) = e14, and we get a42 = a43 = 0 and 2a11 + 1 = 1, i.e., a11 = 0 and thus φ(e11 − e22)

23 is strictly upper triangular. Hence φ(e11 − e22) can be written as a linear combination of elements in φ(N4) = η(N4), contradicting bijectivity of φ.

Despite this, it would still be entirely surprising if a nonstandard solution exists on sln and counterexamples have not been found. Hence we make the following conjecture.

Conjecture 2.1.5. Let n ≥ 4. If φ : sln → sln is a bijective linear map sending rank-one square- zero matrices to square-zero matrices, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ sln.

Note that if n = 2, 3 every square-zero matrix is rank-one, so φ has a standard description in low-dimensional cases.

2.2 On maps sending rank-t idempotents to idempotents

Analogous to square-zero matrices, by considering Jordan forms and the invariance of rank under conjugation, the similarity class of an idempotent matrix P of rank k is the set of all rank-k idempotents. Hence any computation performed on one rank-k idempotent carries over to all rank- k idempotents. This simple technique is the crux of the proof of the idempotent preserver problem in [13] on matrices over commutative rings. By performing computations on speciﬁc idempotents, the authors prove that maps preserving idempotent matrices are Jordan mappings. Only rank-one and rank-two idempotent matrices are necessary for their proof.

More precisely, the proof illustrates a more general fact that a bijective linear map φ sending rank-one and rank-two idempotent matrices to idempotents (i.e. φ(R∩P) ⊆ P, where R = R1 ∪R2

24 and P is the set of idempotents) must in fact preserve all idempotents. Note the similarity to

Theorem 2.1.1.

We sketch this observation. Analogous to Theorem 2.1.1, having a speciﬁed action on rank-one and rank-two idempotents speciﬁes an action on all idempotents. If P is a rank-k idempotent, we can write P = P1 + P2 + ··· Pk, where the Pi are pairwise orthogonal rank-one idempotents. Since

2 2 φ(Pi) = φ(Pi) and φ(Pi + Pj) = φ(Pi + Pj) for all 1 ≤ i ≤ t and i 6= j, then φ(Pi) ◦ φ(Pj) = 0 whenever i 6= j. This implies that φ(P )2 = φ(P ).

In addition, there is a precise relationship between an idempotent P and a square-zero matrix

N such that P + λN is idempotent for all λ ∈ C (see Lemma 2.2.2). This connection between idempotents and square-zero matrices can be used to determine φ on sln, which turns out to be standard. Putting the description on sln together with the orthogonal idempotent relation, the mapping φ is a Jordan automorphism. The natural question is thus: can one obtain a description in case only idempotents of a speciﬁed rank are mapped to idempotents?

Relaxing the hypotheses to φ(Rk ∩ P) ⊆ P oﬀers a surprising conclusion.

Theorem 2.2.1 (Julius [47]). Suppose φ : Mn → Mn is a bijective linear map sending rank-k idempotents to idempotents, with 2 ≤ k ≤ n − 2. Then φ is either of the form

(1) φ(X) = f(X), or

tr(X) (2) φ(X) = I − f(X), k n for all X ∈ Mn, where f is a Jordan automorphism of Mn.

Remark. If φ has form (2) and P is a rank-k idempotent, then φ(P ) = In − f(P ) and, since f(P ) is a rank-k idempotent, it follows that φ(P ) has rank n − k. However, a map of form (2) is not even a Jordan automorphism; for instance, a fairly straightforward computation shows that

25 φ(e11)◦φ(e22) 6= 0. Hence this preserver problem has nonstandard solutions in contrast to Theorem

1.1.6.

We also note that a recent paper [69] considers problems of this type: mapping rank-k projections to other projections of fixed rank. The method is quite different and analytic in flavor, drawing inspiration from applications of preserver problems to mathematical physics.

The theorem is proved in two steps. First, we show that φ induces a transformation of sln preserving square-zero matrices. By invoking the result of Semrl,ˇ we obtain a complete description of the restriction of φ to sln. Then the image of a rank-k idempotent is determined, which extends the description of φ to Mn.

Lemma 2.2.2. Let P be a nontrivial idempotent matrix and N be a square-zero matrix. The matrix P + λN is idempotent for all λ ∈ C if and only if PN + NP = N; consequently, rank(N) ≤ min{rank(P ), rank(I − P )} and rank(P + λN) = rank(P ) for all λ ∈ C.

Proof. Suppose rank(P ) = k with 1 ≤ k ≤ n − 1. We have

(P + λN)2 = P + λN ⇐⇒ PN + NP = N for all λ ∈ C. Multiply the equation PN + NP = N on the right by N and on the left by P to conclude that NPN = 0 and PNP = 0, respectively. It follows that

N = PN + NP = PN(I − P ) + (I − P )NP.

Let A = PN(I − P ) and B = (I − P )NP (so N = A + B). Notice that BA = (I − P )NPN(I −

P ) = 0. Up to similarity, we may assume that P is the idempotent e11 + e22 + ··· + ekk, so

I 0 P = k 0 0 and we can write 0 A N = B 0

26 in block form. Identify A as a k × (n − k) rectangular matrix and B as an (n − k) × k rectangular matrix. The condition BA = 0 implies that every column vector of A is contained ker(B), and so rank(A) ≤ dim ker(B). By rank-nullity theorem, dim ker(B) = k − rank(B). Thus rank(A) ≤ k − rank(B) and equivalently, rank(N) ≤ rank(A) + rank(B) ≤ k = rank(P ). A totally analogous argument using instead the relation AB = 0 also implies that rank(N) ≤ n − k = rank(I − P ).

The trace of an idempotent matrix is its rank. Hence

rank(P + λN) = tr(P + λN) = tr(P ) = rank(P ) for all λ ∈ C.

Let 1 ≤ k ≤ n − 1 and s be a natural number such that s ≤ min{k, n − k}. For any choice of k and s, there is an idempotent matrix P of rank k and a square-zero matrix of rank s such that

P +λN is idempotent for all λ ∈ C. For instance, take P = e11 +···+ekk and N = e1,k+1 +··· es,k+s.

Observe that under the conditions of Lemma 2.2.2, every conjugate of P +λN is idempotent. Hence every square-zero matrix of rank s can be associated to an idempotent of rank k, and vice versa.

Thus a linear map preserving idempotents of a speciﬁed rank will preserve the relationship between a square-zero matrix and a corresponding idempotent, as follows.

Lemma 2.2.3. Fix 1 ≤ k ≤ n − 1. If φ : Mn → Mn is a bijective linear map that sends rank-k idempotents to idempotents, then φ(N)2 = 0 whenever N is a square-zero matrix of rank s ≤ min{k, n − k}.

Proof. Let N be a square-zero matrix of rank s. Let P be an idempotent matrix of rank k such that P + λN is a idempotent for all λ ∈ C. By Lemma 2.2.2, it follows that rank(P + λN) = k for all λ ∈ C. Hence

φ(P + λN)2 = φ(P + λN) ⇐⇒ λφ(P ) ◦ φ(N) + λ2φ(N)2 = λφ(N).

27 The system of equations obtained by taking λ = 1 and λ = −1 imply that φ(N)2 = 0.

Importantly we make no claim about rank(φ(N)); only that φ(N) is square-zero whenever N is square-zero, provided that rank(N) ≤ min{k, n − k}.

Corollary 2.2.4. Fix 1 ≤ k ≤ n − 1. If φ : Mn → Mn is a bijective linear map sending rank-k idempotents to idempotents, then φ(sln) = sln. Moreover, if P and Q are rank-k idempotents, then

φ(P ) and φ(Q) have the same rank.

Proof. By Lemma 2.2.3, φ sends rank-one square-zero matrices to square-zero matrices. Hence the image of any trace-zero matrix under φ is also trace-zero. Since φ is bijective, it follows that

φ(sln) = sln.

If P and Q are rank-k idempotents, then φ(P ) and φ(Q) are idempotents. Since P − Q has zero trace, so does φ(P ) − φ(Q). Hence rank(φ(P )) = tr(φ(P )) = tr(φ(Q)) = rank(φ(Q)).

In the next corollary we insist on 2 ≤ k ≤ n − 2 for the description on sln. Because if k = 1 or k = n − 1 then φ need only send rank-one square-zero matrices to square-zero matrices by

Lemma 2.2.3. Hence a proof of Conjecture 2.1.5 (or, should nonstandard solutions arise, a complete description) resolves the missing cases here.

Corollary 2.2.5. Fix 2 ≤ k ≤ n − 2. If φ : Mn → Mn is a bijective linear map sending rank-k idempotents to idempotents, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

(1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ sln.

Proof. This follows from Lemma 2.2.3 and Theorem 2.1.1.

28 Under the assumption that 2 ≤ k ≤ n − 2, the map φ has a standard description on sln. It is now time to extend the description to all of Mn. It is well-known that Mn = sln ⊕ CIn. In fact Mn can be written as sln ⊕ CA, where A is any matrix of nonzero trace. Thus we need only determine

φ(E), where E = e11 + ··· + ekk.

Lemma 2.2.6. Fix 2 ≤ k ≤ n − 2. If φ : Mn → Mn is a bijective linear map sending rank-k idempotents to idempotents, then rank(φ(E)) = k or rank(φ(E)) = n − k. Furthermore, φ(X) =

−1 t −1 ±UXU or φ(X) = ±UX U for all X ∈ sln.

−1 t −1 Proof. By Corollary 2.2.5, we have φ(X) = cUXU or φ(X) = cUX U for all X ∈ sln. Deﬁne

ψ : Mn → Mn by

ψ(X) = U −1φ(X)U (2.3)

for all X ∈ Mn. If A is a symmetric trace-zero matrix, then ψ(A) = cA. The map ψ is clearly a bijective linear map that sends rank-k idempotents to idempotents.

Suppose ψ(E) = H = (hij). Notice that for all 1 ≤ p ≤ k and k + 1 ≤ q ≤ n, the matrix

E + eqq − epp is idempotent, so

ψ(E + eqq − epp) = H + ψ(eqq − epp) = H + c(eqq − epp) is also idempotent; i.e.,

2 2 H + c(eqq − epp) = H + c (eqq + epp) + cH(eqq − epp) + c(eqq − epp)H.

Cancelling H = H2 and a scalar multiple of c from both sides we obtain the equation

eqq − epp = c(eqq + epp) + H(eqq − epp) + (eqq − epp)H. (2.4)

Multiplying equation (2.4) on the left and right by eqq gives eqq = ceqq + 2hqqeqq and multiplying equation (2.4) on the left and right by epp gives −epp = cepp − 2hppepp. In addition, if u 6= q and

29 u 6= p, then multiplying (2.4) on the left by euu gives the equation 0 = huqeuq − hupeup. It follows that huq = hup = 0. If v 6= q and v 6= p, then multiplying (2.4) on the right by evv gives the equation 0 = hqveqv − hpvepv. It follows that hqv = hpv = 0 as well. Equation (2.4) does not furnish a direction implication that hpq = hqp = 0, but changing indices can show it. Indeed, replace p in

0 0 0 equation (2.4) with a p such that 1 ≤ p ≤ k and p 6= p. Multiply the new equation by epp on the left to get hpq = 0 and multiply the new equation by epp on the right to get hqp = 0.

Combining the above, a complete description of the entries of H is obtained:

1 + c 1 − c h = and h = if 1 ≤ p ≤ k < q ≤ n (2.5) pp 2 qq 2 and hpq = 0 for any distinct indices p and q. Since H is diagonal and idempotent, the diagonal entries must be 0 or 1. According to equation (2.5), the only two possibilities are c = 1 or c = −1.

Therefore H = E when c = 1 and H = In − E when c = −1. Hence H is either rank-k or rank-(n − k), respectively. By Corollary 2.2.4, the bijective map ψ (and therefore φ) sends rank-k idempotents to rank-k idempotents in the former case and sends rank-k idempotents to rank-(n−k) idempotents in the latter case.

All that remains is to extend the description on sln to Mn via rank-k idempotents.

Proof of Theorem 2.2.1. Returning to equation (2.3), it follows that φ(E) is either UEU −1 or

−1 −1 t −1 In −UEU . In the former case, since Mn = sln ⊕CE, clearly φ(X) = UXU or φ(X) = UX U for all X ∈ Mn, when c = 1. This proves (1) of Theorem 2.2.1. What is the description when c = −1?

−1 t −1 Suppose c = −1. Then φ(X) = −UXU or φ(X) = −UX U for all X ∈ sln. Since E is

−1 symmetric, in either case, φ(E) = In − UEU . Hence there is a Jordan automorphism f such that φ(X) = −f(X) for all X ∈ sln and φ(E) = In − f(E).

30 Let B ∈ Mn be an arbitrary matrix and (uniquely) write B = λE + A, where A ∈ sln. Now

φ(B) = λφ(E) + φ(A) = λ(In − f(E)) − f(A) = λIn − f(λE + A) = λIn − f(B)

for all B ∈ Mn, so it remains to determine the linear functional λ = λ(B). Since φ maps sln to sln, λ vanishes on sln. Furthermore λ(E) = 1. It follows from linearity that λ(X) = tr(X)/k for all X ∈ Mn.

2.3 Related problems and conjectures

The blind spot of Theorem 2.2.1 is the case k = 1 and k = n − 1. There are various improvements one can make. First and foremost, if Conjecture 2.1.5 holds, the standard description on sln is the key to the proof of Theorem 2.2.1, with the extension to Mn by some calculations involving speciﬁc idempotents. Without the conjecture; however, the challenges concerning idempotents can be solved by relaxing hypotheses in a natural way.

As an application, we first present a simplified proof of the rank-one idempotent preserver result in finite-dimensional. The theorem can be found, for example, in [49].

Theorem 2.3.1. Let φ : Mn → Mn be a bijective linear map preserving rank-one idempotents. If n ≥ 3, then φ is a Jordan automorphism. If n = 2, then φ is either a Jordan automorphism or of the form φ(X) = tr(X)I2 − f(X) for all X ∈ Mn, where f is a Jordan automorphism of Mn.

Proof. Since φ(R1 ∩ P) ⊆ R1 ∩ P by hypothesis, it follows that φ(R1 ∩ S) ⊆ R1 ∩ S by Lemma

2.2.3 (we get the standard description on sln for free). Hence the restriction of φ to sln is a scalar multiple of a Jordan automorphism regardless of n. If n ≥ 3 emulate the proof of Lemma 2.2.6 to

−1 determine that c = 1 (which does use that n ≥ 3) and conclude that φ(e11) = Ue11U . Hence φ is Jordan.

If n = 2 deﬁne ψ(X) = U −1φ(X)U. Notice that ψ is a bijective linear map preserving rank-one

31 idempotents. Using equation (2.4) of Lemma 2.2.6, we conclude that

1+c 2 a ψ(e11) = 1−c , (2.6) b 2

2 where c is the scalar from the standard description on sln. It is easy to check that ψ(e11 + e12) =

ψ(e11 + e12) implies

ψ(e11)ψ(e12) + ψ(e12)ψ(e11) = ψ(e12). (2.7)

Using equations (2.6) and (2.7) we conclude that b = 0 in case ψ(e12) = ce12 and a = 0 in case

ψ(e12) = ce21. Also considering ψ(e11)ψ(e21) + ψ(e21)ψ(e11) = ψ(e21), we get the reverse, that a = 0 and b = 0, respectively. Hence in any case, ψ(e11) is a diagonal idempotent matrix, and so c = ±1. If c = 1 then ψ(e11) = e11 and so ψ is a Jordan automorphism, hence so is φ. If c = −1, then ψ(e11) = e22 = I2 − e11, and hence φ is of the form tr(X)I2 − f(X).

Here is one improvement for rank-one idempotents. By Corollary 2.2.4, φ(Rk ∩ P) ⊆ P implies that

φ(Rk ∩ P) ⊆ Rs ∩ P (2.8) for some s. Fix k = 1. If s = n − 1, we obtain a complete description of φ.

Corollary 2.3.2. Let n ≥ 3. If φ : Mn → Mn is a bijective linear map sending rank-one idempotent matrices to rank-(n − 1) idempotents, then φ is of the form

φ(X) = tr(X)In − f(X)

where f is a Jordan automorphism of Mn.

Proof. The map ψ(X) = tr(X)In − φ(X) is a bijective linear map satisfying ψ(R1 ∩ P) ⊆ R1 ∩ P , so by Theorem 2.3.1, we have tr(X)In − φ(X) = f(X) for some Jordan automorphism f.

Another improvement is the following. Fix some 1 ≤ k ≤ n−1. Suppose the inclusion (2.8) was upgraded to φ(Rk ∩ P) = Rs ∩ P (reminiscent of an “in both directions” hypothesis). We say then

32 that φ sends rank-k idempotents surjectively onto rank-s idempotents. This hypothesis is indeed quite natural, but up to this point, has not been assumed. If this is incorporated into idempotent problems of this type, we obtain a strong conclusion with no missing cases.

Theorem 2.3.3. If φ : Mn → Mn is a bijective linear map sending rank-k idempotents surjectively onto rank-s idempotents, then φ is of the form

(1) φ(X) = f(X), or

tr(X) (2) φ(X) = I − f(X) k n for all X ∈ Mn, where f is a Jordan automorphism of Mn.

Proof. First note that if 2 ≤ k ≤ n − 2, then certainly Theorem 2.2.1 draws the conclusion that k = s or k = n − s corresponding respectively to the forms (1) and (2) of φ. It is left to consider the cases k = 1 and k = n − 1.

If k = 1, then φ−1 is a bijective linear map sending rank-s idempotents surjectively onto rank- one idempotents. If 2 ≤ s ≤ n − 2, we conclude as in the previous paragraph that φ−1 takes the form (1) or (2). But this implies that s = 1 or s = n − 1, which is a contradiction. The situation for k = n − 1 is analogous.

The case when k = 1 and s = 1 is the standard situation of Theorem 2.3.1 and the case k = 1 and s = n − 1 is Corollary 2.3.2. If k = n − 1 then φ−1 can be obtained by Theorem 2.3.1 and

Corollary 2.3.2 as well, which determines φ.

There are (at least) two approaches that could solve the main Theorem 2.2.1 in the missing cases of k = 1 and k = n − 1. The most desirable is to provide a proof of Conjecture 2.1.5. If alternatively this is too intractable (or worse, false), one could also show that the containment

(2.8) must automatically become equality. However this also seems like a hard problem. There is

33 certainly no obvious set-theoretic reason for equality to be obtained. On the other hand, there is no obvious matrix-theoretic reason for a proper inclusion to be obtained either. Hence we make another reasonable conjecture.

Conjecture 2.3.4. Let 1 ≤ k ≤ n − 1. If φ : Mn → Mn is a bijective linear map sending rank-k idempotents to idempotents, φ sends rank-k idempotents surjectively onto rank-s idempotents for some 1 ≤ s ≤ n − 1.

Remark 2.3.5. This conjecture does bear a striking resemblance preservers of algebraic sets due to Dixon; namely, that bijective linear maps preserving an algebraic subset S preserves it in both directions [Lemma 1, [27]]. Unfortunately, Rk ∩P is not an algebraic set. For brevity, let Rk ∩P be denoted by Pk. In general one needs to form the union Pk ∪Pk−1 ∪... P1 ∪{0} to have an algebraic set of idempotents. But preservers of unions of idempotents just reduces to the usual idempotent preserver problem as soon as k ≥ 2.

Now we explore some potential uses of preserver problems concerning idempotents of a speciﬁed

2 rank. An n × n matrix A is called an involution if A = In. The eigenvalues of an involution A are ±1, hence tr(A) is always an integer. Moreover, every involution can be written as 2P − In, where P is an idempotent of rank equal to the multiplicity of the eigenvalue 1 in A, which induces a one-to-one correspondence between the set of idempotents and the set of involutions. (Just like idempotents, the similarity of two involutions is determined by taking traces.) Linear maps preserving matrices annihilated by a polynomial with two distinct roots are described by Howard

(see the upcoming Theorem 5.0.1). Since involutions are annihilated by the polynomial x2 − 1, linear maps preserving involutions are known.

If n is even, then Mn contains trace-zero involutions. As an application of the idempotent preserver problem above, we obtain the following theorem.

34 Theorem 2.3.6. If n ≥ 4 is even and φ : Mn → Mn is a bijective, linear, and unital map sending trace-zero involutions to involutions, then there is a Jordan automorphism f of Mn such that

(1) φ(X) = f(X), or

2tr(X) (2) φ(X) = I − f(X), n n for all X ∈ Mn.

Proof. Let A be an involution and P the corresponding idempotent such that A = 2P − In. If A

n n 2 is trace-zero, then rank(P ) = 2 . Since n ≥ 4, 2 is not equal to 1 or n − 1. Moreover, φ(A) = In implies that

2 2 2 φ(2P − In) = 4φ(P ) − 4φ(P ) + φ(In) = In.

2 By our assumption that φ(In) = In, it follows that φ(P ) = φ(P ). Since every trace-zero involution

n is sent to an involution, it follows that φ sends rank- 2 idempotents to idempotents. Hence by

n Theorem 2.2.1, φ is either of the form (1) or (2) with t = 2 .

Hence the trace-zero involution preserver problem described above has nonstandard solutions in contrast to a standard theorem of Howard (see Theorem 5.0.1 in Chapter 5). Moreover, one can

2 n replace the assumption that φ(In) = In with φ(In) = In and still preserve rank- 2 idempotents, and both of the ﬁnal forms ensure that φ(In) = In, so the hypotheses can be relaxed slightly.

In general if φ sends involutions of a speciﬁed trace t to involutions, in order to use Theorem

2.2.1 one must have −(n − 4) ≤ t ≤ n − 4. Indeed, if t is the trace of an involution then 2k − n = t with k the rank of its corresponding idempotent. The condition −(n − 4) ≤ t ≤ n − 4 is equivalent to 2 ≤ k ≤ n − 2.

A band B is a set of matrices closed under multiplication that such that every element of B is idempotent. If Ik denotes the k × k identity matrix then Ik Y ⊥ 0 Y Bk = : Y ∈ Mk,n−k and Bk = : Y ∈ Mk,n−k (2.9) 0 0 0 In−k

35 ⊥ are bands of matrices in Mn with constant rank (= k). Clearly Bk = {In − B : B ∈ Bk}. Jordan automorphisms send bands to bands.

It can be shown that bands of a constant rank k are triangularizable, and in some basis, take the form     0 XXY  0 Ik Y  : X ∈ X ,Y ∈ Y  0 0 0  where X and Y are sets of matrices of the appropriate size (note the collection X or Y may be empty, giving bands of the form (2.9)); see [Section 2.3, [73]]. Hence every band of constant rank k can be associated to a rank-k idempotent. Since a band is closed under multiplication, the algebra generated by it is just the linear span of its elements. Given any square-zero element N and rank-k idempotent P in (the span of) a band B, it follows that PN + NP = N. The structure of bands of constant rank k mirrors the structure of rank-k idempotent and square-zero matrices, connecting to the ideas of Section 2.2. In particular if one deﬁnes a bijective linear mapping φ : Mn → Mn by

tr(X) φ(X) = k In − f(X) for all X ∈ Mn, where f is a Jordan automorphism, the φ sends bands of rank k to bands of rank n − k.

Maps preserving bands do not appear to have a treatment in the literature. The above settles bands of constant rank in Mn.

36 CHAPTER 3

Maps preserving nonzero Lie products

We say that a map φ : M → M preserves commutativity if φ(A)φ(B) = φ(B)φ(A) whenever

AB = BA for all A, B ∈ M. Classifying all maps preserving commutativity can be formulated as a relation preserver problem by declaring A ∼ B if and only if AB = BA. Furthermore, deﬁning the Lie product [A, B] = AB − BA (i.e., the additive commutator of A and B), the commutativity preserver problem can be restated as preserving Lie products equal to zero.

The commutativity preserver problem is an attractive problem to researchers both inside and outside of linear preservers. There are two main directions: algebraic and analytic.

In the algebraic direction, the systematic study of Lie structures and Lie homomorphisms on associative algebras/rings was initiated by Herstein in the 1950s. One problem was to classify all bijective Lie homomorphisms (see Deﬁnition 3.1.2) of simple algebras/rings. Since every Lie homomorphism preserves commutativity, a complete description of commutativity preservers also solves Herstein’s Lie isomorphism problems. Optimal results for arbitrary algebras/rings can be found in Section 3.1. One can also put Jordan product structures on algebras/rings and ask the same questions.

In [66], Omladiˇcstudied bijective linear maps preserving commutativity in both directions on the algebra of bounded operators on an inﬁnite-dimensional Banach space, an analogue to the

ﬁnite-dimensional case below. Also, in the realm of mathematical physics, quantum structures are typically described as operators acting on an underlying Hilbert space. Observable physical

37 quantities are described by hermitian operators. Two bounded observables are jointly measurable if and only if their corresponding hermitian operators commute. Thus a linear transformation of the operator space (often called a “symmetry” of the underlying quantum system) preserves joint measurability if and only if it preserves commutativity in both directions. Hence the description of commutativity preservers are relevant to quantum mechanics. See [64] for other applications of preservers to the mathematical formulation of quantum mechanics.

In this chapter, we present a variant of the commutativity preserver problem in which Lie products equal to a nonzero matrix are preserved, and consider two low-rank cases. The results obtained here suggest that describing maps preserving Lie products equal to nonzero matrices appears to be a hard problem.

Recall Theorem 1.1.11 due to Watkins. Commutativity preserving maps on Mn(F ), with F algebraically closed, have the form

−1 t −1 φ(X) = cUXU + f(X)In or φ(X) = cUX U + f(X)In.

for all X ∈ Mn. Maps of the form above are called standard commutativity preserving maps. The map φ is not assumed to preserve commutativity in both directions. The proof given by Watkins reduces the commutativity preserver problem to the rank-one preserver problem via centralizers.

A sketch of the proof, including a useful lemma, is as follows.

2 Lemma 3.0.1 (Lemma 1, [83]). If dim C(A) = n − 2n + 2, then there is a scalar λ ∈ C and

2 rank-one matrix E such that A = E + λIn. If dim C(A) > n − 2n + 2, then A = λIn for some scalar λ.

Proof. If pA is the characteristic polynomial of A, then pA splits into linear factors over C. Suppose

{λ1, . . . , λr} are its eigenvalues and let {n1, . . . , nr} denote the number of blocks in the Jordan form of A corresponding to the eigenvalues {λ1, . . . , λr}. We are free to arrange the eigenvalues such that

38 n1 ≥ n2 ≥ · · · ≥ nr. Let dij denote the multiplicity of the linear factor of pA corresponding to the

Pr Pni ith eigenvalue (i = 1, . . . , r) and the jth Jordan block (j = 1, . . . , ni). Note that i=1 j=1 dij = n since the characteristic polynomial is degree n. A formula due to Frobenius is

r n X Xi dim C(A) = min{dij, dik}. i=1 j,k=1

Pr Pni Hence dim C(A) ≤ i=1 j,k=1 dij by making the trivial estimate min{dij, dik} ≤ dij. As k runs from 1 to ni, dij gets counted ni times. The estimate ni ≤ n1 also gives

r n r n r n X Xi X Xi X Xi dij ≤ ni dij ≤ n1 dij = n1n. i=1 j,k=1 i=1 j=1 i=1 j=1

2 2 2 Hence if n + 2n − 2 = dim C(A), then n + 2n − 2 ≤ n1n, or equivalently, n − 2 + n ≤ n1. So either n1 = n or n1 = n − 1. If n1 = n (that is, there are n Jordan blocks corresponding to the eigenvalue

λ1), then the Jordan form of A is merely the scalar λ1In. If n1 = n − 1, then A has either 1 or 2 distinct eigenvalues. If A only has one eigenvalue, say λ1 = λ, then the Jordan form of A consists of n − 2 blocks of size 1 × 1 plus a 2 × 2 Jordan block all corresponding to λ; which we recognize as a scalar matrix λIn plus an oﬀ-diagonal matrix unit. Hence A = λIn + E, where E is a rank-one nilpotent. If A has two eigenvalues, then n − 1 of the diagonal entries of the Jordan form are λ1 while the other is λ2. This is the scalar matrix λ1In plus a diagonal matrix unit with coeﬃcient

λ2 − λ1, so A = λ1In + (λ2 − λ1)E where E is a rank-one idempotent.

Notice that if B ∈ C(A) and φ preserves commutativity, then φ(B) ∈ C(φ(A)), i.e.,

φ(C(A)) ⊆ C(φ(A)) (A ∈ Mn).

Hence the commutativity preserver problem can be rephrased such that φ “commutes” with taking centralizers. So if φ is bijective,

dim φ(C(A)) ≤ dim C(φ(A)) (A ∈ Mn). (3.1)

39 Considering Lemma 3.0.1, we have φ(In) = cIn for some c ∈ C. By bijectivity it follows that every rank-one matrix A must satisfy dim C(φ(A)) = n2 − 2n + 2. By Lemma 3.0.1 there is a corresponding scalar λ and rank-one matrix E (depending on A) such that φ(A) = E + λI.A linear functional f can be constructed such that f(A) is the corresponding scalar λ. Thus the linear bijective map φ(A) − f(A)In preserves the set of rank-one matrices, and applying the result of Marcus and Moyls (Theorem 1.1.1) completes the proof.

Watkins’s original proof depended heavily on the hypotheses that n ≥ 4 and the underlying

ﬁeld was algebraically closed. The assumption that n ≥ 4 facilitated the computations in deﬁning the linear functional f, while the second was imposed in order to discuss Jordan canonical forms in the proof of Lemma 3.0.1. Shortly after the original paper was published, joint work by Pierce and

Watkins (see [72]) weakened both hypotheses to n ≥ 3 and arbitrary ﬁelds (see Theorem 1.1.13).

The proof uses the fundamental theorem of projective geometry. Other preserver problems using projective techniques include Theorems 1.1.4 and 1.1.5. A description of the technique is included in the expository paper [52].

However, 2 × 2 matrices have comparatively “small” centralizers. It is not hard to show that, because every nonscalar matrix A ∈ M2(F ) is algebraic of degree 2, we have C(A) = F [A] = F +FA, the F -subalgebra of M2(F ) generated by In and A (see [11] for a treatment of algebras with this centralizer property). Hence if φ : M2(F ) → M2(F ) is a linear mapping that preserves scalar matrices; in particular if φ is a unital mapping, then φ trivially preserves commutativity. Consider the following concrete example.

Example 3.0.2 ([83]). The linear map L : M2(F ) → M2(F ) deﬁned by

a b a a − d L : 7→ . (3.2) c d c b + d is bijective and satisﬁes [L(A),L(B)] = 0 whenever [A, B] = 0 (straightforward to verify). Notice

40 also L is a unital map; that is L(I2) = I2. However, L does not have a standard form. Assume to

−1 the contrary that L(X) = cSXS + f(X)I2 (see Theorem 1.1.11). On the one hand, by equation

(3.2) we have that L(e12) = e22. On the other,

−1 L(e12) = cSe12S + f(e12)I2.

−1 Considering traces, it must be that tr(L(e12)) = tr(e22) = 1 while tr(cSe12S +f(e12)I2) = 2f(e12),

1 so f(e12) = 2 . However, considering determinants, it must be that det(L(e12)) = det(e22) = 0 while

−1 2 det(cSe12S + f(e12)I2) = f(e12) . This implies that f(e12) = 0, a contradiction. The case when

t −1 L(X) = cSX S + f(X)I2 is analogous.

Finally, we mention the classical commutativity preserver problem under the relaxed hypothesis that φ is merely assumed linear (not necessarily bijective). Clearly any linear mapping whose image

0 tr(A) is commutative is now a solution. Also along these lines, for each matrix A ∈ Mn let A = A− n In denote its trace-zero part. Note that [A, B] = 0 if and only if [A0,B0] = 0, so the linear mapping

0 from Mn into sln deﬁned by A 7→ A preserves commutativity but is certainly not bijective. We still recognize this as a “standard” solution in the sense that the map is bijective on sln.

Theorem 3.0.3 (Omladiˇc,Radjavi, Semrlˇ [67]). Let n ≥ 3. If φ : Mn → Mn is a linear map preserving commutativity, then either the image of φ is commutative or there exists an invertible matrix U ∈ Mn, scalar c ∈ C, and linear functional f such that

−1 (1) φ(X) = cUXU + f(X)In, or

t −1 (2) φ(X) = cUX U + f(X)In

for all X ∈ Mn.

Hence the image of φ is either commutative, sln, or Mn. What do the maximal commutative subalgebras of Mn look like? In fact, the maximum dimension of a commutative subalgebra of Mn

41 n2 is 1 + b 4 c. This can be realized as an algebra generated by matrices of the form

0 ∗ 0 0

n−1 together with the identity matrix In, where the nonzero block is the full vector space of d 2 e ×

n+1 b 2 c matrices. In general, every maximal commutative subalgebra is conjugate to this upper triangular example. Every pair of nonscalar matrices in a commutative subalgebra therefore has trivial multiplication. A simple proof of these observations can be found in [63].

3.1 Commutativity preservers and Herstein’s Lie isomorphism problem

Here we describe the generalization of the commutativity preserver problem to simple rings. De- velopments in this algebraic setting enabled a solution of Herstein’s Lie isomorphism problems.

Deﬁnition 3.1.1. The standard polynomial of degree n is

X sn(x1, x2, . . . , xn) = sgn(σ)xσ(1)xσ(2) ··· xσ(n), σ∈Sn where sgn(σ) is the sign of the permutation σ and Sn is the symmetric group on n letters.

The famous Amitsur-Levitzki theorem states that s2n is a polynomial identity of Mn(F ); that is, for any collection of n × n matrices A1,A2,...,A2n, we have s2n(A1,A2,...,A2n) = 0. Polynomial identity theory (PI theory) of rings is a powerful tool for studying large classes of rings and makes a brief appearance in the upcoming remarks. Standard references are [12, 74].

Deﬁnition 3.1.2. Let R be a ring and let A and B denote two R-algebras. An R-linear map

φ : A → B is a Lie homomorphism if φ([a, b]) = [φ(a), φ(b)] for all a, b ∈ A.A Lie isomorphism is a bijective Lie homomorphism. A Lie automorphism is a Lie isomorphism with B = A.

One can replace the Lie product with the Jordan product and deﬁne analogous Jordan homomorphisms of R-algebras.

42 Herstein initiated the study of the relationship between the ordinary, Lie, and Jordan product structure of associative simple rings in the 1950’s [38, 40, 41]. He described the Lie and Jordan ideal structure of simple rings, but many open questions remained concerning Lie isomorphisms (Jordan isomorphisms were addressed directly by Herstein and Jacobson, Rickart in [39], [45], respectively).

Lie structure is generally considered to be more diﬃcult than the Jordan structure.

Two straightforward families of Lie isomorphisms are given by (ordinary) isomorphisms and the negative of antiisomorphisms. By “the negative of an antiisomorphism” we mean a map of the form −α(x), where α(xy) = α(y)α(x) for all x, y in an R-algebra A with α bijective. Indeed, if

φ = −α, then

φ([a, b]) = −α(ab − ba)

= −α(b)α(a) + α(a)α(b)

= [α(a), α(b)]

= [−α(a), −α(b)]

= [φ(a), φ(b)] for all a, b ∈ A. Hence α is a Lie isomorphism. Moreover one can add any map into the center of the target algebra and still obtain a Lie isomorphism.

The description on matrices was known for a long time. In 1951, Hua [43] showed that every Lie automorphism of Mn(D), where D is a division ring and n ≥ 3, is of the form α + τ, where α is an automorphism or the negative of an antiautomorphism and τ is an additive map into the center of

Mn(D) vanishing on Lie products. Around this time the problems on simple rings were generalized to the larger class of prime rings (see the remarks following Theorem 3.1.4). Martindale and his students extended Hua’s result to prime rings containing nontrivial idempotents and found the same standard description with some mild technical restrictions (see [9] and the references therein).

However, an idempotent-free result would be optimal.

43 Some years later, in his influential paper, Breˇsardefinitively removed such a dependence on idempotents and effectively solved Herstein’s Lie isomorphism problem. The description of commutativity preserving maps was the key to the solution. For brevity we will state the theorems for simple rings.

Theorem 3.1.3 (Breˇsar[9]). Let A and A0 be central simple algebras over a ﬁeld F . Suppose that

0 0 the characteristic of A is not 2, and suppose that F 6= F3. Let θ : A → A be a bijective linear

2 0 mapping satisfying [θ(x ), θ(x)] = 0 for all x ∈ A. If neither A nor A satisﬁes s4 then

θ(x) = cα(x) + f(x)(x ∈ A) where c ∈ F , c 6= 0, α is an isomorphism or antiisomorphism of A onto A0, and f is a linear mapping from A into the center of A.

0 The condition that A and A do not satisfy s4 is analogous to the requirement that n ≥ 3 for commutativity preserving mappings on Mn. Indeed, every simple ring satisfying s4 is either commutative or can be isomorphically embedded in M2(F ) for a ﬁeld F . This is a folklore result in PI theory; see [Theorem C.2, [12]].

Every commutativity preserving map θ satisﬁes [θ(x2), θ(x)] = 0, so in fact, the hypotheses of Theorem 3.1.3 are substantially weaker. Restricting focus to Lie isomorphisms, many of the technical hypotheses are relaxed substantially to give an optimal result, as follows.

Theorem 3.1.4 (Breˇsar[9]). Let R and R0 be simple unital rings such that the characteristic of

0 0 0 R is not 2. Let θ : R → R be a Lie isomorphism. If neither R nor R satisﬁes s4, then θ is of the form α + τ, where α is a homomorphism or negative of an antihomomorphism of R into R0, α is one-to-one, and τ is an additive mapping of R into the center of R0 sending Lie products to zero.

As mentioned above, these theorems are stated for prime rings, which are a generalization of simple rings. A ring R is called prime if aRb = 0 implies a = 0 or b = 0 with a, b ∈ R. Equivalently,

44 R is prime if IJ = 0 implies I = 0 or J = 0 for all (two-sided) ideals I and J of R. One related construction is the Martindale (right) ring of quotients Q(R) of a prime ring R. The center of Q(R) is called the extended centroid of R. The maps into the center appearing in Theorems 3.1.3 and 3.1.4 can be replaced with maps into the extended centroid with some other technical considerations.

For a comprehensive treatment of rings of quotients and the extended centroid, see [Section 2.3,

[4]] or [Chapter 7, [10]].

3.2 On maps preserving Lie products equal to e12

This section presents an original result Theorem 3.2.6 intimately related to commutativity preservers. The ﬁrst subsection contains a brief interlude on triangularizability with a simple result

(Corollary 3.2.5) with surprising implications for the main theorem.

3.2.1 Some results on triangularization

Recall Schur’s triangularization theorem: every complex square matrix is similar to a matrix in upper-triangular form (in fact, unitarily equivalent, but this is not needed). When can two matrices be written in upper-triangular forms relative to the same basis?

Deﬁnition 3.2.1. Two matrices A and B are called simultaneously triangularizable if there is an invertible matrix U such that UAU −1 and UBU −1 are upper-triangular matrices.

The theory of simultaneous triangularization is a rich subject with applications in both the

ﬁnite- and inﬁnite-dimensional cases. We refer the reader to the aptly named textbook [73] for more interesting results in this subject. It is a standard fact that two commuting matrices are simultaneously triangularizable. For our purposes, we must consider simultaneous triangularizability for pairs of matrices with a nonzero Lie product.

Theorem 3.2.2 ((Laﬀey’s Theorem) Theorem 1.3.6, [73]). If A and B are complex matrices such that rank([A, B]) = 1, then A and B are simultaneously triangularizable.

45 A generalization to algebraically closed ﬁelds is due to Guralnick [35].

Example 3.2.3. It is easy to see that if rank([A, B]) ≥ 2, then A and B need not be simultaneously triangularizable. Take A = e12 and B = e21. Their Lie product is e11 − e22 which is clearly not nilpotent. If both e12 and e21 were simultaneously triangularizable, there would be some basis in which both are represented by strictly upper-triangular matrices. However the Lie product of two strictly-upper triangular matrices is strictly upper-triangular and hence nilpotent, giving a contradiction.

Standard results in triangularization theory gives conditions for a pair of matrices A and B to be simultaneously triangularizable in terms of nilpotent elements. This connection to Laﬀey’s theorem is immediate since every rank-one trace-zero matrix is nilpotent. In fact more is true:

Theorem 3.2.4 (Theorem 1.3.8, [73]). A collection E of n × n matrices is simultaneously triangularizable if and only if for every m ≥ 1 and subset {R1,R2,...,Rm, S, T } ⊆ E, the matrix

R1R2 ··· Rm(ST − TS) is nilpotent.

Take E = {A, B}. The theorem says that A and B are simultaneously triangularizable if and only if w(A, B)[A, B] is nilpotent for every monomial w(A, B) in A and B. Combining the above results, we obtain the following.

Corollary 3.2.5 (Ginsburg, Julius, Velasquez [34]). If A = (aij) and B = (bij) are n × n matrices such that [A, B] = e12, then a21 = b21 = 0.

Proof. Since rank([A, B]) = 1, A and B are simultaneously triangularizable by Theorem 3.2.2.

Hence the matrices A[A, B] and B[A, B] are both nilpotent by Theorem 3.2.4, so tr(A[A, B]) = a21 = 0 and tr(B[A, B]) = b21 = 0.

Consequently, it is not too diﬃcult to show that the linear subspace

A = span{A ∈ Mn | [A, B] = e12}

46 2 is (n − 1)-dimensional over C, consisting of matrices of the form ∗ ∗ ∗ · · · ∗ 0 ∗ ∗ · · · ∗   ∗ ∗ ∗ · · · ∗   . ......  . . . . . ∗ ∗ ∗ · · · ∗

3.2.2 The main result

Recall that a Lie homomorphism preserves all Lie products while a commutativity preserving map preserves zero Lie products. In other words, if φ is linear and preserves commutativity, then φ(0) = 0 and [φ(A), φ(B)] = 0 whenever [A, B] = 0. Consider a variant of this problem by substituting the zero matrix with a nonzero matrix.

Let φ : Mn → Mn denote a bijective linear map such that

φ(e12) = e12 and [φ(A), φ(B)] = e12 whenever [A, B] = e12. (3.3)

Unlike the zero case, maps preserving Lie products equal to e12 do not have the same standard description. The next theorem stands in contrast to the rich literature surrounding commutativity preserving mappings.

Theorem 3.2.6 (Ginsburg, Julius, Velasquez [34]). If n ≥ 5 and φ : Mn → Mn is a bijective linear map such that φ(e12) = e12 and [φ(A), φ(B)] = e12 whenever [A, B] = e12, then there exists an invertible matrix P and a linear functional f on Mn such that, for (i, j) 6= (2, 1),

−1 (1) φ(eij) = P eijP + f(eij)In, or

−1 (2) φ(eij) = −P ejiP + f(eij)In.

In either case, φ(e21) = xe21 + X, where x ∈ C is nonzero and X ∈ Mn has a zero (2, 1)-entry.

Corollary 3.2.5 explains why φ(e21) completely avoids description. It is impossible to involve a nonzero multiple of e21 in any pair of matrices such that [A, B] = e12. An alternate characterization

47 of the theorem is that φ is a standard commutativity preserving map restricted to the subspace

∼ A = span{A ∈ Mn | [A, B] = e12} = Mn/Ce21.

Recalling the terminology of Section 1.3, Mn/Ce21 is the preserved subspace of φ and is not closed under the ordinary, Lie, or Jordan product. And the description on A need not extend to a commutativity preserving map on Mn.

Example 3.2.7. Let φ : Mn → Mn be the linear map deﬁned by φ(eij) = eij for (i, j) 6= (2, 1) and

φ(e21) = e12 + e21. Such a map is clearly bijective and in view of Corollary 3.2.5 preserves all Lie products equal to e12. However, [φ(e21), φ(e23)] = e13 while [e21, e23] = 0, so φ does not preserve commutativity.

For maps satisfying Property (3.3) it is generally false that φ(C(A)) ⊆ C(φ(A)) if A is a rank- one matrix (see Lemma 3.0.1). Thus the map must be built up from scratch, using Lemma 3.0.1 only where applicable. The following lemma establishes some basic facts about φ acting on the basis of matrix units.

Lemma 3.2.8. If φ : Mn → Mn is a bijective linear map satisfying Property (3.3), then

(1)[ φ(A), e12] = 0 whenever [A, e12] = 0,

(2)[ φ(A), φ(e11)] = 0 whenever [A, e11] = 0, and

(3)[ φ(A), φ(e22)] = 0 whenever [A, e22] = 0.

In other words, φ(C(e12)) ⊆ C(e12), φ(C(e11)) ⊆ C(φ(e11)), and φ(C(e22)) ⊆ C(φ(e22)). Moreover,

(4)[ φ(eii), φ(ejk)] = 0 if i∈ / {j, k} and (j, k) 6= (2, 1),

(5)[ φ(eii + ejj), φ(eij)] = 0 if (i, j) 6= (2, 1),

(6) φ(In) = λIn for some nonzero λ ∈ C,

48 and there exist rank-one matrices E11,E22 and scalars λ11, λ22 such that φ(e11) = E11 + λ11In and

φ(e22) = E22 + λ22In, with E11 consisting of precisely one row vector (resp. column vector) and

E22 consisting of precisely one column vector (resp. row vector).

Proof. If [A, e12] = 0, then [e11 + A, e12] = e12. By Property (3.3), [φ(e11 + A), φ(e12)] = e12, and so [φ(e11), φ(e12)] + [φ(A), φ(e12)] = e12 by linearity. The ﬁrst summand is equal to e12, hence [φ(A), φ(e12)] = 0. Statements (2) and (3) follow from a similar analysis of the Lie products [e11, e12 + A] = e12 and [A + e12, e22] = e12, respectively. Hence φ “commutes” with taking centralizers of e12, e11, and e22.

To prove (4), suppose ﬁrst that i∈ / {j, k}. Since ejk commutes with eii, the cases when i = 1 or i = 2 follow directly from statements (2) and (3) above. In particular, if j = k, then

[φ(eii), φ(ejj)] = 0 whenever i = 1, 2 and i 6= j. Assume that i ≥ 3. Since [e11 + e1k, e12 + eii] = e12, we have that [φ(e11 + e1k), φ(e12 + eii)] = e12. But e1k commutes with e12 and e11 commutes with eii, so we conclude by (1) and (2) that [φ(e1k), φ(eii)] = 0. Hence (4) holds if j = 1. Likewise, since

[eii + e22, ej1 − e12] = e12 whenever j 6= 2, we conclude that [φ(eii), φ(ej1)] = 0. Hence (4) also holds if k = 1.

Assume now that j 6= 1 and k 6= 1. Since [e11 + eii, e12 + ejk] = e12, we have that [φ(e11 + eii), φ(e12 + ejk)] = e12. But ejk commutes with e11 and eii commutes with e12, so it must be that

[φ(eii), φ(ejk)] = 0. The proof of (4) is complete.

Now we will show that [φ(eii + ejj), φ(eij)] = 0. If i = j, there is nothing to prove, so assume i 6= j. Suppose ﬁrst that i = 1 and j = 2. Since [φ(e11), φ(e12)] = e12 and [φ(e22), φ(e12)] = −e12, the linearity of the Lie product shows that [φ(e11 + e22), φ(e12)] = 0 directly. Suppose that i = 1 and j ≥ 3. Since [e11 + ejj, e1j + e12] = e12, we have that [φ(e11 + ejj), φ(e1j) + e12] = e12. But ejj commutes with e12, so we conclude that [φ(e11 + ejj), φ(e1j)] = 0. Replacing e1j with ej1 and repeating this argument also shows that [φ(e11 + ejj), φ(ej1)] = 0.

49 Assume that i = 2 and j ≥ 3. Since [e22 + ejj, e2j − e12] = e12, it follows from linearity that

[φ(e22 + ejj), φ(e2j)] = 0. An analogous argument also shows that [φ(e22 + ejj), φ(ej2)] = 0.

Lastly, if both i ≥ 3 and j ≥ 3, observe that [e11 + eii + ejj, e12 + eij] = e12. Since eij commutes with e11 and eii, ejj commute with e12, we have by (1) and (2) that [φ(eii + ejj), φ(eij)] = 0. Thus statement (5) holds for all pairs (i, j) except (2, 1).

Consider φ(eij), where (i, j) 6= (2, 1). Since

X [φ(In), φ(eij)] = [φ(eii + ejj), φ(eij)] + [φ(ekk), φ(eij)], k∈{ / 1,2}

2 it follows from (4) and (5) that [φ(In), φ(eij)] = 0. Hence dim C(φ(In)) ≥ n − 1. By Lemma 3.0.1,

φ(In) is a nonzero scalar multiple of the identity.

The remaining forms of φ(e11) and φ(e22) also make use of Lemma 3.0.1. The centralizers of

2 φ(e11) and φ(e22) have dimension exactly n − 2n + 2, so there are corresponding rank-one matrices

E11 and E22 and scalars λ11 and λ22 such that

φ(e11) = E11 + λ11In, φ(e22) = E22 + λ22In. (3.4)

If φ(e11) = (aij) and φ(e22) = (bij), then it is routine to verify that [φ(e11), e12] = e12 and

[e12, φ(e22)] = e12 imply that     1 + a11 a12 a13 ··· a1n b22 b12 b13 ··· b1n  0 a11 0 ··· 0   0 1 + b22 0 ··· 0       0 a32 a33 ··· a3n  0 b32 b33 ··· b3n φ(e11) =   , φ(e22) =   .  ......   ......   . . . . .   . . . . .  0 an2 an3 ··· ann 0 bn2 bn3 ··· bnn

By equation (3.4),

  1 + a11 − λ11 a12 a13 ··· a1n  0 a11 − λ11 0 ··· 0     0 a32 a33 − λ11 ··· a3n  E11 =    ......   . . . . .  0 an2 an3 ··· ann − λ11

50 and   b22 − λ22 b12 b13 ··· b1n  0 1 + b22 − λ22 0 ··· 0     0 b32 b33 − λ22 ··· b3n  E22 =   .  ......   . . . . .  0 bn2 bn3 ··· bnn − λ22

However, since E11 and E22 are rank-one matrices, then either     1 a12 a13 ··· a1n 0 a12 0 ··· 0 0 0 0 ··· 0  0 −1 0 ··· 0     0 0 0 ··· 0  0 a32 0 ··· 0 E11 =   , or E11 =   (3.5) ......  ......  . . . . .  . . . . . 0 0 0 ··· 0 0 an2 0 ··· 0 or     0 b12 0 ··· 0 −1 b12 b13 ··· b1n 0 1 0 ··· 0  0 0 0 ··· 0      0 b32 0 ··· 0  0 0 0 ··· 0  E22 =   , or E22 =   . (3.6) ......   ......  . . . . .  . . . . .  0 bn2 0 ··· 0 0 0 0 ··· 0

But [φ(e11), φ(e22)] = [E11,E22] = 0, so if E11 and E22 are both row matrices, then [E11,E22] = 0 implies that E22 is a multiple of E11, which contradicts bijectivity of φ. Hence if E11 is the row

(resp. column) matrix in equation (3.5), E22 must be the column (resp. row) matrix in equation

(3.6). The proof of the lemma is complete.

Deﬁnition 3.2.9. Let V denote the subalgebra of matrices generated by matrices of the form 0 0 0 ··· 0 0 0 0 ··· 0   0 0 ∗ · · · ∗   . ......  . . . . . 0 0 ∗ · · · ∗

Clearly V is isomorphic to Mn−2.

Lemma 3.2.10. C(e12) ∩ C(e11) ∩ C(e22) = V + CIn.

Proof. The centralizers of e12, e11, and e22 are generated by matrices of the form x ∗ ∗ · · · ∗ ∗ 0 0 ··· 0 ∗ 0 ∗ · · · ∗ 0 x 0 ··· 0 0 ∗ ∗ · · · ∗ 0 ∗ 0 ··· 0       0 ∗ ∗ · · · ∗ 0 ∗ ∗ · · · ∗ ∗ 0 ∗ · · · ∗   ,   , and   ,  ......  ......  ......   . . . . . . . . . . . . . . . 0 ∗ ∗ · · · ∗ 0 ∗ ∗ · · · ∗ ∗ 0 ∗ · · · ∗

51 respectively, where x ∈ C. A matrix A is contained in all three centralizers if and only if it is of the form x 0 0 ··· 0 0 x 0 ··· 0   0 0 ∗ · · · ∗ A =   .  ......   . . . . . 0 0 ∗ · · · ∗

Hence A can be uniquely identiﬁed as V + xIn with V ∈ V and x ∈ C.

Since φ preserves the centralizers of e12, e11, and e22, φ preserves V up to addition of a scalar matrix; that is, φ(V) ⊆ V + CIn.

Theorem 3.2.11 (Theorem 6, [34]). Let φ : Mn → Mn be a bijective linear map satisfying Property

(3.3). The restriction of φ to V is a standard commutativity preserving map.

Proof. Let σ1 : Mn → Mn be the linear map deﬁned by σ1(A) = φ(A) − g1(A)In, where g1 is a linear functional on Mn chosen so that σ1 is bijective with g1(e11) = λ11, g1(e22) = λ22, and g1(e12) = 0 (see Lemma 3.2.8 above); in other words,

σ1(e11) = E11, σ1(e22) = E22, σ1(e12) = e12,

respectively. It also follows immediately that [σ1(A), σ1(B)] = [φ(A), φ(B)] for all A, B ∈ Mn, so

σ1 certainly satisﬁes Property (3.3). We may also insist that σ1(In) = In by choosing g1(In) = λ−1 for λ as in part (6) of Lemma 3.2.8.

Lemma 3.2.8 presents two possible deﬁnitions for the map σ1 based on the forms of E11 and

E22. Ideally, one would like to pass to a map that ﬁxes e11, e22, and e12 directly, which can be done as follows. If E11 is the row matrix in equation (3.5) and E22 is the column matrix in equation

Pn (3.6), the condition [E11,E22] = 0 implies that a12 + b12 + k=3 a1kbk2 = 0 for the matrix entries.

Using this relation, there is an invertible matrix U1 such that

−1 −1 −1 U1E11U1 = e11,U1E22U1 = e22 and U1e12U1 = e12, (3.7)

52 given by     1 a12 a13 a14 ··· a1n 1 b12 −a13 −a14 · · · −a1n 0 1 0 0 ··· 0  0 1 0 0 ··· 0      0 −b32 1 0 ··· 0  0 b32 1 0 ··· 0    −1   U1 = 0 −b 0 1 ··· 0  ,U1 = 0 b 0 1 ··· 0  .  42   42  ......  ......  ......  ......  0 −bn2 0 0 ··· 1 0 bn2 0 0 ··· 1

If E11 is the column matrix in equation (3.5) and E22 is the row matrix in equation (3.6), the

Pn condition [E11,E22] = 0 implies that a12 + b12 − k=3 b1kak2 = 0. Using this relation, there is an invertible matrix U2 such that

−1 −1 −1 − U2E11U2 = e11, −U2E22U2 = e22, and − U2e12U2 = e21 (3.8) given by     0 −1 0 0 ··· 0 a12 1 b13 b14 ··· b1n 1 −b12 −b13 −b14 · · · −b1n −1 0 0 0 ··· 0      0 a32 1 0 ··· 0  a32 0 1 0 ··· 0    −1   U2 = 0 a 0 1 ··· 0  ,U2 = a 0 0 1 ··· 0  .  42   42  ......   ......  ......   ......  0 an2 0 0 ··· 1 an2 0 0 0 ··· 1

−1 t If U3 = (U2 ) , taking the transpose through equation (3.8) yields

t −1 t −1 t −1 − U3(E11) U3 = e11, −U3(E22) U3 = e22, and − U3(e12) U3 = e12. (3.9)

−1 Therefore the automorphism σ2 deﬁned by A 7→ U1σ1(A)U1 or the negative antiautomorphism

t −1 σ2 deﬁned by A 7→ −U3σ1(A) U3 satisﬁes

σ2(e11) = e11, σ2(e22) = e22, and σ2(e12) = e12, (3.10)

With either determination of E11 and E22, equation (3.10) guarantees that σ2 satisﬁes Property

(3.3); in particular, that σ2 satisﬁes (1), (2), and (3) of Lemma 3.2.8. However, since σ2 ﬁxes the matrix units e11, e22, and e12, the map σ2 must preserve their respective centralizers.

By Lemma 3.2.10, if A ∈ V, then [σ2(A), e11] = [σ2(A), e22] = [σ2(A), e12] = 0, which shows an inclusion σ2(V) ⊆ C(e11) ∩ C(e22) ∩ C(e12) = V + CIn. Deﬁne a bijective linear map σ3 : Mn → Mn

53 by

σ3(A) = σ2(A) − g3(A)In,

where g3 is a linear functional on Mn chosen so that σ3 is bijective, fixes e11, e22, e12, and σ3(V) ⊆ V with σ3(In) = In. With this definition, σ3 has the property that [σ3(A), σ3(B)] = [σ2(A), σ2(B)] for all A, B ∈ Mn, so certainly σ3 satisfies Property (3.3).

Choose A, B ∈ V such that [A, B] = 0. Since [e11 +A, e12 +B] = e12, it follows immediately that

[σ3(A), σ3(B)] = 0. Thus σ3 preserves commutativity on a subalgebra V, which by assumption, has degree at least 3. By Theorem 1.1.11 and Theorem 1.1.13, the restriction of σ3 to V is a standard commutativity preserving map of the form

−1 −1 t σ3(A) = cS AS + f(A)IV or σ3(A) = cS A S + f(A)IV ,

where c is a nonzero scalar, S ∈ V is an invertible matrix, f is a linear functional on V, and IV is the matrix e33 + e44 + ··· + enn.

Since σ3 was obtained solely by applying inner automorphisms or antiautomorphisms to φ, it follows that φ is also a standard commutativity preserving map. The proof is complete.

The computations performed in Lemma 3.2.8 demonstrated that φ preserves some speciﬁc centralizer subalgebras and that φ sends scalars to scalars. Next we demonstrate that φ preserves more zero Lie products.

Lemma 3.2.12. Let φ : Mn → Mn be a bijective linear map that satisﬁes Property (3.3), with n ≥ 4. If [eij, ekl] = 0 where neither (i, j) nor (k, l) is equal to (2, 1), then [φ(eij), φ(ekl)] = 0.

The lemma gives more evidence to the claim that φ preserves commutativity on an (n2 − 1)- dimensional subspace of Mn. The full proof is omitted since the technique is similar to Lemma

3.2.8.

54 The proof of Theorem 3.2.6 involves extending the description of φ on V as in Theorem 3.2.11 to Mn/Ce21. There are several computational details to check. The overall procedure is: ﬁnd a pair of matrices A and B involving the matrix units in the computations that satisﬁes [A, B] = e12 and use the previous results (in particular, Lemma 3.2.12) to isolate the desired property. All details can be found in [34]. We will sketch the remainder of the proof.

Proof of Theorem 3.2.6. Pass from φ back to σ3 as in the conclusion of Theorem 3.2.11. Deﬁne

σ4 : Mn → Mn by

−1 σ4(A) = Qσ3(A)Q ,

where Q = e11 + e22 + S, which is clearly invertible because S is invertible as a matrix in V. Since

σ3 ﬁxes e11, e22 and e12, it is easy to see that so does σ4. If A ∈ V, then σ4(A) = cA + f(A)IV by deﬁnition. One can show through direct computation that c = 1 and f(A) ≡ 0, so σ4 acts as the identity on V along with e12, e11, and e22. From here, the descriptions of images of matrix units outside V are obtained:

σ4(e1j) = a1je1j + a11In, (j ≥ 3) (3.11)

σ4(ek2) = bk2ek2 + b11In, (k ≥ 3) (3.12)

σ4(e2l) = c2le2l + c11In, (l ≥ 3) (3.13)

σ4(ei1) = di1ei1 + d11In, (i ≥ 3) (3.14)

where a1j, bk2, c2l, di1 6= 0. It can also be shown through several more computations that a1j =

−1 −1 bk2 = c2l = di1 for all such combinations 3 ≤ i, j, k, l ≤ n. Let σ5 be the map deﬁned by

σ5(A) = σ4(A) − g4(A)In, where g4(A) is a linear functional such that g4(eij) is the (1,1)-entry of σ4(eij). The map is linear, bijective, and satisﬁes Property (3.3). Moreover, if β = a13, by the

55 above equations we get  βeij i ≤ 2, j ≥ 3  −1 σ5(eij) = β eij i ≥ 3, j ≤ 2   eij i, j ≥ 3, and obviously σ5(e12) = e12, σ5(e11) = e11, and σ5(e22).

−1 The matrix C = β(e11 + e22) + e33 + ··· + enn satisﬁes C σ5(eij)C = eij for all (i, j) 6= (2, 1).

−1 So if σ6 is the bijective map deﬁned by σ6(A) = C σ5(A)C, it follows that σ6 is a linear bijective map that not only satisﬁes Property (3.3), but also acts as the identity map on the linear span of {eij | (i, j) 6= (2, 1)}. As σ6 was obtained solely by applying various inner automorphisms or antiautomorphisms to φ, it follows that φ has the form of a standard commutativity preserving map on span{eij | (i, j) 6= (2, 1)}.

Lastly, we must have that φ(e21) = xe21 + X, where x 6= 0, since otherwise we would contradict bijectivity. The proof is complete.

As a consequence, we have the following corollary.

Corollary 3.2.13. If n ≥ 5 and φ : Mn → Mn is a bijective linear map such that φ(N) = M and

[φ(A), φ(B)] = M whenever [A, B] = N, where M,N are rank-one nilpotent matrices, then φ is a

2 standard commutativity preserving map on an (n − 1)-dimensional subspace of Mn.

−1 −1 Proof. Let U, V be invertible matrices such that UNU = e12 = VMV and suppose that

[A, B] = N implies that [φ(A), φ(B)] = M, with φ(N) = M. Deﬁne ψ(X) = V φ(U −1XU)V −1 for all X ∈ Mn. Clearly ψ(e12) = e12. Also,

−1 −1 [UAU ,UBU ] = e12 =⇒ [A, B] = N =⇒ [φ(A), φ(B)] = M

−1 =⇒ V [φ(A), φ(B)]V = e12

−1 −1 −1 −1 −1 =⇒ V [φ(U UAU U), φ(U UBU U)]V = e12

−1 −1 =⇒ [ψ(UAU ), ψ(UBU )] = e12. (3.15)

56 Since ψ preserves Lie products equal to e12, it follows from Theorem 3.2.6 that ψ has the form of a standard commutativity preserving map on Mn/Ce21. Thus φ is a standard commutativity

−1 −1 preserving map from Mn/CU e21U into Mn/CV e21V .

A crucial detail of the proof of Theorem 3.2.6 is that the restriction of a map φ to a subalgebra isomorphic to Mn−2(C) preserves commutativity. Thus the condition that n ≥ 5 imposed to invoke

Theorem 1.1.13, but one may be interested in this preserver problem on matrices of lower degree, since Corollary 3.2.5 holds for all square matrices of degree greater than 1.

Corollary 3.2.14. If n 6= 4 and φ : Mn → Mn is a bijective linear map that preserves Lie products equal to a rank-one nilpotent, then φ is a standard commutativity preserving map on an (n2 − 1)- dimensional subspace of Mn.

Proof. In view of Corollary 3.2.13, assume without loss of generality that φ preserves Lie products equal to e12. If n = 2 and φ preserves Lie products equal to e12, pass from φ to σ2 (see equation

(3.10)) and the result is immediate. If n = 3, then V as deﬁned is simply the linear span of e33.

Pass from φ to σ3 and we see that σ3(e33) = ce33. Deﬁne σ4 analogously. A similar procedure in generating the equations (3.11) − (3.14) in this case is much simpler, from which one passes to σ5 and σ6 in exactly the same way.

Recall that commutativity preserving maps on M2 do not have a standard description (see

Example 3.0.2 for one such map). So if n = 4, it is not clear if a nonstandard commutativity preserving map on the lower 2 × 2 corner induces a linear transformation on M4(C) preserving Lie products equal to e12.

3.3 On maps preserving Lie products equal to e11 − e22

The main result of the previous section was largely formulated as a test result to the previously unexplored questions concerning maps preserving nonzero Lie products. Every trace-zero matrix

57 over a ﬁeld is expressible as a single Lie product (see [77] for characteristic zero and [1] for arbitrary characteristic); that is, if N ∈ sln, then there exist at least one pair of matrices A and B such that

[A, B] = N. It is therefore a natural question to consider the same problem as in the previous section but replace e12 with any trace-zero matrix.

Notice that the observation of Corollary 3.2.5 was a consequence of Lie products being nilpotent.

Naturally we ask: if φ : Mn → Mn preserves Lie products equal to N, where N is not nilpotent, must φ be a standard commutativity preserving map? Up to similarity, the matrix e11 − e22 is the only trace-zero matrix of rank 2 that is not nilpotent, so we take N = e11 − e22 as another test case.

On the one hand, preserving Lie products equal to a trace-zero matrix of higher rank could con- ceivably be more complicated than the rank-one case. On the other hand, in view of Example 3.2.3 following Laﬀey’s Theorem 3.2.2, if rank([A, B]) ≥ 2 then A and B need not be triangularizable.

It may then be possible then to avoid the subtle problems of the N = e12 case.

Indeed, the situation for N = e11 − e22 is considerably nicer than N = e12.

Theorem 3.3.1 (Julius [46]). If n ≥ 5 and φ : sln → sln is a bijective linear map such that

φ(e11 − e22) = e11 − e22 and [φ(A), φ(B)] = e11 − e22 whenever [A, B] = e11 − e22, then there exists an invertible matrix U ∈ Mn such that

(1) φ(X) = UXU −1, or

(2) φ(X) = −UXtU −1

for all X ∈ sln.

Also by extending φ to Mn, we obtain:

Corollary 3.3.2 (Julius [46]). If n ≥ 5 and Φ: Mn → Mn is a bijective linear map such that

Φ(e11 − e22) = e11 − e22 and [Φ(A), Φ(B)] = e11 − e22 whenever [A, B] = e11 − e22, then there exists

58 an invertible matrix U ∈ Mn and a linear functional f : Mn → C such that

−1 (1)Φ( X) = UXU + f(X)In, or

t −1 (2)Φ( X) = −UX U + f(X)In

for all X ∈ Mn.

Preserver problems like these can be meaningfully stated on Lie algebras; however, only a handful of papers do so. See Lin [57] for notions of standard commutativity preserving maps in the context of other Lie ideals of Mn, and [85] for more general results. More will be discussed in

Section 3.4.

The proofs of these two results is similar to the case N = e12. As such the mapping must once again be built up from scratch. First, observe that any two rank-one trace-zero matrices satisfying

[A, B] = e11 − e22 can be simultaneously transformed to e12 and e21.

Lemma 3.3.3. If A and B are rank-one trace-zero matrices such that [A, B] = e11 − e22, there

−1 −1 −1 exists an invertible matrix P ∈ Mn such that P AP = e12,PBP = e21, and P (e11 − e22)P = e11 − e22.

−1 Proof. Let U ∈ Mn be an invertible matrix such that UAU = e12. Then

0 0 [A, B] = e11 − e22 ⇐⇒ [e12,B ] = C

0 −1 0 −1 0 where B = UBU and C = U(e11 − e22)U . This means C = (aij) is of the form   a11 a12 a13 ··· a1n  0 −a11 0 ··· 0    0  0 a32 0 ··· 0  C =    ......   . . . . .  0 an2 0 ··· 0

0 Since a11 is an eigenvalue of C , a conjugate of e11 −e22, it follows that a11 := a = ±1. For i 6= j and x ∈ C, denote εij(x) = In +xeij. Each εij(x) can be thought of as an elementary row operation

59 0 −1 and thus the (i, j)-th entry of εij(aij)C εij(aij) is zero for all i = 1, j ≥ 3 or j = 2, i ≥ 3. Hence

0 conjugating C by εij(aij) as many times as necessary produces a matrix of the form a t 0 ··· 0 0 −a 0 ··· 0   00 0 0 0 ··· 0 C =   , ......  . . . . . 0 0 0 ··· 0

00 with t ∈ C. Each εij(x) with i = 1, j ≥ 3 or j = 2, i ≥ 3 commutes with e12 so we have [e12,B ] =

00 00 t C for some rank-one trace-zero matrix B . The matrix W = e11 − e22 + 2 e12 + e33 + ··· + enn

00 −1 −1 satisﬁes WC W = a(e11 − e22) and W e12W = −e12. Moving minus signs around if necessary, it follows that B000 = aW B00W −1 satisﬁes

000 [e12,B ] = e11 − e22.

000 000 000 If B = (bij), expanding [e12,B ] = e11 − e22 along with rank(B ) = 1 implies that  2  b11 b11 0 ··· 0  1 b11 0 ··· 0   000  0 0 0 ··· 0 B =   ,  ......   . . . . . 0 0 0 ··· 0

000 000 But since B is trace-zero, we have B = e21 which completes the proof.

Using the common similarity transformation X 7→ PXP −1 for the invertible P in Lemma 3.3.3, one can pass from φ to a bijective linear map ψ preserving Lie products equal to e11 − e22 such that ψ(e12) = e12, ψ(e21) = e21, and ψ(e11 − e22) = e11 − e22. Observe that C(e12) and C(e21) are preserved by ψ and C(e12) ∩ C(e21) = V + CIn (as in Lemma 3.2.10) is also satisﬁed. Again utilizing a standard description on the trace-zero elements of V, one extends the description to the rest of sln. The proof of Theorem 3.3.1 is very similar to the proof of Theorem 3.2.6. The image may be determined on each matrix unit since eij, i 6= j belongs to the union of C(e12) and C(e21) and of course the action on e11 − e22 is speciﬁed. Corollary 3.3.2 follows from an extension up to

Mn via the rule Φ(A + cIn) = φ(A) + cIn for all A ∈ sln and c ∈ C.

60 Remark 3.3.4. As a last remark, we note that the assumption φ(e11 − e22) = e11 − e22 is rather necessary; without it, one cannot determine the image of e11 − e22 (unlike the N = e12 case).

For example, in the 2 × 2 matrices, let A = (aij) and B = (bij) be trace-zero matrices satisfying

[A, B] = e11 − e22. Then

[A, B] = a12[e12,B] + a21[e21,B] + a11[e11 − e22,B] b21 −2b11 −b12 0 0 2b12 = a12 + a21 + a11 0 −b21 2b11 b12 −2b21 0

= e11 − e22 yields a system of equations  a12b21 − a21b12 = 1  a11b12 − a12b11 = 0 .   a11b21 − a21b11 = 0

Multiply the ﬁrst equation by a11 to get

a11 = a11a12b21 − a11a21b12 = a12(a21b11) − a21(a12b11) = 0.

The argument is analogous for b11. Hence the diagonal entries of A and B are zero. Consequently,

[A, B] = e11 − e22 =⇒ [φ(A), φ(B)] = e11 − e22 alone implies nothing about φ(e11 − e22); in which case, the preserved subspace of φ (on sl2) is merely the 2-dimensional subspace span{e12, e21}. On

M2, one can include the identity matrix in the span for a 3-dimensional preserved subspace.

3.4 Related problems and conjectures

Along with the zero Lie product, the discussion above for N = e12 and N = e11 − e22 motivates the following general problem for maps preserving Lie products equal to N.

I. Characterize bijective linear maps φ : Mn → Mn such that φ(N) = N and [φ(A), φ(B)] = N

whenever [A, B] = N for any trace-zero matrix N.

II. Find conditions on an arbitrary trace-zero matrix N for such maps to be standard commu-

tativity preserving maps.

61 It is not too hard to deduce that the reason we encountered LPPs with proper preserved subspaces is because nilpotent Lie products can be related to simultaneous triangularizability, which in turn limits the matrix pairs A and B satisfying [A, B] = N. An additional related question concerns the characterization of the subspace

AN = span{A ∈ Mn | [A, B] = N}.

Are there nonzero matrices N for which AN = Mn? This seems unlikely but a systematic approach to solving the equation AB − BA = N for arbitrary N is challenging in and of itself.

The comments following Corollary 3.3.2 indicate that preserver problems have generally not yet transferred to the theory of Lie algebras. However it is well-known that every ﬁnite-dimensional

Lie algebra over a ﬁeld of characteristic zero can be faithfully embedded into square matrices with the commutator bracket [·, ·] (Ado’s Theorem). For arbitrary characteristic, the same result is due to Iwasawa [pg. 202 - 204, [44]]. Hence Lie algebra preserver problems could be converted into matrix-theoretic preserver problems.

There has been considerable interest in (Lie) algebras of upper-triangular matrices. Let Tn and

Ln denote the full algebra of upper-triangular matrices and lower-triangular matrices, respectively.

See [5] for rank-one preservers and [65] for rank-one idempotent and commutativity preservers

(also [59]) on Tn. For linear preservers on Tn it is invaluable to have a characterization of Jordan automorphisms of Tn.

Pn 2 Let D = k=1 ek,n+1−k be the so-called “exchange matrix”. Note that D = In and conjugation by D maps Ln to Tn.

Theorem 3.4.1 (Molnar, Semrlˇ [65]). If φ : Tn → Tn is a Jordan automorphism, there exists an invertible matrix U ∈ Tn or V ∈ Ln such that

(1) φ(X) = UXU −1, or

62 (2) φ(X) = DVXtV −1D

for all X ∈ Tn.

A characterization of Jordan automorphisms of strictly upper-triangular matrices is given by

Wang and You in [82]; the description is considerably more complicated than Mn and Tn. In these cases, the solutions to LPPs could behave very diﬀerently than solutions on Mn. Despite this, we note that commutativity preservers on Tn are identical in form to standard commutativity preservers on Mn.

63 CHAPTER 4

Maps preserving nonzero ordinary products

The matrix equation AB = 0 is another relation on matrix pairs which has been treated extensively in the preserver literature. Recalling the proof of the standard commutativity preserver problem, any rank-one matrix is characterized (up to perturbation by a scalar) by the dimension of its centralizer; hence a bijective linear map preserving commutativity reduces to the rank-one preserver problem. The condition AB = 0 also has a special connection to rank-one matrices.

For any B ∈ Mn, let ann`(B) = {A ∈ Mn | AB = 0} denote the left annihilator subspace of B.

2 It is straightforward to verify that dim ann`(B) = n − rank(B)n. Any linear mapping satisfying

φ(A)φ(B) = 0 whenever AB = 0 “commutes” with taking left annihilators; i.e.,

φ(ann`(B)) ⊆ ann`(φ(B)).

With bijectivity,

2 2 n − rank(B)n = dim φ(ann`(B)) ≤ dim ann`(φ(B)) = n − rank(φ(B))n reduces to

rank(φ(B)) ≤ rank(B) (4.1)

for all B ∈ Mn. The condition (4.1) is referred to as “rank non-increasing” and has been studied in its own right (see [50]); however, notice that rank(B) = 1 implies that rank(φ(B)) ≤ 1. Since

φ is bijective it follows that rank(φ(B)) = 1 and so there exists invertible U, V ∈ Mn such that

t φ(X) = UXV or φ(X) = UX V for all X ∈ Mn by Theorem 1.1.1. It is easy to see that in the

64 case X 7→ UXV , the zero product property AB = 0 =⇒ UAV UBV = 0 implies further that

t VU = cIn for some nonzero scalar c ∈ C. The case X 7→ UX V always fails to preserve the zero product since transposition reverses the order of multiplication. In essence these observations prove the following.

Theorem 4.0.1 (Wong [84]). If φ : Mn → Mn is a bijective linear map preserving zero products,

−1 there exists and invertible matrix U ∈ Mn and scalar c ∈ C such that φ(X) = cUXU for all

X ∈ Mn.

Recall Theorem 3.0.3, which states that if φ is merely assumed to be a linear map preserving commutativity then either the image of φ is commutative, sln, or Mn. We now mention its analogue to the zero product (from [Corollary 2.4, [22]]).

Theorem 4.0.2 (Chebotar, Ke, Lee, Wong [22]). Let φ : Mn → Mn be a linear map preserving zero products. Either the image of φ has trivial multiplication, or there exists an invertible matrix

−1 U ∈ Mn and scalar c ∈ C such that φ(X) = cUXU for all X ∈ Mn.

From the discussion following Theorem 3.0.3, maximal commutative subalgebras of Mn carry trivial multiplication. The authors in [22] show that if the linear map φ preserves zero products then φ also preserves commutativity. Using the description provided by Theorem 3.0.3, one must also drop the transpose case and perturbation by a scalar. The description follows.

4.1 Maps preserving products equal to ﬁxed invertible matrices

What are some nonzero variants of Theorem 4.0.1? We say that φ : Mn → Mn preserves equal products if φ(A)φ(B) = φ(C)φ(D) whenever AB = CD. Taking C = 0, this problem immediately reduces to the zero product preserver problem. Over arbitrary algebras, the problem is more interesting (see [21] for results in this area). We also say that φ : Mn → Mn preserves equal ﬁxed products if φ(A)φ(B) = M whenever AB = K, where K,M ∈ Mn are ﬁxed matrices. If K = M,

65 we say φ preserves products equal to K. The solution set to LPPs of this type may be much more restrictive than the zero product preserver problem. Indeed, if K is not similar to M, then inner automorphisms do not preserve equal ﬁxed products.

However, when comparing the zero product preserver problem to a nonzero product preserver problem, the transpose map may preserve products of matrices. From elementary linear algebra, we know that AB = In implies BA = In, so

AB = In =⇒ BA = In

t =⇒ (BA) = In

t t =⇒ A B = In. The case of bijective linear maps preserving products equal to the identity matrix has been completely characterized. By [Theorem 1.3, [58]], such maps are scalar multiples of automorphisms or antiautomorphisms (over C the scalar must be ±1). In fact the result holds on matrices over a division ring.

For possibly nonscalar matrices, consider the following.

Example 4.1.1. Let D be an invertible matrix. Suppose there is a nonzero b ∈ C and invertible

U ∈ Mn such that the equation

b2UDt = D−1U (4.2)

t −1 holds. Let φ : Mn → Mn be the bijective linear map deﬁned by φ(X) = bDUX U . We claim that φ preserves products equal to D. Indeed, if AB = D, then A and B must be invertible and

B = A−1D. Thus φ(A)φ(A−1D) = (bDUAtU −1)(bDU(A−1D)tU −1)

= b2DUAtU −1DUDtA−tU −1

= DUAtU −1D(b2UDt)A−tU −1

= DUAtU −1DD−1UA−tU −1 = D.

66 Note that this mapping is a rigid mapping but, if D is nonscalar, not generally a scalar multiple of a Jordan automorphism. It diﬀers signiﬁcantly from the classical zero product preserver problem.

In this sense, this map is a nonstandard solution.

Remark 4.1.2. It is certainly possible for equation (4.2) to not hold. Let σ(D) denote the spectrum of D. Rearrange the equation as b2UDtU −1 = D−1. Since the spectrum is invariant under both transposition and conjugation, we have

σ(b2D) = σ(D−1).

It is easy to see also that σ(b2D) = b2σ(D); hence the equality of sets

2 2 2 −1 −1 −1 {b λ1, b λ2, . . . , b λr} = {λ1 , λ2 , . . . , λr }

r is a necessary condition for equation (4.2), where {λi}i=1 are the distinct eigenvalues of D. This

“spectral condition” can be used to rule out the possibility of nonstandard mappings preserving products to D.

For an example, let λ1, λ2, and λ3 be distinct complex numbers and x a nonzero complex number

(to be determined). We want to solve the set equation

−1 −1 −1 {xλ1, xλ2, xλ3} = {λ1 , λ2 , λ3 }.

−1 −1 −1 −1 Suppose xλ1 = λ1 . Then either xλ2 = λ2 or xλ2 = λ3 . In the former, this implies xλ3 = λ3 .

−2 −2 −2 2 2 2 Hence x = λ1 = λ2 = λ3 which simpliﬁes to λ1 = λ2 = λ3. This contradicts distinctness.

−1 2 The latter case must hold: xλ2 = λ3 . Consequently λ1 = λ2λ3. Proceeding this way over all the remaining choices, the spectral condition is satisﬁed if and only if at least one of the equations

2 2 2 λ1 = λ2λ3, λ2 = λ1λ3, or λ3 = λ1λ2 hold. Solving the spectral condition is even easier in case σ(D) has at most 2 distinct eigenvalues.

67 −2 1 For if σ(D) = {λ}, then clearly x = λ is the unique solution. If σ(D) = {λ, µ}, then x = λµ is also the unique solution.

Hence if the eigenvalues of D do not satisfy any of the equations imposed by the spectral condition, such as in the case σ(D) = {1, 2, 3}, equation (4.2) fails and the maps appearing in

Example 4.1.1 do not preserve products equal to D.

Regardless, if D satisﬁes equation (4.2) for an appropriate b and U, then the collection of bijective linear maps preserving products equal to D certainly contains scalar multiples of inner automorphisms and maps of the (nonstandard) form appearing in Example 4.1.1.

A recent paper [16] veriﬁes that these two families of mappings are in fact the only two examples.

As an added beneﬁt the result holds over a division ring D.

Theorem 4.1.3 (Catalano, Hsu, Kapalko [16]). Let D be a division ring with characteristic not 2 and center F . Let Mn(D) be the ring of n × n matrices over D with n ≥ 2. Let D,E ∈ Mn(D) be

ﬁxed invertible matrices and let φ : Mn(D) → Mn(D) be a bijective additive map satisfying

φ(A)φ(B) = E whenever AB = D.

Then φ(X) = φ(In)f(X) for all X ∈ Mn(D), where f : Mn(D) → Mn(D) is either an automorphism or antiautomorphism. Moreover,

(1) if f is an automorphism, then φ(In) = cIn with c ∈ F , and

−1 (2) if f is an antiautomorphism, then φ(In) = φ(D) E and φ(D) = cIn with c ∈ F .

The ultimate goal is to obtain a characterization of bijective linear maps preserving products equal to an arbitrary ﬁxed matrix, D. Only the noninvertible case needs resolved. A method for treating the case D diagonalizable will be described in Section 4.2 with observations on the general case provided in Section 4.3.

68 4.2 Maps preserving products equal to a diagonalizable matrix

The main goal of this section is to address maps preserving products equal to a diagonalizable matrix. If D is also invertible, then the result may be deduced from Theorem 4.1.3. Thus we devote this section to proving the following theorem.

Theorem 4.2.1 (Catalano, Julius [17]). Let D ∈ Mn be a ﬁxed diagonal matrix and φ : Mn → Mn a bijective linear map such that φ(A)φ(B) = D whenever AB = D. If rank(D) < n, there exists

−1 an invertible matrix U ∈ Mn and scalar c ∈ C such that φ(X) = cUXU for all X ∈ Mn.

Note that if D = 0, we recover the zero product preserver problem, Theorem 4.0.1. Furthermore if D is diagonalizable and rank-one, then D is a scalar multiple of a rank-one idempotent. Multi- plying φ by the square root of this scalar multiple yields a bijective linear map preserving products equal to a rank-one idempotent. This case has been explicitly addressed by Catalano in [14] (the solutions are scalar multiples of automorphisms). Since rank-one idempotent matrices are diagonalizable, Theorem 4.2.1 generalizes the zero product preserver problem and the rank-one idempotent product preserver problem. Going forward we make the assumption that, if d = rank(D), then

2 ≤ d ≤ n − 1.

We use a combinatorial approach to generate factorizations of D. Let Sn denote the symmetric group on the letters {1, 2, . . . , n}. There is a natural action of Sn on Mn(C) given by permuting columns. If σ ∈ Sn, the action of σ on the n × n identity matrix In produces a permutation matrix. The correspondence between a permutation and its permutation matrix is clearly one-to- one. Multiplying a permutation matrix by its inverse gives a factorization of the identity. In this simple way, we have obtained several factorizations of In of minimal complexity, in the sense that each row and column of each factor has exactly one nonzero entry.

It is easy to see that this permutation approach can produce several factorizations of an arbitrary diagonal matrix D having any set of eigenvalues (see equation (4.3) below). Since D may have

69 arbitrary rank in what follows, we introduce some concise terminology.

Deﬁnition 4.2.2. Let d and n be positive integers such that d < n. A permutation σ ∈ Sn is called a (d, n)-permutation if σ sends the ﬁrst d letters to the last d letters; that is, σ(i) = j whenever

1 ≤ i ≤ d and n − d + 1 ≤ j ≤ n.

A simple counting argument shows that the number of (d, n)-permutations is d!(n − d)!. Fac- torizing a diagonal matrix D of rank d in Mn corresponding to a (d, n)-permutation is convenient for the upcoming computations.

Remark 4.2.3. Suppose D has d nonzero eigenvalues x11, x22, . . . , xdd, counting multiplicities. Let

−1 S be an invertible n × n matrix such that SDS = diag(x11, x22, . . . , xdd, 0,..., 0) ∈ Mn, where the number of zero diagonal entries is equal to n − d. If φ is a linear map such that φ(A)φ(B) = D whenever AB = D, it follows that

−1 −1 (S φ(A)S)(S φ(B)S) = diag(x11, x22, . . . , xdd, 0,..., 0)

−1 −1 −1 whenever (S AS)(S BS) = diag(x11, x22, . . . , xdd, 0,..., 0). Since the map A 7→ S AS is linear and bijective, it is suﬃcient to prove Theorem 3.2.6 in the special case

D = diag(x11, x22, . . . , xdd, 0,..., 0).

Without loss of generality, assume going forward that D = diag(x11, x22, . . . , xdd, 0 ..., 0).

First, it will be shown that φ preserves pairwise zero products of matrix units.

Lemma 4.2.4. Let D ∈ Mn be a ﬁxed diagonal matrix of rank d < n. If a linear map φ : Mn → Mn satisﬁes the property φ(A)φ(B) = D whenever AB = D, then φ(eij)φ(ekl) = 0 whenever j 6= k.

0 Proof. Let d = n − d and for convenience, denote by Mpq the vector space of all rectangular p × q matrices over C. Partition Mn(C) into subspaces in two ways:  (1) (1)  (2) (2)  M 0 M M 0 M 0 0 dd dd and d d d d ,  (1) (1)  (2) (2)  Md0d0 Md0d Mdd Mdd0

70 We show that φ(eij)φ(ekl) = 0 by cases depending on where eij is in the ﬁrst partition and where ekl is in the second partition.

For any permutation σ ∈ Sn, deﬁne

Aσ = e1σ(1) + e2σ(2) + ··· + edσ(d),Bσ = x11eσ(1)1 + x22eσ(2)2 + ··· + xddeσ(d)d. (4.3)

(1) We have AσBσ = D for all σ ∈ Sn. However, if σ is a (d, n)-permutation, Aσ ∈ Mdd and

(2) Bσ ∈ Mdd . Written in block form,

0 A 0 0 σ = D, 0 0 Bσ 0 where the matrix on the left is written according to partition (1) and the matrix on the right is according to partition (2).

(1) (2) Fix some eij ∈ Mdd0 . If ekl ∈ Md0d and j 6= k, then (λeij + Aσ)(µekl + Bσ) = D for all λ, µ ∈ C.

Written in block form according to the two partitions,

λe A µe 0 ij σ kl = D. 0 0 Bσ 0

It follows that φ(λeij + Aσ)φ(µekl + Bσ) = D. Using linearity and the fact that φ(Aσ)φ(Bσ) = D, we obtain the equation

λφ(eij)φ(Bσ) + µφ(Aσ)φ(ekl) + λµφ(eij)φ(ekl) = 0. (4.4)

Since equation (4.4) holds for arbitrary λ, µ ∈ C, in particular it holds for λ = ±1 and µ = ±1.

The systems of equations obtained by varying parameters over ±1 clearly imply that

φ(eij)φ(Bσ) = φ(Aσ)φ(ekl) = φ(eij)φ(ekl) = 0. (4.5)

Hence φ(eij)φ(ekl) = 0 in this case.

0 0 Equation (4.5) says more. Let Aσ and Bσ denote the matrices Aσ and Bσ but with the ﬁrst

0 0 terms negated. Then clearly AσBσ = D. Performing the same calculations as above, we derive an

71 analog to equation (4.5), and in particular:

0 φ(eij)φ(Bσ) = 0. (4.6)

Considering equations (4.5) and (4.6), it follows that φ(eij)φ(eσ(1)1) = 0 for every (d, n)- permutation σ. This process can be repeated by negating any term appearing in Aσ and its

(2) counterpart in Bσ to conclude that, in fact, φ(eij)φ(ekl) = 0 whenever ekl ∈ Mdd .

(2) Now suppose ekl ∈ Md0d0 and j 6= k. Clearly for all λ, µ ∈ C, we have

λe A 0 µe ij σ kl = D. 0 0 Bσ 0

Employing a similar argument as above, it follows that φ(Aσ)φ(ekl) = φ(eij)φ(ekl) = 0.

(2) Lastly, if ekl ∈ Mdd0 , pick a (d, n)-permutation σ such that σ(i) = k. Observe that for λ = ±1,

e λA 0 −λe ij σ jl = D. 0 0 λBσ ekl

Hence

φ(eij + λAσ)φ(ekl + λBσ − λejl) = D.

By the previous cases, including the remark φ(Aσ)φ(ejl) = 0, it follows that

φ(eij)φ(ekl) + λφ(Aσ)φ(ekl) − λφ(eij)φ(ejl) = 0.

Varying parameters over λ = ±1 implies that φ(eij)φ(ekl) = 0 in this case.

(1) Hence for eij ∈ Mdd0 , it has been established that φ(eij)φ(ekl) = 0 for every matrix unit ekl such that j 6= k. The remainder of the proof uses identical tools to cover the remaining cases (some of which can already be deduced from the above discussion) and is therefore omitted for brevity.

The proof of Lemma 4.2.4 did not require bijectivity. The two partitions of Mn found above facilitated computations. A diﬀerent partition will be used for the remainder of this discussion.

72 Partition Mn into subspaces as follows:   Mdd Mdd0   , (4.7) Md0d Md0d0

0 where d = rank(D) and d = n − d. Notice that the blocks Mdd and Md0d0 are isomorphic to Md and Mn−d, respectively.

Lemma 4.2.5. Let D ∈ Mn be a ﬁxed diagonal matrix. If a bijective linear map φ : Mn → Mn satisﬁes the property φ(A)φ(B) = D whenever AB = D, then φ(Md0d0 ) = Md0d0 and φ(Mdd) = Mdd.

√ 0 Proof. Suppose D = diag(x11, x22, . . . , xdd, 0,..., 0), with d zero diagonal entries. Let D denote √ √ √ the matrix diag( x11, x22,..., xdd, 0,..., 0). Regarding the square roots of complex numbers, the continuity of the complex-valued square root function is not important for this proof, so no √ matter how square-roots are chosen, ( D)2 = D. However, we will insist that the selection of square roots of the ﬁxed complex eigenvalues xii should be well-deﬁned in the sense that if i 6= j √ √ and xii = xjj, then xii = xjj. √ √ Let A ∈ Mdd0 ∪Md0d0 and B ∈ Md0d ∪Md0d0 such that AB = 0. Observe that A D = DB = 0. √ √ √ √ Now ( D + λA)( D + µB) = D for all λ, µ ∈ C. Hence (φ( D) + λφ(A))(φ( D) + µφ(B)) = D √ by hypothesis. Since φ( D)2 = D, it follows that

√ √ λφ(A)φ( D) + µφ( D)φ(B) + λµφ(A)φ(B) = 0.

Varying the parameters λ and µ over ±1 reveal that, in fact,

√ √ φ(A)φ( D) = φ( D)φ(B) = φ(A)φ(B) = 0. (4.8)

Hence φ preserves zero products for pairs of matrices A ∈ Mdd0 ∪ Md0d0 and B ∈ Md0d ∪ Md0d0 . √ Moreover, multiplying through equation (4.8) by φ( D) on the right implies that

√ φ(A)φ( D)2 = φ(A)D = 0.

73 Since D is a diagonal matrix, it follows that φ(A) ∈ Mdd0 ∪ Md0d0 for all A ∈ Mdd0 ∪ Md0d0 , hence

φ(Mdd0 ∪Md0d0 ) ⊆ Mdd0 ∪Md0d0 . An analogous argument shows that φ(Md0d ∪Md0d0 ) ⊆ Md0d ∪Md0d0 .

Hence φ sends the intersection of these two block partitions, namely Md0d0 , into itself. Equality follows from bijectivity. This proves one of the claims.

Secondly, since φ(Md0d0 ) = Md0d0 , it follows from Lemma 4.2.4 that φ(eij)Md0d0 = Md0d0 φ(eij) =

0 for all 1 ≤ i, j ≤ d. In particular, φ(eij)ekl = 0 for all ekl ∈ Md0d0 . Thus φ(Mdd) ⊆ Mdd with equality from bijectivity.

Proof of Theorem 4.2.1. By Lemma 4.2.5, we know that φ(Mdd) = Mdd and φ(Md0d0 ) = Md0d0 . D is invertible as a matrix in Mdd. It follows from Theorem 4.1.3 that the restriction of φ to Mdd

−1 t −1 takes the form φ(X) = bP XP or φ(X) = bDP X P for all X ∈ Mdd, where P is an invertible

t −1 d × d matrix and b ∈ C is nonzero. However, we will quickly show that φ(X) = bDP X P is not actually a possibility.

t −1 We can choose i, j, k < d with j 6= k. If φ(X) = bDP X P for all X ∈ Mdd, then

−1 −1 φ(eij)φ(eki) = (bDP ejiP )(bDP eikP )

2 −1 −1 = b DP ejiP DP eikP .

−1 −2 −1 Note that P DP (= b D ) is a diagonal matrix (and invertible in Mdd); hence

2 −1 φ(eij)φ(eki) = b rDP ejkP ,

−1 where r ∈ C \{0} is the (i, i)-entry of P DP . By Lemma 4.2.4, we know that φ(eij)φ(eki) = 0,

−1 and in particular, this means P ejkP = 0 for all j 6= k, which is impossible. Therefore, we must

−1 have that φ(X) = bP XP for all X ∈ Mdd.

Now, the restriction of φ to Md0d0 preserves the zero product. Indeed, if A, B ∈ Md0d0 and

AB = 0, then

(Id + λA)(D + µB) = D

74 for all λ, µ ∈ C. Varying parameters over λ = ±1 and µ = ±1 implies that φ(A)φ(B) = 0.

Hence the restriction map φ| preserves the zero product and by Theorem 4.0.1 has the form Md0d0

−1 0 0 φ(X) = cW XW for all X ∈ Md0d0 , where W is some invertible d × d matrix and c is a nonzero scalar.

Deﬁne Q = P ⊕ W . Clearly Q is invertible as an n × n matrix and b2QD = DQ. The linear map ψ : Mn → Mn deﬁned by

ψ(X) = b−1Q−1φ(X)Q is a bijective linear map that preserves products equal to D. By design, it also follows that

−1 ψ(eij) = eij for all eij ∈ Mdd and ψ(eij) = b ceij for all eij ∈ Md0d0 .

Pick some eij ∈ Mdd0 ∪ Md0d and ekl ∈ Mdd. If j 6= k, then ψ(eij)ψ(ekl) = ψ(eij)ekl = 0 implies that the kth column of ψ(eij) is zero. Furthermore, if i 6= l, then ψ(ekl)ψ(eij) = eklψ(eij) = 0 implies that the lth row of ψ(eij) is zero. This can be repeated analogously for ekl ∈ Md0d0 . Since k and l can be arbitrary, we have ψ(eij) = βijeij for all eij ∈ Mdd0 ∪ Md0d, where βij is some nonzero scalar. Hence the map ψ sends matrix units to scalar multiples of themselves.

2 The remainder of the proof is a short series of claims about the n scalars βij. By examining various factorizations of D, it will follow that βij takes on constant values on each of the blocks in the partition (4.7).

The ﬁrst claim is that b = c. Recall Deﬁnition 4.2.2. Let σ be a (d, n)-permutation such that

σ(1) = n and deﬁne

Aσ = e1σ(1) + e2σ(2) + ··· + edσ(d),Bσ = x11eσ(1)1 + x22eσ(2)2 + ··· + xddeσ(d)d.

It follows that

(e11 + Aσ)(Bσ − e1n + enn) = D

75 and so

ψ(e11 + Aσ)ψ(Bσ − e1n + enn) = D.

Expanding out according to Lemma 4.2.4, and noting that e1n is a matrix unit in Aσ, we have

−ψ(e11)g(e1n) + ψ(e1n)ψ(enn) = 0.

By deﬁnition,

−1 −β1n + b cβ1n = 0

and so b = c. Hence ψ(eij) = eij whenever eij ∈ Mdd ∪ Md0d0 .

−1 Next we claim that βij = βji for 1 ≤ i, j ≤ n. To each permutation σ ∈ Sn, deﬁne the matrix

√ √ √ Xσ = x11e1σ(1) + x22e2σ(2) + ··· + xddedσ(d).

t Clearly XσXσ = D for all σ ∈ Sn. Fix some pair (i, j). Pick σ such that σ(i) = j so that Xσ

√ t √ t contains the term xiieij and Xσ contains the term xiieji. Hence the product ψ(X)ψ(X ) = D contains the term ψ(eij)ψ(eji) = βijβjixiieii. The (i, i)-th entry of D is clearly xii so βijβji = 1.

Since i and j can be arbitrary, the claim holds.

We claim further that βij = βil for all 1 ≤ i ≤ d and d + 1 ≤ j, l ≤ n (this puts both eij and eil in the subspace Mdd0 ). Let Id = e11 + ··· + edd. Notice that (Id + eij)(D + eil − ejl) = D, or in block matrix form, I e D e d ij il = D. 0 0 0 −ejl

Using Lemma 4.2.4, we are left with ψ(eii)ψ(eil) − ψ(eij)ψ(ejl) = 0. By deﬁnition, we have

βil − βij = 0 and so βil = βij as claimed.

So the βij coeﬃcients are equal across the rows of Mdd0 and by the above inverse relationship, also equal across the columns of Md0d. We claim further that βij = βkj for all 1 ≤ i, k ≤ d and d + 1 ≤ j ≤ n (this puts eij and ekj in Mdd0 ). Here we use a slightly diﬀerent factorization

76 (ik) (ik) of D. Let Id and D respectively denote the matrices Id and D with the ith and kth rows interchanged (i.e., the usual action of a transposition (ik) ∈ Sn on the rows of each matrix). Notice

(ik) (ik) (ik) (ik) that Id D = D and (Id + eij)(D + ekj − ejj) = D, or in block matrix form, " # I(ik) e D(ik) e d ij kj = D. 0 0 0 −ejj

Again using Lemma 4.2.4, we are left with ψ(eik)ψ(ekj) − ψ(eij)ψ(ejj) = 0. Thus βkj − βij = 0 implies that βkj = βij. Analogous computations for each transposition (ik) with 1 ≤ i, k ≤ d imply that the βij are equal across the columns of Mdd0 and hence across the rows of Md0d.

Thus there is a single scalar β ∈ C such that ψ(eij) = βeij for all eij ∈ Mdd0 and correspondingly,

−1 ψ(eji) = β eji for all eji ∈ Md0d. At this point, it is evident that  eij eij ∈ Mdd ∪ Md0d0  ψ(eij) = βeij eij ∈ Mdd0 .   −1 β eij eij ∈ Md0d

−1 Hence if C = βId ⊕ Id0 , where Id0 = ed+1,d+1 + ··· + enn, it follows that C ψ(X)C is the identity map. Thus

C−1ψ(X)C = b−1C−1Q−1φ(X)QC = X implies that φ(X) = b(QC)X(QC)−1. Take U = QC and the theorem is proved.

4.3 Related problems and conjectures

The ultimate goal is to describe linear maps φ : Mn → Mn preserving equal ﬁxed products (i.e.,

AB = D implies φ(A)φ(B) = E) for arbitrary matrices. The results of Section 4.2 treated the case when φ preserves equal ﬁxed products of diagonalizable matrices with the same eigenvalues.

In the spirit of LPPs being rigid mappings, it is very reasonable to expect that AB = D =⇒

φ(A)φ(B) = E also yields a nice relationship between D and E. One straightforward example is the following. If φ : Mn → Mn is a linear map such that AB = 0 implies φ(A)φ(B) = E, then

77 −1 φ(X) = cUXU for all X ∈ Mn. The proof of this is immediate; take A = B = 0, conclude that

E = 0, and use Theorem 4.0.2. Hence linear maps sending zero products to nonzero products do not exist.

However, we do not wish to assume that D and E have a prescribed relationship, such as being similar. Obtaining a description of maps preserving equal ﬁxed products therefore needs an abstract technique. We now describe a technique that ﬁrst appeared in [15] that reduces nonzero product preservers to zero product preservers.

Deﬁnition 4.3.1. Let X be the set of all matrices with all rows zero except (possibly) one, and let Y be the set of all matrices with all columns zero except (possibly) one. Let X2 be a subset of

X such that each element in X2 has at most 2 nonzero entries, and analogously, let Y2 be a subset of Y such that each element of Y2 has at most 2 nonzero entries. By a zero-2 pair, we mean a pair

(X2,Y2) ∈ X2 × Y2 such that X2Y2 = 0. Additionally, we say that f preserves zero products on zero-2 pairs if f(X2)f(Y2) = 0 for all zero-2 pairs (X2,Y2).

Typical elements of X2 and Y2 are matrices that can be written as aeij + beij0 and cekl + dek0l, respectively, with a, b, c, d ∈ C. The condition

(aeij + beij0 )(cekl + dek0l) = 0 holds either by virtue of trivial multiplicaiton of the matrix units or by a zero relation among the coeﬃcients. Examples of zero-2 pairs include (e12, e34) and (e12+e13, e24−e34). The key observation is:

Theorem 4.3.2 (Catalano, Chang-Lee [15]). If φ : Mn → Mn is a linear map that preserves zero products on zero-2 pairs, then φ preserves the zero product.

Recall Lemma 4.2.4 of the previous section; we showed explicitly that φ(eij)φ(ekl) = 0 whenever j 6= k. In the language of zero-2 pairs, we showed that the map φ of that section preserved the

78 trivial zero products of zero-2 pairs. However, this is not enough to conclude that all zero-2 pairs are preserved. One should also show that φ(eij)φ(ejl) = φ(eik)φ(ekl) holds in order to preserve zero products on zero-2 pairs arising from a zero relation on its coeﬃcients. To use this technique it is therefore enough to show that matrix units behave like matrix units in the image of φ. We have essentially proved the following.

Lemma 4.3.3. Let φ : Mn → Mn be a linear map such that

(1) φ(eij)φ(ekl) = 0 whenever j 6= k, and

(2) φ(eij)φ(ejl) = φ(eik)φ(ekl) for all i, j, k, l. Then φ preserves zero products on zero-2 pairs, and hence preserves the zero product.

Here is a heuristic description of how to use this technique eﬀectively for maps preserving products equal to ﬁxed elements. Suppose AB = D implies φ(A)φ(B) = E. If matrix units eij and ekl (with j 6= k) can be inserted into the equation AB = D as

(A + λeij)(B + µekl) = D (4.9) for all λ, µ ∈ C then it clearly follows that φ(eij)φ(ekl) = 0. Likewise, if the equations

(A + λ(eij + eik))(B + µ(ejl − ekl)) = D (4.10) and

A(ejl − ekl) = (eij + eik)B = 0 (4.11) hold for all λ, µ ∈ C, then by linearity, φ(eij)φ(ejl) = φ(eik)φ(ekl). Hence φ preserves the zero product by Lemma 4.3.3 and we obtain that φ(X) = cUXU −1. This sets up a relationship c2UDU −1 = E. Note that φ(A)φ(B) = E being constant on pairs A and B satisfying AB = D is used here; not any properties of E itself! The diﬃculty, of course, is proving that equations (4.9)

− (4.11) actually hold for some pair A and B.

79 Recently, Catalano and the author utilized the zero-2 pairs approach to eﬀectively resolve the arbitrary case of maps preserving equal ﬁxed products. However, the stipulation is that the matrix

D needed to be of rank ≤ n − 2. The rank condition provided enough freedom to solve equations

(4.9) − (4.11).

Theorem 4.3.4 (Catalano, Julius [18]). Let D ∈ Mn be a ﬁxed matrix with rank(D) ≤ n − 2, and let E ∈ Mn be any ﬁxed matrix. If φ : Mn → Mn is a bijective linear map such that φ(A)φ(B) = E whenever AB = D, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that φ(X) =

−1 cUXU for all X ∈ Mn.

Hence rank(D) ≤ n − 2 is covered by Theorem 4.3.4 while rank(D) = n is covered by Theorem

4.1.3. The only remaining question is: what to do when rank(D) = n − 1?

Example 4.3.5. Let N be the rank-(n − 1) nilpotent matrix 0 1 0 ··· 0 0 0 1 ··· 0    ..   .  .   0 0 0 ··· 1 0 0 0 ··· 0 If AB = N, then rank(A) = n − 1 or rank(A) = n. The same goes for rank(B). However, at least one of A and B must be rank-(n − 1). The rank of a matrix determines the dimension of its left annihilator subspace as per the comments at the beginning of this chapter. Hence if rank(A) = n − 1, then both A and N have n-dimensional left annihilators. Moreover, if CA = 0, then CAB = CN = 0. So ann`(A) = ann`(N). The left annihilators of N are precisely the matrix units of the form ejn with j = 1, 2, . . . , n. Hence ejnA = 0 implies that the bottom row of A is also zero.

Analogously on the right, if rank(B) = n − 1 then N and B have common right annihilators, so the ﬁrst column of B is also zero. This puts extreme restrictions on the solutions to AB = N.

Namely, if the bottom row of A is not all zeros, then the ﬁrst column of B must be all zeros to

80 accommodate the factorization AB = N. So it is impossible to insert all the matrix units into the equation AB = N. This immediately disqualiﬁes the zero-2 pair method.

A different approach is needed for maps preserving equal fixed products in the case rank(D) = n−1. Catalano and the author had extreme difficulty in solving this problem keeping E completely arbitrary. However, if one makes the totally natural and reasonable hypothesis that rank(D) = rank(E) = n − 1 then the description of maps satisfying AB = D =⇒ φ(A)φ(B) = E is fully resolved.

Theorem 4.3.6 (Catalano, Julius [18]). Let D,E ∈ Mn be ﬁxed matrices of rank-(n − 1). If

φ : Mn → Mn is a bijective linear map such that φ(A)φ(B) = E whenever AB = D, there exists

−1 an invertible matrix U ∈ Mn and scalar c ∈ C such that φ(X) = cUXU for all X ∈ Mn.

The proof can be sketched as follows. Assume without loss that D and E are already in Jordan canonical form. Since rank(D) = rank(E) = n − 1 there is precisely one row and one column of all zeros. Using elementary tools of linear algebra one can show that if B is a rank-(n − 1) matrix with a column of zeros then there exists a matrix A such that AB = D. The condition φ(A)φ(B) = E implies that, if rank(φ(B)) = n − 1, then φ(B) also has a column of zeros. Indeed this is true; for if rank(B) = n − 1 there exists a matrix C such that CB = 0, hence (A + C)B = D implies that φ(A + C)φ(B) = E. With bijectivity and linearity, it follows that φ(B) has a nonzero left annihilator and so φ(B) cannot be invertible.

Hence the subspace generated by rank-(n − 1) matrices with a common column of zeros are all mapped to matrices with a common column of zeros (here the assumption that rank(E) = n − 1 is used). Now if AB = D for some invertible B, then φ(B) cannot have a column of zeros (otherwise

B is a linear combination of rank-(n − 1) matrices with a common column of zeros and is not invertible). The equation DB−1B = D is of course solvable for every invertible B so we conclude that φ(B) is invertible whenever B is invertible. By Theorem 1.1.3 we obtain the description

81 X 7→ UXV and X 7→ UXtV as possibilities for φ. With some eﬀort, one can show that the transpose case is impossible and VU = cIn. This completes the proof. All details can be found in

[18].

Next we discuss some related problems. First, the assumption of bijectivity can be dropped in

Theorem 4.3.4. This is because preserving zero products on zero-2 pairs does not require bijectivity.

Invoking Theorem 4.0.2, either φ carries trivial multiplication (E = 0) or is bijective (E 6= 0). It is less certain; however, if bijectivity can be dropped in Theorem 4.3.6. There is also the remaining treatment of the case rank(D) = n − 1 and E arbitrary but ﬁxed. There are plenty of reasons to suspect the map is as well-behaved as every other case.

Hence we make a conjecture.

Conjecture 4.3.7. Let D ∈ Mn be a ﬁxed matrix with rank(D) = n − 1 and let E ∈ Mn be any ﬁxed matrix. If φ : Mn → Mn is a bijective linear map such that φ(A)φ(B) = E whenever

−1 AB = D, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that φ(X) = cUXU for all X ∈ Mn.

Now, consider the same problem stated for an arbitrary matrix product ?. We say that φ preserves equal ﬁxed ?-products if φ(A) ? φ(B) = M whenever A?B = K. Theorems 4.2.1, 4.3.4, and 4.3.6 represent the incremental progress toward resolving the case of ordinary multiplication.

Barring the minor gap of Conjecture 4.3.7, maps preserving equal fixed (ordinary) products are fully resolved. The handful of cases of equal fixed Lie products is the subject of Chapter3, and as mentioned there, it is difficult to characterize the preserved subspace. However, there is good news for the Jordan product.

Theorem 4.3.8 (Catalano, Hsu, Kapalko [16]). Let M,K ∈ Mn be ﬁxed matrices. If φ : Mn → Mn is a bijective linear map such that φ(A) ◦ φ(B) = M whenever A ◦ B = K, there exists an invertible matrix U ∈ Mn and scalar c ∈ C such that

82 (1) φ(X) = cUXU −1, or

(2) φ(X) = cUXtU −1

for all X ∈ Mn.

The proof is quite technical but some brief observations can be made in special cases. Suppose

2 1 that K has a square root L; that is, L = K. Then 2 φ(L) ◦ φ(L) = M. Now let S be any square-zero matrix. Then 1 (L − S) ◦ (L + S) = K 2 and hence 1 1 (φ(L) − φ(S)) ◦ (φ(L) + φ(S)) = M − φ(S)2 = M. 2 2

Thus φ preserves square-zero matrices. Using Theorem 2.0.1, the description on sln is immediately obtained. This illustrates the utility of square-zero preserving mappings whenever the Jordan product is involved.

83 CHAPTER 5

Maps preserving roots of nonhomogeneous polynomials

The previous preserver problems largely concern maps preserving some multiplicative structure: in

Chapter2, Chapter3, and Chapter4, we examined linear maps preserving exponent structures

(idempotent and square-zero matrices), Lie product, and ordinary product structures, respectively.

In this chapter, we turn our attention to multiplicative structure induced by preservers of matrix polynomials. Reframing a linear preserver problem in terms of polynomials turns out to be quite a powerful approach.

Let p(x1, x2, . . . , xk) be a complex polynomial in the noncommuting variables {x1, x2, . . . , xk} and let A = Mn. Denote

V (p) = {X ∈ Ak : p(X) = 0}.

One can ask for a description of bijective linear maps preserving V (p); that is, the mapping φ :

Mn → Mn satisﬁes

p(φ(A1), φ(A2), . . . , φ(Ak)) = 0 whenever p(A1,A2,...,Ak) = 0.

Many classical preserver problems discussed thus far can be stated as maps preserving the zeros of a polynomial p in either one variable two variables. For instance, the idempotent preserver problem asks for maps preserving V (x2 − x) and the commutativity preserver problem asks for maps preserving V (xy − yx).

Restrict attention to the single variable setting. An important result is due to Howard.

84 Theorem 5.0.1 (Howard [42]). Let p(x) be a complex polynomial with at least two distinct roots.

Suppose n ≥ 3 and φ : Mn → Mn is a bijective linear map preserving V (p). There is an invertible matrix U ∈ Mn such that

(1) φ(X) = UXU −1, or

(2) φ(X) = UXtU −1, or, if p(x) = x`q(xk) for some ` ≥ 0, k ≥ 2, and q(x) a complex polynomial, φ may take the form

(3) φ(X) = ωUXU −1, or

(4) φ(X) = ωUXtU −1, where ω ∈ C is a kth root of unity.

Example 5.0.2. To illustrate the possibility of (3) and (4), take p(x) = x2 − 1. This means

2 2 φ(A) = In whenever A = In, and so we say that φ preserves involution matrices. Notice that p(x) = x0q(x2), where q(x) = x − 1 so X 7→ ±UXU −1 and X 7→ ±UXtU −1 preserve involutions.

In addition, the theorem cannot be improved to include n = 2. We furnish an easy counterexample.

Note that 2 × 2 involution matrices are either trace-zero or ±I2. Deﬁne φ : M2(C) → M2(C) be deﬁned by

φ(X) = X − tr(X)I2. (5.1)

Clearly φ is bijective and linear. The trace-zero involutions are preserved but the identity is mapped to its negative. Hence φ is not a scalar multiple of a Jordan automorphism.

Remark 5.0.3. The repeated root case of Theorem 5.0.1 is known as well. Let p(x) = (x − a)r,

r where r ≥ 2 and a ∈ C. Take a = 0. Then any map preserving V (x ) preserves the nilpotent matrices of nilindex ≤ r. Such maps are of the form X 7→ cUXU −1 or X 7→ cUXtU −1 by Theorem

85 2.0.2. The special case of r = n is Theorem 1.1.5 while the case r = 2 is Theorem 2.0.1. The preserved domain is sln.

r We will brieﬂy explain the case a 6= 0. Suppose A = (aij) satisﬁes (A − aIn) = 0. Without loss of generality, assume that A is upper-triangular. Since A − aIn is nilpotent it follows that aii = a

r for i = 1, . . . , n. Hence a bijective linear map φ : Mn → Mn preserving the roots of p(x) = (x − a)

0 0 has the property that φ(A) = aIn + M whenever A = aIn + M, where M and M are nilpotent of nilindex less than or equal to r.

r Let N be any nilpotent matrix of nilindex ≤ r. Then aIn + λN is a root of (x − a) for all

λ ∈ C. If φ(aIn) = aIn + M, then

r r 0 = (φ(aIn + λN) − aIn) = (M + λφ(N)) (5.2)

r for all λ ∈ C. Expanding (M + λφ(N)) gives a linear combination of products of M and φ(N), each with a coeﬃcient λi for i = 1, . . . , r. This generates a homogeneous system of r equations for all λ ∈ C. Choose λ to be a primitive rth root of unity. Adding all such equations eliminates any term containing M leaving only the relation λrφ(N)r = 0. In other words, φ preserves nilpotent

−1 matrices of nilindex ≤ r and thus the restriction of φ to sln takes the form X 7→ cUXU or

t −1 X 7→ cUX U for all X ∈ sln by Theorem 2.0.2.

Returning to equation (5.2), notice that M + φ(N) is nilpotent for all N such that N r = 0.

Pick the nilpotent matrix N such that φ(N) = M ∗ (recall the image of N is determined by the previous paragraph). Next recall some standard spectral theory of matrices. The matrix M + M ∗ is nilpotent but also hermitian, hence M +M ∗ = 0, or equivalently, M ∗ = −M. Thus M commutes

∗ with M , so M is a normal matrix and hence diagonalizable, so in fact M = 0. Hence φ(aIn) = aIn.

r Thus every bijective linear map preserving V ((x − a) ) is standard nilpotent preserver on sln and unital.

86 Bijective linear maps preserving roots of any single variable polynomial are completely classiﬁed.

Moreover we can classify the preserved subspace generated by V (p).

Proposition 5.0.4. Let p(x) be a nonconstant complex polynomial and let A = hA ∈ Mn | p(A) =

0i. Then  0, if p(x) = x   CIn, if p(x) = x − λ, λ 6= 0 A = s sln, if p(x) = x , s ≥ 2   Mn, otherwise Proof. The cases when p(x) has degree one are both obvious. If p(x) = xs for s ≥ 2, then A

s contains every square-zero matrix, and thus contains sln. Linear combinations of roots of x must all be trace-zero, so A = sln.

Suppose that a, b ∈ C are distinct roots of p(x) with a 6= 0. Then aIn ∈ A implies that In ∈ A.

Notice that (x − a)(x − b) divides p(x) and consequently

diag(a, a, . . . , a, b, b, . . . , b) ∈ A where the number of a’s on the diagonal is, say, equal to t. Thus

diag(a, a, . . . , a, b, b, . . . , b) = (a − b)diag(1, 1,..., 1, 0, 0,..., 0) + bIn ∈ A.

Hence diag(1, 1,..., 1, 0, 0,..., 0) ∈ A. Since every conjugate of diag(1, 1,..., 1, 0, 0,..., 0) is also a root of p(x), it follows that A contains every rank-t idempotent. Since t is arbitrary, A contains all idempotents, so A = Mn.

r Suppose now that p(x) = (x − a) , with a 6= 0 and r ≥ 2. As before, aIn ∈ A implies In ∈ A.

If S is a square-zero matrix, then aIn + S ∈ A implies that S ∈ A. Hence A contains all trace-zero matrices and all scalar matrices, so A = Mn.

Linear maps preserving the roots of speciﬁc polynomials in two variables have also been well- studied. Taking p(x, y) = xy, p(x, y) = xy − yx, or p(x, y) = xy + yx, preserving the zeros of

87 p(x, y) amounts to classifying maps preserving the zero, Lie, or Jordan product (see Section 1.1.3).

Other preserver problems include maps preserving products of matrices equal to idempotents or nilpotents; in particular, the authors in [31], [81], and [70] considered maps preserving the zeros of p(x, y) = (xy)2−xy, while the authors in [53] considered maps preserving the zeros of p(x, y) = (xy)n

(without the assumption of linearity). The authors in [36] take a uniﬁed approach to preserving the zeros of multivariable polynomials to obtain an analogue of Howard’s single-variable theorem.

Following the theme of the previous section we consider the problems of characterizing maps preserving the roots of a polynomials but replace the zero matrix with a nonzero matrix. That is, if B is a ﬁxed matrix and p(x) is a complex polynomial, characterize bijective linear mappings

φ : Mn → Mn such that

p(φ(A)) = B whenever p(A) = B. (5.3)

Hence we say that φ preserves the roots of the nonhomogeneous polynomial p(A) = B. Immediately we encounter a potential obstacle that complicates the situation as follows.

Proposition 5.0.5. Let p(x) be a complex polynomial and let φ : Mn → Mn be a bijective linear map such that p(φ(A)) = B whenever p(A) = B. The preserved subspace of φ is contained in C(B).

Proof. If p(A) = B, then AB = Ap(A) = p(A)A = BA.

Notice that if B = bI is a scalar matrix then one can use Proposition 5.0.4 to determine the restricted domain of φ since φ preserves the roots of p(x) − b. If B is a nonscalar matrix, the restricted domain of φ drops dramatically in dimension (see, for instance, Lemma 3.0.1). At ﬁrst glance it may be that the preserved subspace is too irregular to expect a reasonable description.

The results in this chapter both concern p(x) = x2, perhaps the simplest nonlinear polynomial.

In Section 5.1 we present a nice situation occuring when B is idempotent and in Section 5.2 we present a not-so-nice situation when B = e12.

88 5.1 Maps preserving the square roots of an idempotent matrix

2 Fix 1 ≤ t ≤ n−1. Suppose φ : Mn → Mn be a bijective linear map such that φ(A) = e11 +···+ett

2 whenever A = e11 + ··· + ett. A simple computation shows that if n = 2, the only square roots of e11 are ±e11. In this case any linear map sending e11 to ±e11 preserves the square roots of e11.

For convenience let Et = e11 + ··· + ett denote the canonical rank-t idempotent. By Proposition

5.0.5 every square root of Et is of the form

C 0 = C ⊕ S 0 S

2 where C is a t × t complex matrix and S is an (n − t) × (n − t) matrix satisfying C = Et and

S2 = 0; in other words, C is an involution matrix and S is a square-zero matrix. Conversely, given any t × t involution C and (n − t) × (n − t) square-zero matrix S, the n × n matrix C ⊕ S is a square root of Et. In light of Proposition 5.0.4 the span of square roots of Et (and consequently the preserved subspace of φ) is M 0 A = t . (5.4) 0 sln−t

A is of codimension 1 in C(Et). However it is a direct sum of two very “nice” spaces.

As hinted above, the square roots of Et depend intimately on involution matrices and square- zero matrices. This observation leads to a remarkable situation in which φ “decomposes” into two maps that preserve diﬀerent sets of matrices. Consequently φ can be built up in terms of two diﬀerent preservers.

Theorem 5.1.1 (Borisov, Julius, Sikora [6]). Let n ≥ 3. If φ : A → A is a bijective linear map

2 2 such that φ(A) = e11 + ··· + ett whenever A = e11 + ··· + ett, with 1 ≤ t ≤ n − 1, then φ is of the form φ = α ⊕ β, where α : Mt → Mt is a bijective linear map preserving involutions and

β : sln−t → sln−t is a bijective linear map preserving square-zero matrices.

Proof. As an abuse of notation we refer to Mt as the t × t upper corner block of Mn and sln−t as

89 2 the (n − t) × (n − t) lower corner block of Mn. Given a C ∈ Mt and S ∈ sln−t such that C = Et

2 2 and S = 0, then (C + xS) = Et for all x ∈ C. Thus

2 2 2 2 φ(C + xS) = φ(C) + xφ(C) ◦ φ(S) + x φ(S) = Et.

2 Canceling φ(C) = Et, the equation reduces to

xφ(C) ◦ φ(S) + x2φ(S)2 = 0.

Subtracting the two equations obtained by taking x = 1 and x = −1, it follows that φ(C)◦φ(S) = 0

2 and φ(S) = 0. Since φ(C) ◦ φ(S) is bilinear in C and S, we conclude that φ(Mt) ◦ φ(sln−t) = 0.

We now show that φ preserves the square-zero matrices in A. Given a square-zero matrix

2 T ∈ A it may be written as a direct sum of square-zero matrices in Mt and sln−t. Since φ(S) = 0

2 whenever S ∈ sln−t is a square-zero matrix, it suﬃces to show that φ(T ) = 0 whenever T ∈ Mt is a square-zero matrix.

Let Tˆ be the Jordan normal form of T in Mt and P be an invertible matrix such that T =

ˆ −1 ˆ ˆ t P TP . Arrange the blocks of T such that T = e12 + e34 + ··· + ei,i+1, where i ≤ 2 . The involution matrix Cˆ = (e11 − e22) + (e33 − e44) + ··· + (eii − ei+1,i+1) + ei+2,i+2 + ··· + ett. satisﬁes

ˆ ˆ 2 ˆ −1 (C + xT ) = Et for all x ∈ C. Write C = P CP . Since Et is the identity in Mt, it follows

2 that (C + xT ) = Et for all x ∈ C. Using the same linearity argument as above, it follows that

φ(T )2 = 0. Hence for all A ∈ A, we have tr(φ(A)) = 0 whenever tr(A) = 0.

Next we show that φ(Mt) = Mt and φ(sln−t) = sln−t. Write Et = φ(U) + φ(V ), where U ∈ Mt and V ∈ sln−t. Let Cu,Su ∈ Mt and Cl,Sl ∈ sln−t be the matrices such that φ(U) = Cu ⊕ Cl and

φ(V ) = Su ⊕ Sl. Thus

C + S 0 E 0 φ(U) + φ(V ) = u u = t (5.5) 0 Cl + Sl 0 0

Clearly Cl = −Sl. We know that φ(U) ◦ φ(V ) = 0, and so Cu ◦ Su = Cl ◦ Sl = 0. We conclude that

90 2 2 Cl = Sl = 0. Squaring equation (5.5), we get

2 2 Cu + Su = Et.

Substituting Su = Et − Cu into the above,

2 2 2 Cu + (Et − Cu) = 2Cu − 2Cu + Et = Et.

Hence Cu is idempotent. Taking traces,

t = tr(Et) = tr(φ(U) + φ(V )) = tr(φ(U)) = tr(Cu + Cl) = tr(Cu),

and consequently rank(Cu) = t. The only rank-t idempotent in Mt is Et itself. Thus Cu = Et and Su = 0. Hence φ(Mt) contains the matrix Et. Since φ(Mt) ◦ φ(sln−t) = 0, this implies that

φ(sln−t) = sln−t. Hence the lower corner block is completely preserved by φ. Returning now to the relation φ(Mt) ◦ sln−t = 0, it follows that φ(Mt) = Mt as well.

2 Deﬁne α : Mt → Mt as the restriction φ to Mt. Since φ(C) = Et whenever C ∈ Mt satisﬁes

2 C = Et, it follows that α preserves involutions. Deﬁne β : sln−t → sln−t as the restriction of φ to

2 2 sln−t. Since φ(S) = 0 whenever S ∈ sln−t satisﬁes S = 0, it follows that β preserves square-zero matrices.

Remark 5.1.2. In general α and β may be of opposite types; one could be an automorphism and the other an antiautomorphism, so φ need not be an automorphism or antiautomorphism (it must be a Jordan automorphism of Mt ⊕ sln−t, however). This nonstandard solution is due to the decomposition of the preserved subspace into two components.

There are some extremal cases of interest. If t = 1, then α(e11) = ±e11. If t = 2, recall that the involution preserver problem does not have a standard description on M2 (see the map in equation

(5.1)). If t = n − 1 then A is simply Mn−1 and φ reduces to the involution preserver problem.

91 Hence the problem (5.3) is solved when B is idempotent (since every idempotent B is similar to an Et with t = rank(B)).

5.2 Maps preserving the square roots of a rank-one nilpotent

In this section, we attempt to classify bijective linear maps preserving the square roots of a rank-one nilpotent matrix; in particular, e12. Notice that e12 has no square roots in M2, so we implicitly take n ≥ 3 going forward. Column vectors will be denoted by boldface letters. By Proposition

5.0.5, it is easy to verify the following.

2 Lemma 5.2.1. If A = e12, then 0 ∗ ut A = 0 0 0  , 0 v S n−2 t t where u, v ∈ C satisfy u v = 1 and S ∈ sln−2 is a square-zero matrix such that S u = Sv = 0.

Consider the converse problem of constructing a square root of e12 given a particular square-zero

n−2 matrix S. Do there exist column vectors u, v ∈ C satisfying the conditions of Lemma 5.2.1?

k k Lemma 5.2.2. Let S ∈ slk be a square-zero matrix. If rank(S) = 2 and u, v ∈ C are column

t t k vectors such that S u = Sv = 0, then u v = 0. Otherwise, there exist column vectors u, v ∈ C such that Stu = Sv = 0 and utv = 1.

k k ˆ Proof. Let {ej}j=1 be the standard basis vectors of C . Let S be the Jordan form of S and P ∈ Mk

ˆ −1 k be the invertible matrix such that S = P SP . Suppose rank(S) = 2 . If v ∈ ker S, then it may

t be written v = P vˆ, where vˆ ∈ span{e1, e3,..., ek−1} = ker Sˆ. Similarly, u ∈ ker S implies

−1 k t u = (P ) uˆ, where uˆ ∈ span{e2, e4,..., ek} = ker Sˆ . Then

utv = uˆtP −1P vˆ = uˆtvˆ = 0.

k ˆ ˆt If rank(S) < 2 , the kernels of S and S have at least one vector in common, in particular, ek.

92 −1 t Take vˆ = uˆ = ek and deﬁning v = P vˆ and u = (P ) uˆ, we get

t t −1 t u v = uˆ P P vˆ = (ek) ek = 1, as claimed.

What does this mean for the subspace generated by square roots of e12?

2 Lemma 5.2.3. Let A = span{A ∈ Mn | A = e12}. If n = 4, then 0 ∗ ∗ ∗ 0 0 0 0 A =   . 0 ∗ 0 0 0 ∗ 0 0 Otherwise, 0 ∗ ∗ ... ∗ 0 0 0 ... 0   0 ∗  A =   . . .  . . sln−2  0 ∗ 2 2 Proof. For any n ≥ 3, we have (e1r +er2) = e12 and (ie1r −ier2) = e12, for all r ≥ 3. Consequently

2 e13, e14, . . . , e1n and e32, e42, . . . , en2 lie in A. In addition, (e12 + e13 + e32) = e12 so e12 ∈ A as well.

Let n = 4 and A be a square root of e12. If S ∈ sl2 is a square-zero matrix in the lower right

2 t corner of A, then the column vectors u, v ∈ C in A must satisfy S u = Sv = 0. By Lemma 5.2.2

t we must have u v = 0. But then A is not a square root of e12. Consequently the only square-zero matrix in the lower right corner of A is the zero matrix.

If n = 3, the form of A is immediate from Lemma 5.2.1. If n > 4, then for every rank-one square-

n−2 zero matrix S ∈ sln−2, we can ﬁnd (by Lemma 5.2.2) corresponding column vectors u, v ∈ C such that the n × n matrix 0 0 ut 0 0 0  0 v S is a square root of e12. Hence A contains all rank-one square-zero matrices in the lower right corner, and thus sln−2.

93 Let C = span{e12, e13, . . . , e1n, e32, e42, . . . , en2}. The previous lemma demonstrates that A = C if n = 4 and A = C ⊕ sln−2 otherwise (here “⊕” denotes the direct sum of vector spaces, not the

2 direct sum of matrices). Let φ : A → A be a bijective linear map such that φ(A) = e12 whenever

2 A = e12. Unlike the previously explored idempotent case, the space of square roots of e12 does not decompose into block subspaces. What can possibly be said about φ?

2 2 Lemma 5.2.4. Let φ : A → A be a bijective linear map such that φ(A) = e12, whenever A = e12.

Then

(1) φ(e12) ∈ Ce12,

2 (2) φ(C) ◦ φ(S) = 0 whenever (C + S) = e12, with C ∈ C and S ∈ sln−2, and

2 n−2 (3) φ(S) = 0 whenever S ∈ sln−2 is a square-zero matrix, except possibly if rank(S) = 2 .

2 Proof. If A = e12, then A + xe12 is a square root of e12 for all x ∈ C and

2 2 2 2 e12 = φ(A + xe12) = φ(A) + xφ(e12) ◦ φ(A) + x φ(e12) .

2 2 Using φ(A) = e12 and the standard trick of varying over values of x, conclude that φ(e12) = 0 with

φ(e12) ∈ A, as well as φ(e12) ◦ φ(A) = 0 for all A ∈ A. By linearity, it follows that φ(e12) ◦ A = 0.

Writing 0 ∗ ut t φ(e12) = 0 0 0  ∈ A, 0 v S we get that if n = 3, 4, then S = 0, and if n ≥ 5, then S◦sln−2 = 0 implies S = 0. Using e1j, ej2 ∈ A for j ≥ 3, we get that φ(e12) ◦ e1j = φ(e12) ◦ ej2 = 0 implies u = 0 and v = 0. Thus φ(e12) ∈ Ce12.

By Lemma 5.2.1, every square root of e12 may be written as C + xS, where x ∈ C, subject to

2 2 2 2 the constraints C = e12,S = 0, and C ◦ S = 0. Then φ(C) = e12 and φ(C + xS) = e12 imply that

xφ(C) ◦ φ(S) + x2φ(S)2 = 0.

94 By the usual argument, it follows that φ(C) ◦ φ(S) = 0 and φ(S)2 = 0. This proves (2).

2 By Lemma 5.2.2, one can always ﬁnd a C ∈ C such that (C + xS) = e12 for all x ∈ C, provided

n−2 that rank(S) < 2 . Hence (3) is proved as well.

But φ need not preserve rank.

Example 5.2.5. If g : A → C is a linear functional, then the linear map ψ deﬁned by ψ(A) =

A + g(A)e12 preserves the square roots of e12 since e12A = Ae12 = 0. However, every bijective linear map that preserves square roots of e12 restricts to a bijection on the quotient space A/Ce12.

This can be seen as follows. Given such a map ψ, deﬁne ζ : A → A by A 7→ φ(A) − g(A)e12, where g : Mn(C) → C is a linear functional such that the (1, 2)-entry of ζ(A), denoted [ζ(A)]1,2, is zero for all A ∈ A. Deﬁne

ˆ ∼ A = {A ∈ A | [A]1,2 = 0} = A/Ce12.

Clearly, ζ(Aˆ) ⊆ Aˆ by the deﬁnition of ζ. If A ∈ Aˆ is such that ζ(A) = φ(A) − g(A)e12 = 0, then

φ(A) ∈ Ce12. Since Ce12 is an invariant subspace of φ by Lemma 5.2.4, it follows from bijectivity

ˆ ˆ of φ that A ∈ Ce12 ∩ A = 0. Hence ker ζ|Aˆ = 0 and ζ acts bijectively on A.

Example 5.2.6. Consider the linear map φ : A → A defined by 0 a ut 0 a ut 0 0 0t 7→ 0 0 0t , 0 v S 0 v + s1 S where s1 is the first column of S. The map φ is bijective. Suppose A is a square root of e12. Then utv = 1, Sv = Stu = 0 and S2 = 0. Then observe that  t t t t  t t 0 u v + u s1 (S u) 0 1 + u s1 0 2 φ(A) = 0 0 0t  = 0 0 0t 2 0 Sv + Ss1 S 0 0 0 t t and φ preserves square roots of e12 because u s1 is the first entry of the vector S u = 0.

t Analogously, adding any row of S to u yields a map that preserves the square roots of e12. In fact, adding any ﬁxed linear combination of the columns of S to v or the rows of S to ut also yields such a map.

95 Suppose φ(X) = cUXU −1 for all X ∈ A, where U is some invertible matrix leaving A invariant.

−1 Then cUeijU = eij for i, j ≥ 3 and i 6= j. This implies that for all such pairs (i, j), the oﬀ- diagonal entries of U in the ith column and jth row are zero. Hence U is a linear combination of

−1 the n × n identity matrix, e12, and e21. But then cU sln−2U ⊆ sln−2 while φ(sln−2) 6⊆ sln−2, a contradiction. The proof for X 7→ cUXtU −1 follows by similar reasoning. Hence φ does not take the form of a Jordan automorphism.

Considering the examples above, there is little hope of obtaining a complete description of bijective linear maps preserving square roots of e12. Despite this, observe that in each example, the space C was preserved and the image of sln−2 completely contained sln−2. In fact these two properties are equivalent, as the next theorem shows. Note that if n = 3 or n = 4, the theorem trivially holds since A = C in both cases. We also discuss the case n = 6 in Remark 5.2.8.

Theorem 5.2.7 (Borisov, Julius, Sikora [6]). Let n ≥ 5 and n 6= 6. If φ : A → A is a bijective linear

2 2 map such that φ(A) = e12 whenever A = e12, then φ(C) = C if and only if φ(sln−2)|sln−2 = sln−2.

Moreover, if either situation holds, then there is a bijective linear map f : sln−2 → sln−2 preserving square-zero matrices that extends to φ.

Proof. If φ(C) = C, then φ bijective implies φ(sln−2)|sln−2 = sln−2. Conversely, assume for a

contradiction that φ(sln−2)|sln−2 = sln−2 but φ(C) 6= C. Using this and the fact that φ(e12) ∈

Ce12 ⊆ C, there must be a j ≥ 3 so that A = e1j +ej2 or A = ie1j −iej2 satisﬁes φ(A)|sln−2 = R 6= 0.

2 Since φ(A) = e12, we get that R is a square-zero matrix. By hypothesis there is a bijective linear

map f : sln−2 → sln−2 that extends to φ; that is, f(X) = φ(X)|sln−2 for all X ∈ sln−2. If n = 5, then f sends rank-one square-zero matrices to square-zero matrices. Since every square-zero matrix in sl3 is rank-one, it follows that f preserves square-zero matrices. If n ≥ 7, then rank-one and rank-two square-zero matrices in sln−2 are mapped to square-zero matrices, and so by Proposition

96 2.1.1, f preserves square-zero matrices as well. So we have

f(X) = cUXU −1 or f(X) = cUXtU −1

for all X ∈ sln−2, where U ∈ Mn−2(C) is invertible and c ∈ C\{0}.

By statement (2) of Lemma 5.2.4, we have that φ(A) ◦ φ(ekl) = 0 whenever j 6= k, l and

−1 k, l ≥ 3 are distinct. Hence R ◦ φ(ekl)|sln−2 = R ◦ f(ekl) = R ◦ (cUeklU ) = 0. Equivalently,

−1 −1 U RU ◦ ekl = 0, and so the oﬀ-diagonal entries of U RU in the kth column and lth row are zero.

By varying k, l over all possible choices, we conclude that U −1RU is a diagonal square-zero matrix.

The only diagonal square-zero matrix is the zero matrix, and so R = 0, a contradiction. Thus the assumption that φ(C) 6= C is false, proving the converse.

Remark 5.2.8. In the case n = 6, the space sl4 contains only rank-one and rank-two square-zero matrices; but the rank-two square-zero matrices have maximum possible rank in sl4. By Lemma

2 5.2.2, there is no matrix C ∈ C such that (C + S) = e12 when S ∈ sl4 and rank(S) = 2. Hence it may be that φ(S)2 6= 0, which explains the above exclusion. Note that, once again, Conjecture

2.1.5 would assist here since rank-one square-zero matrices are mapped to square-zero matrices. In particular, if such maps turn out to be standard, then there is no need to exclude n = 6 in Theorem

5.2.7 above.

One could likely expect maps preserving the square roots of a nilpotent of rank greater than

1 to be more complicated. The results in this section demonstrate concretely that over nonsquare subspaces of Mn, the descriptions of linear preserver problems can become quite wild.

It would interesting to ﬁnd more complex polynomials p and matrices B such that the preserved subspace A = span{A ∈ Mn | p(A) = B} decomposes nicely. Otherwise the problem of characterizing the bijective linear maps preserving these roots may prove too intractable.

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