––––––––––––––––––––––––––––— Popular and Democratic Republic of Algeria

University of Oran 1 Ahmed Ben Bella

Faculty of exact and applied Sciences

Department of Mathematics

Operators Similar To Their Adjoints

THESIS SUBMITTED FOR THE DEGREE OF DOCTORATE IN MATHEMATICS Presented by :

Dehimi Souheyb

Doctorate’sThesis Committee :

President: Professor C.Bouzar University of Oran 1 Ahmed Ben Bella

Thesis Supervisor: Professor M.H.Mortad University of Oran 1 Ahmed Ben Bella

Examiners: Professor K.Belghaba University of Oran 1 Ahmed Ben Bella

Dr. M.Meftah (M.C.A) University of Oran 1 Ahmed Ben Bella

Professor B.Benahmed National Polytechnic School of Oran (ENPO Ex: ENSET)

Professor M.Tlemçani University of Sciences and Technology of Oran (M.B)

Academic year 2016 2017 − Acknowledgement

I would like to thank first and for most my supervisor, Professor Mohammed Hichem Mortad, for his many suggestions, helpful discussions, patience and constant support during this research.

I sincerely thank Professor C.Bouzar for giving me the honor of being president of the jury.

I thank Professors K.Belghaba, B.Benahmed and M.Tlemçani and Doctor M.Meftah for their time and effort participating in my thesis committee.

Of course, I am grateful to my parents for their support, encouragement, and patience. Without them this work would never have come into existence. Contents

Introduction vi

1 Essential background 11 1.1 Banachalgebra ...... 11 1.1.1 Introduction...... 11 1.1.2 Basic properties of spectra ...... 12

1.2 C∗-algebra...... 13 1.3 Boundedoperators ...... 15 1.3.1 Definitions and properties ...... 17 1.3.2 Approximate point spectrum ...... 18 1.3.3 Resolutions of the identity ...... 19 1.4 Polar decomposition of an operator ...... 21 1.4.1 Isometry and partial isometry ...... 22 1.4.2 Polar decomposition of an operator ...... 22 1.5 Positiveoperators...... 25 1.6 Numericalrange...... 27

2 Non-normal operator classes 29 2.1 Compactoperators ...... 29 2.2 Hyponormal operators ...... 30 2.2.1 Definitions and properties ...... 31

i 2.2.2 Some conditions implying normality or self-adjointness ...... 33 2.2.3 p-Hyponormal operators ...... 35 2.3 Normaloid operators ...... 39 2.4 Paranormal operators ...... 42 2.4.1 Definitions and properties ...... 42 2.4.2 k-paranormal operators ...... 44 2.5 Convexoid operators ...... 46 2.5.1 Examples ...... 47 2.6 ClassAoperators...... 51 2.6.1 Quasi-class A operators ...... 54

3 Similarities involving bounded operators 56 3.1 Introduction...... 56 3.2 Operators similar to their adjoints ...... 58 3.2.1 Conditions implying self-adjointness of operators ...... 58 3.2.2 Operators similar to self-adjoint ones ...... 63 3.3 Operators with inverses similar to their adjoints ...... 64 3.3.1 Operators similar to unitary ones ...... 64 3.3.2 Operators with left inverses similar to their adjoints ...... 69 3.4 Similarities involving normal operators ...... 73 3.5 Quasi-similarity of operators ...... 76

4 Similarities involving unbounded operators 78 4.1 Preliminaries ...... 78 4.1.1 Adjoint Operators ...... 80 4.1.2 Self-adjoint Operators ...... 82 4.2 Quasi-similarity of unbounded operators ...... 86 4.3 Similarities involving unbounded normal operators ...... 90

ii 4.4 Unbounded operators similar to their adjoints ...... 94

Bibliography 99

iii Résumé

Le but de ce travail est d’étudier la similarité entre les opérateurs linéaires et leurs adjoints dans un espace de Hilbert. Le travail présenté est organisé selon le plan suivant: Dans les deux premiers chapitres, un rappel sur des notions essentiel les sur les opérateurs linéaires bornés. Dans les deux derniers chapitres nous donnons quelques conséquences de la similarité et quasi-similarité entre les opérateurs linéaires. L’un des principaux objectifs de cette thèse est de généraliser le théorème de Sheth.

Abstract

The aim of this work is to study the similarity between linear operators and their adjoints in a Hilbert space. The work is organized according to the following plan. In the first two chapters, a reminder on essential notions about bounded linear operators. In the last two chapters we give some consequences of the similarity and quasi-similarity between the linear operators. One of the main objectives of this thesis is to generalize Sheth’stheorem.

Keywords: similarity, bounded and unbounded operators, closed, self-adjoint, normal, hyponomal operators, unitary cramped operators, Sheth.

iv Notations

k The field of real or complex numbers. σ(T ) The spectrum of T. r(T ) The spectral radius of T. (H) The Banach algebra of all bounded linear operators. B ran(T ) The range of a linear operator T.

T ∗ The adjoint operator of T.

σp (T ) The point spectrum of T.

σapp (T ) The approximate point spectrum of T.

PM Initial projection

PN Final projection W (T ) The numerical range of T. (X,Y ) The set of all compact linear operators. B∞ co σ (T ) The convex hull of σ (T ) .

σe (T ) The essential spectrum of T. D (T ) The domain of a linear operator T. (T ) The graph of an operator T. G T The closure of the closable operator T.

v Introduction

The theory of Banach algebras is an abstract mathematical theory which is the synthesis of many specific cases from different areas of mathematics. They are named after the Polish mathematician Stefan Banach (1892—1945) who had introduced the concept of Banach space, but Banach had never studied Banach algebras. In fact, the first one who have defined them is Mitio Nagumo in 1936 ([Nag36]) under the name “linear metric rings”. In 1941, I.M.Gelfand (1913—2009) introduced them under the name “normed rings”([Gel41]). In the classical monograph [BD73], the authors write that, if they had it their way, they would rather speak of “Gelfand algebras”. But in 1945, Warren Ambrose (1914—1995) came up with the name “Banach algebras”([Amb45]) . Banach algebras show up naturally in many areas of analysis: Let X be a Banach space. then (X) , the algebra of all bounded linear operators on X, is a Banach algebra, B with respect to the usual operator norm. If we have an analytic object that has a Banach algebra naturally associated with it, then this algebra can provide us with further insight into the nature of the underlying object. A notion which will be of great use in this thesis is that of hyponormal operator. A bounded hyponormal operator is a bounded operator T on a Hilbert space H such that

T ∗T TT ∗ . This definition was introduced by Paul Halmos [Halm50] in 1950 and ≥ generalizes the concept of a normal operator (where T ∗T = TT ∗). The important thing is that there is a prominent example of a hyponormal operator, the unilateral shift. If l2

vi is the Hilbert space of square summable sequences and T is defined on l2 by

T (x0, x1,...) = (0, x0, x1,...) .

Then T is called the unilateral shift and is the most basic example of hyponormal operators. The unilateral shift is a well understood non normal operator; it is arguably the best understood non normal operator on an infinite dimensional space. Paul Halmos began a strategic attack on operator theory by extracting two properties of the shift in [Halm50]. One was the definition of hyponormal operators and the other the idea of a subnormal operator. A subnormal operator is one that has a normal extension; every subnormal operator is hyponormal. One of the first important results in the theory of hyponormal operators, due to C. R. Putnam, is the fact that if T is a pure hyponormal operator, then its real and imaginary parts, X and Y , must be absolutely continuous self- adjoint operators [Put63]. That is, the spectral measures for X and Y must be absolutely continuous with respect to Lebesgue measure on the real line. Thus the Spectral Theorem for self-adjoint operators can be applied to X, and this operator can be represented as a multiplication operator on L2 [a, b] for some interval in R. The operator Y also has such a representation, but on a different L2 space. Can Y be represented on the same space L2 [a, b] in a way that is intimately connected with the representation of X ? Indeed, a result of Kato [Kat68], though it is not directly related to hyponormal operators, implies that this can be done. About the same time, many authors began investigating hyponor- mal operators from this perspective [Pin68, Rus68, Xia63][Rus68][Xia63]. In addition if T is hyponormal, then

π T ∗T TT ∗ Area (σ (T )) . k − k ≤ This was proved by Putnam in [Put70] and is well known as Putnam’sinequality. An operator T (H) is said to be p-hyponormal if ∈ B

p p (TT ∗) (T ∗T ) ≤

vii for a positive number p. In fact, semi-hyponormal operators were first introduced by Professor D. Xia in [Xia80]. He also provides an example of a semi-hyponormal operator which is not hyponormal. In [Xia80] Xia proved that, if T is p-hyponormal, then

2p 1 π T ∗T TT ∗ p ρ − dρdθ k − k ≤ ZZσ(T ) for p 1 . Ch¯oand Itoh proved that Putnam’sinequality holds for p hyponormal oper- ≥ 2 − ators in [CI95] . The notion of a paranormal operator dates back to 1960s and is due to V. Istra¸tescu˘ in [Ist66] he named them “operators of class N ”. T. Furuta in [Fur67] introduced the term “paranormal operators”. The class of paranormal operators can be seen as a generalization of other important classes: hyponormal operators and subnormal and normal operators . In subsequent years paranormal operators have been the subject of further research. For example, we know that a paranormal operator T is compact if and only if T n is compact for some n N. Moreover, compact paranormal operator is normal . Paranormality ∈ appears also to be an important property when studying various problems in operator theory. Let A and B be a bounded operators. We say that A is similar to B iff

SA = BS

for some bounded invertible operator S. In 1956 Beck and C.R.Putnam showed that if

T is a bounded operator which is unitarily equivalent to its adjoint T ∗, via cramped unitary operator U, necessarily T is self-adjoint. Our main work is to answer the next question: suppose that T is a bounded operator and S is an invertible operator for which

0 / W (S) and ST = T ∗S, where W (S) is the numerical range, then when does it follow ∈ that necessary T is self-adjoint?. In 1966, Sheth had proved that, if a bounded hyponormal operator T satisfies the above relation, then T is self-adjoint. In 1969 J. P. Williams, in his paper Operators Similar to Their Adjoints, had proved an important theorem which is : If T is a bounded operator

viii 1 such that S− TS = T ∗, where 0 / W (S), then the spectrum of T is real. Williams ∈ result is considered a motivation of our thesis. In [Cas83], Castern had given a necessary and suffi cient conditions for a bounded operator T being similar to unitary or self-adjoint operator. Embry showed that, If S and T are commuting normal operators and AS = TA, where 0 is not in the numerical range of A, then S = T . Her result includes a slight improvement of a result of J. P.Williams. In [Wil69] Williams proved that σ (E) is real if AE = E∗A, where 0 is not in the closure of W (A). Thus if E is normal, then E is self-adjoint. If H and K are complex Hilbert spaces, the bounded linear operator X : H K −→ is said to be quasi-invertible iff it is one-to-one and has dense range. Two operators A : H H and B : K K are quasi-similar provided there exist quasi-invertible −→ −→ operators X : H K and Y : K H such that XA = BX and YB = AY . Quasi- −→ −→ similarity was first introduced by Sz-Nagy and Foia¸s(see, for example [NF70]) in their theory of an infinite dimensional analogue of the Jordan for certain classes of operators. It replaces the familiar notion of similarity. Quasi-similarity is the same thing as similarity in finite dimensional spaces, but in infinite-dimensional spaces it is a much weaker relation, so weak that two operators can be quasi-similar and yet have unequal spectra [NF70, p.262] For normal operators this cannot happen! It follows from the Fuglede-Putnam commu- tativity theorem that if two normal operators are quasi-similar, they are actually unitarily equivalent [Dou69] and therefore have equal spectra. This result does not generalize to hyponormal operators. Sarason has given an example of two hyponormal operators which are similar but not unitarily equivalent [Hal67]. Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators

ix which fail to be bounded turn out to be closed. Let X,Y be two Banach spaces. A linear operator T : D(T ) X Y is closed if for every sequence xn in D(T ) converging to ⊂ → { } x in X such that T xn y Y as n one has x D(T ) and T x = y. → ∈ → ∞ ∈ A densely defined operator T is said to be hyponormal if: D (T ) D (T ∗) , and ⊆ T ∗x T x for all x D (T ) . In [OS89], Ôta and Schmüdgen proved that quasi- k k ≤ k k ∈ similar closed hyponormal operators have equal spectra, and in [Mor10], Mortad gener- alized Embry’s famous theorem. Mortad’s result considered as a further motivation to generalize Sheth’stheorem.

x Chapter 1

Essential background

This chapter is divided into two parts. In the first one, we collect fundamental results on Banach Algebra, mostly without proof. The second part will be dedicated to the study of bounded linear operators. Most of the material covered in this chapter is from [Con90].

1.1 Banach algebra

1.1.1 Introduction

A complex algebra is a vector space over the complex field C in which a multiplication A is defined and satisfies x (yz) = (xy) z,

(x + y) z = xz + yz, and α (xy) = (αx) y = x (αy) for all x, y, and z in and for all scalars α. A if, in addition, is a Banach space with respect to a norm that satisfies the multiplication A inequality xy x y (x , y ) k k ≤ k k k k ∈ A ∈ A

11 and if has an identity e such that A xe = ex = x (x A) ∈ and e = 1 k k then A is called a Banach algebra. The presence of a unit is very often omitted from the definition of a Banach algebra. However, when there is a unit it makes sense to talk about inverses, so that the spectrum of an element of can be defined in a more natural way than is otherwise possible. A

1.1.2 Basic properties of spectra

Definition 1.1.1 Let be a Banach algebra. The spectrum σ (x) of x is the set of A ∈ A all complex numbers λ such that λe x is not invertible. The resolvent set ρ(x) of x is − 1 the complement of σ (x); it consists of all λ C for which (λe x)− exists. ∈ −

The spectral radius of x is the number

r (x) = sup λ : λ σ (x) . {| | ∈ } Of course, r (x) makes no sense if σ (x) is empty. But this never happens, as we shall see.

Theorem 1.1.2 If is a Banach algebra and x , then A ∈ A 1) the spectrum σ (x) of x is compact and nonempty. 2) The spectral radius r (x) of x satisfies

1 r (x) = lim xn n . n →∞ k k Definition 1.1.3 A subset J of a commutative complex algebra is said to be an ideal A if :

12 (1) J is a subspace of (in the vector space sense), and A (2) xy J whenever x and y J. ∈ ∈ A ∈ If J = , J is a proper ideal. Maximal ideals are proper ideals which are not contained 6 A in any larger proper ideal.

Proposition 1.1.4 Let be a commutative complex algebra, and let ∆ be the set of all A complex homomorphisms of A (1) An element x is invertible in if and only if h(x) = 0 for every h ∆. ∈ A A 6 ∈ (2) An element x is invertible in if and only if x lies in no proper ideal of . ∈ A A A (3) λ σ(x) if and only if h(x) = λ for some h ∆. ∈ ∈

1.2 C∗-algebra

A C∗-algebra is a particular type of Banach algebra that is intimately connected with the theory of operators on a Hilbert space. Some of the general theory developed in this section will be used in the next section to prove the spectral theorem, which reveals the structure of normal operators.

Definition 1.2.1 If is a Banach algebra, an involution is a map x x∗ of into A 7→ A A such that the following properties hold for x and y in and α in C : A (1) (x∗)∗ = x

(2)(xy)∗ = y∗x∗

(3) (αx + y)∗ = αx∗ + y∗

Note that if has involution and an identity e, then e∗x = (e∗x)∗∗ = (x∗e)∗ = x, A similarly, xe∗ = x. Since the identity is unique, e∗ = e.

Definition 1.2.2 A C∗-algebra is a Banach algebra with an involution such that for A every x in : A 2 x∗x = x . k k k k

13 Any x for which x = x∗ is called hermitian, or self-adjoint. ∈ A

Proposition 1.2.3 If is C∗-algebra and x , then A ∈ A

x∗ = x k k k k

2 Proof. Note that x = x∗x x∗ x , so x x∗ . Since x = x∗∗, then k k k k ≤ k k k k k k ≤ k k x∗ x , which means x∗ = x . k k ≤ k k k k k k Definition 1.2.4 Let ∆ be the set of all complex homomorphisms of a commutative Ba- nach algebra . The formula A xˆ(h) = h(x)(h ∆) ∈ assigns to each x a function xˆ : ∆ C; we call xˆ the Gelfand transform of x. ∈ A →

Theorem 1.2.5 Suppose is commutative C∗-algebra, with maximal ideal space ∆. The A Gelfand transform is then an isometric isomorphism of onto C (∆), which has the A additional property that

h(x∗) = h(x)(x , h ∆) ∈ A ∈ or, equivalently, that

(x∗) ˆ = xˆ (x ) ∈ A In particular, x is hermitian if and only if xˆ is a real-valued function.

The next theorem is a special case of the previous theorem. We shall state it in a form that involves the inverse of the Gelfand transform, in order to make contact with the symbolic calculus.

Theorem 1.2.6 If is commutative C∗-algebra, which contains an element x such that A the polynomials in x and x∗ are dense in , then the formula A (Ψf) ˆ = f xˆ ◦

14 defines an isometric isomorphism Ψ of C (σ(x)) onto , which satisfies A

Ψf = (Ψf)∗ for every f C (σ(x)) . Moreover , if f(λ) = λ on σ(x), then Ψf = x. ∈

Definition 1.2.7 In a Banach algebra with involution, the statement ”x 0” means that ≥ x = x∗ and that σ(x) [0, [. ⊂ ∞

Theorem 1.2.8 Every C∗-algebra has the following properties: A (1) Hermitian elements have real spectra.

(2) If x is normal (xx∗ = x∗x), then r(x) = x . ∈ A k k 2 (3) If x , then r(xx∗) = x . ∈ A k k (4) If x , y , x 0, and y 0, then x + y 0. ∈ A ∈ A ≥ ≥ ≥ (5) If x , then xx∗ 0. ∈ A ≥

Proposition 1.2.9 Suppose is C∗-algebra, is a closed subalgebra of , e . Then A B A ∈ B

σ (x) = σ (x) for every x . B A ∈ B

1.3 Bounded operators

In conformity with notations used earlier, (H) will now denote the Banach algebra of B all bounded linear operators T on a Hilbert space H = 0 , normed by 6 { }

T = sup T x : x H, x 1 k k {k k ∈ k k ≤ }

We shall see that (H) has an involution which makes it into a C∗-algebra. B

Theorem 1.3.1 If T (H) and if (T x, x) = 0 for every x H, then T = 0 ∈ B ∈

15 Proof. Since (T (x + y) , x + y) = 0, we see that

(T x, y) + (T y, x) = 0 (x H, y H) . (1.1) ∈ ∈ If y is replaced by iy in (1.1) , the result is

i (T x, y) + i (T y, x) = 0 (x H, y H) . (1.2) − ∈ ∈ Multiply (1.2) by i and add to (1) , to obtain

(T x, y) = 0. (1.3)

With y = T x, (1.3) gives T x 2 = 0. Hence T x = 0. k k Corollary 1.3.2 If S,T (H) , such that ∈ B (Sx, x) = (T x, x)

for every x H, then S = T. ∈ Definition 1.3.3 Let T (H) , then the unique operator S (H) satisfying ∈ B ∈ B (T x, y) = (x, Sy)(x H, y H) ∈ ∈

is called the adjoint of T and is denoted by S = T ∗.

We claim that T T ∗ is an involution on (H) , that is, that the following properties → B hold :

(T + S)∗ = T ∗ + S∗

(αT )∗ = αT ∗

(ST )∗ = T ∗S∗

T ∗∗ = T Since

2 T ∗T = T k k k k holds for every T (H) , then (H) is a C∗-algebra, relative to the involution T T ∗. ∈ B B →

16 1.3.1 Definitions and properties

We recall that ker (T ) = x H : T x = 0 { ∈ } ran (T ) = y H : y = T x, x H . { ∈ ∈ } Theorem 1.3.4 Let T (H) , then ∈ B

ker (T ∗) = ran(T )⊥ and ker (T ) = ran(T ∗)⊥.

Definition 1.3.5 An operator T (H) is said to be ∈ B (1) normal if T ∗T = TT ∗,

(2) self-adjoint (or hermitian) if T ∗ = T ,

(3) unitary if T ∗T = I = TT ∗, where I is the identity operator on H,

(4) an isometry if T ∗T = I (5) a projection if T 2 = T.

It is clear that self-adjoint operators and unitary operators are normal.

Theorem 1.3.6 An operator T (H) is normal if and only if ∈ B

T x = T ∗x k k k k for every x H. Normal operators T have the following properties: ∈ (1) ker (T ) = ker (T ∗) . (2) ran (T ) is dense in H if and only if T is one to-one. (3) T is invertible if and only if there exist c > 0 such that T x c x for every x H. k k ≥ k k ∈

Theorem 1.3.7 If U (H) , the following three statements are equivalent: ∈ B (1) U is unitary . (2) ran (U) = H and (Ux, Uy) = (x, y) for all x H, y H. ∈ ∈ (3) ran (U) = H and Ux = x for all x H. k k k k ∈

17 Theorem 1.3.8 Each of the following four properties of a projection P (H) implies ∈ B the other three: (1) P is self-adjoint. (2) P is normal.

(3) ran (P ) = ker (P )⊥ . (4) (P x, x) = P x 2 . k k Property 3) is usually expressed by saying that P is an orthogonal projection.

Theorem 1.3.9 (1) If U is unitary and λ σ (U) , then λ = 1. ∈ | | (2) If S is self-adjoint and λ σ (S) , then λ is a real number. ∈

1.3.2 Approximate point spectrum

Definition 1.3.10 Let T (H) , the point spectrum of T , σp (T ) is defined by ∈ B

σp (T ) = λ C : ker (T λI) = 0 . { ∈ − 6 { }}

Definition 1.3.11 Let T (H) , the approximate point spectrum of T, σap (T ) is de- ∈ B fined by

σap (T ) = λ C : there is a sequence xn in H such that xn = 1 and (T λI) xn 0. . { ∈ { } k k k − k → }

Note that σp (T ) σap (T ) . ⊂ Proposition 1.3.12 Let T (H) , the following statements are equivalent: ∈ B (1) λ / σap (T ) . ∈ (2) ker (T λI) = 0 and ran(T ) is closed. − (3) There is a constant c > 0 such that (T λI) x c x for all x H. k − k ≥ k k ∈

Theorem 1.3.13 The approximate point spectrum σap(T ) is a nonempty closed subset of C that includes the boundary ∂σ(T ) of the spectrum σ(T ).

18 Theorem 1.3.14 (Fuglede-Putnam-Rosenblum) Assume that M,N,T (H) ,M ∈ B and N are normal, and MT = TN. (1.4)

Then M ∗T = TN ∗.

Note that the hypotheses of Theorem (1.3.14) do not imply that MT ∗ = T ∗N, even when M and N are self-adjoint and T is normal. For instance, if

1 0 0 1 1 1 M = ,N = ,T = ,  0 1   1 0   1 1  − −       then MT = TN but MT ∗ = T ∗N. 6

1.3.3 Resolutions of the identity

Definition 1.3.15 Let be a σ-algebra in a set Ω, and let H be a Hilbert space. In this R setting, a resolution of the identity is a mapping

E : (H) R → B with the following properties: (1) E ( ) = 0,E (Ω) = 1. ∅ (2) Each E (w) is a self-adjoint projection for all w . ∈ R (3) E w0 w00 = E w0 E w00 for all w0 , w00 . ∩ ∈ R (4) If w0 w =
00 , then
E w 0
w00 = E w0 + E w00 for all w0 , w00 . ∩ ∅ ∪ ∈ R (5) For every x H and y H, the
set functions
Ex,y defined
by ∈ ∈

Ex,y(w) = (E(w)x, y)

is a complex measure.

19 When is the set of all Borel sets on a compact or locally compact Hausdorf space, R it is customary to add another requirement to 4) : each Ex,y should be a regular borel measure. Here are some immediate consequences of these properties.

Since each Ex,y is self-adjoint projection, we have

2 Ex,x(w) = (E(w)x, x) = E(w)x (w , x H) . k k ∈ R ∈

By (3) , any two of the projections E (w) commute with each other.

Theorem 1.3.16 If is a closed normal subalgebra of (H) which contains the identity A B operator I and if ∆ is the maximal ideal space of , then the following assertion are true: A (1) There exists a unique resolution E of the identity on the Borel subsets of ∆ which satisfies T = Tˆ dE Z∆ for every T , where Tˆ is the Gelfand transform of T. ∈ A (2) An operator S (H) commutes with every T if and only if S commutes with ∈ B ∈ A every projection E(w).

We now specialize this theorem to a single operator.

Theorem 1.3.17 If T (H) and T is normal, then there exists a unique resolution of ∈ B the identity E on the Borel subsets of σ (T ) which satisfies

T = λdE(λ). Zσ(T ) Furthermore, every projection E(w) commutes with every S (H) which commutes ∈ B with T.

Proof. Let be the smallest closed subalgebra of (H) that contains I,T, and T ∗. Since A B T is normal , Theorem (1.3.16) applies to . By Theorem (1.2.6), the maximal ideal A

20 space of can be identified with σ (T ) in such way that Tˆ(λ) = λ for every λ σ (T ) . A ∈ The existence of E follow now from Theorem (1.3.16).

If ST = TS, then also ST ∗ = T ∗S, by Theorem (1.3.14) ; hence S commutes with every member of . By (2) of Theorem 1.3.16 SE(w) = E(w)S for every Borel set w σ (T ) . A ∈

Remark 1.3.18 If E is the spectral decomposition of normal operator T (H), and if ∈ B f is a bounded Borel function on σ (T ) , it is customary to denote the operator

Ψ(f) = fdE Zσ(T ) by f (T ) . Using this notation, part of the content of Theorems (1.3.16) and (1.3.17) can be sum- marized as follows: The mapping f f (T ) is a homomorphism of the algebra of all bounded Borel functions → on σ (T ) into (H) , which carries the function 1 to I, and carries the identity function B on σ (T ) to T. If S (H) and ST = TS, then Sf (T ) = f (T ) S for every bounded Borel function f. ∈ B

Proposition 1.3.19 A normal T (H) is ∈ B (1) self-adjoint if and only if σ (T ) lies in the real axis. (2) unitary if and only if σ (T ) lies on the unit circle.

1.4 Polar decomposition of an operator

This section ([Na07]) consists of two subsections. In the first one we introduce the concept of isometry and partial isometry. In the second subsection, we discuss in some detail the polar decomposition of an operator.

21 1.4.1 Isometry and partial isometry

What is the appropriate analog of a complex number of length one ? If z is a complex number such that z = 1, then it lies on the unit circle in the complex plane. Since | | zw = z w = w | | | | | | | | for all w, we see that multiplication by z preserves length (it does not stretch). We are therefore led to consider operators that preserve norms. That is for an arbitrary inner product space H, we shall consider an operator U in (H) such that for any x H we B ∈ would have, Ux = x . k k k k

Definition 1.4.1 A bounded linear operator U on a complex Hilbert space H is said to be a partial isometry operator if there exists a closed subspace M such that Ux = x k k k k for all x M, and Ux = 0 for any x M ⊥, where M is called the initial space of U, and ∈ ∈ the range of U, ran (U) is called the final space, and the projections onto the initial space and the final space are said to be initial projection and final projection,respectively.

Theorem 1.4.2 ([Na07]) Let U be a partial isometry operator on a complex Hilbert space H with initial space M and final space N , PM and PN are initial projection and final projection ,respectively. Then the following hold :

1) UPM = U and U ∗U = PM .

2) U ∗ is a partial isometry operator with initial space N and final space M, that is

U ∗PN = U ∗ and U ∗U = PN .

1.4.2 Polar decomposition of an operator

We can write a complex number z = a + ib in polar form using the formulas :

a = r cos θ and b = r sin θ.

.

22 In other words, z = r (cos θ + i sin θ) .

Using this analogy, how do we write an arbitrary bounded linear operator acting on a Hilbert space in ”polar form”? If we use the complex numbers themselves to do this, then, the motivation for polar decomposition would be the following equation:

z z z = z = √zz. z | | z | | | | √ z We see that zz is a positive real number, and z is a complex number of absolute value | | equals one. Since adjoints are the operator analog of complex conjugation, we expect that an arbitrary operator T (H) can be written in the form ∈ B

T = U√T ∗T

where U is an isometry. Amazingly, this works! Since √T ∗T is well-defined. This is the basic idea behind the polar decomposition theorem.

Theorem 1.4.3 Let M be a dense subspace of a normed space X . Let T be a bounded linear operator from M to a Banach space Y. Then there exist T˜ which is the unique extension of T from X to Y , with T x = T˜ x . k k

Theorem 1.4.4 Let S and T be bounded linear operators on a complex Hilbert space H.

If T ∗T = S∗S, then there exists a partial isometry operator U with initial space M = ran (T ) and final space N = ran (S), and S = UT.

Theorem 1.4.5 Let T be a bounded linear operator on a complex Hilbert space H. Then the following hold : 1) There exists a partial isometry operator U such that

T = U T | |

23 1 where T = (T ∗T ) 2 , M and N are initial and final spaces of U, respectively, can be | | expressed as follows: M = ran ( T ) and N = ran (U). | | 2) ker (U) = ker ( T ) and U ∗U T = T . | | | | | | 3) ran ( T ) = ran (T ). | | ∗

Proof. 1) Follows immediately from Theorem 1.4.3.

2) Let x M ⊥, then Ux = 0, so x ker (U). Hence M ⊥ ker (U). ∈ ∈ ⊆ Conversely, let x ker (U) then Ux = 0, but M is closed then x can be written ∈ uniquely as, x = y + z : y M and z M ⊥. Thus ∈ ∈

0 = Ux = Uy = y . k k k k k k

Hence, y = 0 and x = z M ⊥. Therefore, ker (U) M ⊥, hence ker (U) = M ⊥. By using ∈ ⊆ the fact that T is self-adjoint and | |

ran ( T ) = ker ( T )⊥ | | | |

we have ker (U) = ker ( T ) . | | By using (1) above, and by Theorem1.4.2 we have, U ∗T = U ∗U T = PM T . But M | | | | ⊃ ran ( T ), therefore, PM T = T . Hence U ∗U T = T . | | | | | | | | | | 3) Since T = U T , then | | U ∗T = U ∗U T = T . | | | | Then,

ran ( T ) = ran ( T ∗) = ran (T ∗U) ran (T ∗) . | | | | ⊆

Conversely, since T ∗ = T U ∗, then | |

ran (T ∗) = ran ( T U ∗) ran ( T ) . | | ⊆ | |

Therefore, ran (T ∗) =ran ( T ). | |

24 Lemma 1.4.6 Let H be a complex Hilbert space, and S (H) be a positive operator. ∈ B Then 1) Sx, x = 0 holds for some x H if and only if Sx = 0. h i ∈ 2) ker (Sq) = ker (S) holds for any positive real number q.

Theorem 1.4.7 Let T (H), H is a complex Hilbert space, and let T = U T be the ∈ B | | polar decomposition of T . Then the following holds : 1) ker (T ) = ker ( T ) . | | q q 2) T ∗ = U T U ∗ for any positive number q. | | | |

Theorem 1.4.8 Let (H), where H is a complex Hilbert space, and let T = U T be ∈ B | | the polar decomposition of T . Then T ∗ = U T ∗ is the polar decomposition of T ∗. | |

Corollary 1.4.9 Let (H), where H is a complex Hilbert space, and let T = U T be ∈ B | | the polar decomposition of T . Then

q q T = U ∗ T ∗ U for any positive number q. | | | |

1.5 Positive operators

Theorem 1.5.1 Suppose T (H) , then ∈ B (1) (T x, x) 0 for every x H if and only if ≥ ∈ (2) T = T ∗ and σ (T ) [0, [. ⊂ ∞

If T (H) satisfies (1), we call T a positive operator and write T 0. ∈ B ≥ Proof. In general, (T x, x) and (x, T x) are complex conjugates of each other. But if (1) holds, then (T x, x) is real , so that

(x, T ∗x) = (T x, x) = (x, T x)

25 for every x H. By Corollary (1.3.2), T = T ∗, and thus σ (T ) lien in the real axis. If ∈ λ > 0, (1) implies that

λ x 2 = (λx, x) ((T + λI) x, x) (T + λI) x x k k ≤ ≤ k k k k

so that (T + λI) x λ x . k k ≥ k k By Theorem (1.3.6), T + λI is invertible in (H) , and λ is not in σ (T ) . It follows that B − (1) implies (2). Assume now that (2) holds, and let E be the spectral decomposition of T , so that

(T x, x) = λdEx,x(λ)(x H) . ∈ Zσ(T )

Since each Ex,x is a positive measure, and since λ 0 on σ (T ) , we have (T x, x) 0. ≥ ≥ Thus (2) implies (1).

Theorem 1.5.2 Every positive T (H) has a unique positive square root S (H) . ∈ B ∈ B If T is invertible, so is S.

Theorem 1.5.3 If T (H) , then the positive square root of T ∗T is the only positive ∈ B operator P (H) that satisfies P x = T x for every x H. ∈ B k k k k ∈

The fact that every complex number λ can be factored in the form λ = α λ , where | | α = 1, suggests the problem of trying to factor T (H) in the form T = UP, with U | | ∈ B is unitary and P 0. When this possible we call UP a polar decomposition of T. ≥

Theorem 1.5.4 Let T (H) , then ∈ B (1) If T is invertible, then T has a unique polar decomposition T = UP. (2) If T is normal, then T has a polar decomposition T = UP in wich U and P commute which each other and with T .

26 In (1), no two of T, U, P need to commute. For example

0 1 0 1 2 0 =  2 0   1 0   0 1        The polar decomposition leads to an interesting result concerning similarity of normal operators.

Theorem 1.5.5 Suppose M,T,N (H) ,M and N are normal, T is invertible, and ∈ B

1 M = TNT − . (1.5)

If T = UP is the polar decomposition of T, then

1 M = UNU − . (1.6)

Two operators which satisfy (1.5) are usually called similar. If U is unitary and (1.6) holds, M and N are said to be unitarily equivalent.

1.6 Numerical range

In this section we will study the basic properties of the numerical range of an operator. As the numerical range and radius of an operator are intimately connected, we will draw more information about the numerical radius in this section. We begin with the definition of the numerical range.

Definition 1.6.1 Let T (H) , The numerical range of T , denoted W (T ), is the non- ∈ B empty set W (T ) = (T x, x) for some x = 1 { k k }

Proposition 1.6.2 Let T,S (H) , then ∈ B (1) W (T ∗) = W (T )∗ .

27 (2) W (T ) contains all of the eigenvalues of T .

(3) If U (H) is unitary then W (UTU ∗) = W (T ) . ∈ B (4) W (T ) R if and only if T is self-adjoint. ⊂ (5) If H is finite dimensional, W (T ) is closed and thus compact. (6) W (T + S) W (T ) + W (S) . ⊂

Theorem 1.6.3 (Toeplitz-Hausdorff [Hau19][Toe18]) Let T (H) , then W (T ) ∈ B is convex.

Theorem 1.6.4 Let T (H) , then σ (T ) W (T ). ∈ B ⊂

The numerical radius of an operator T (H) on a nonzero complex Hilbert space H ∈ B is the nonnegative number

w (T ) = sup λ = sup (T x, x) . λ W (T ) | | x =1 | | ∈ k k It is ready verified that

2 w (T ∗) = w (T ) and w (T ∗T ) = T . k k

The numerical radius is a norm on (H). That is, 0 w (T ) for every T (H) B ≤ ∈ B and 0 < w (T ) if T = 0, w (αT ) = α w (T ) , and w (T + S) w (T ) + w (S) for every 6 | | ≤ α C and T,S (H) . However, the numerical radius does not have the operator norm ∈ ∈ B property in the sense that the inequality w (TS) w (T ) w (S) is not true for all operators ≤ T,S (H) . Moreover, the numerical radius is a norm equivalent to the operator norm ∈ B of (H) ,as in the next theorem. B

Theorem 1.6.5 Let T (H) , then ∈ B

0 r (T ) w (T ) T 2w (T ) . ≤ ≤ ≤ k k ≤

28 Chapter 2

Non-normal operator classes

In this chapter, we will investigate some classes of bounded operators such as hyponormal operators, normaloid operators, and convexoid operators. We will also be discussing the relations between them.

2.1 Compact operators

Let X and Y be normed spaces. A linear transformation T : X Y is compact if its → maps bounded sets into relatively compact subsets of Y. That is, T is compact if T (A) is compact in Y whenever A is bounded in X. Let (X,Y ) denote the collection of B∞ all compact linear transformation of a normed space X into a normed space Y so that (X,Y ) (X,Y ). Set (X) = (X,X) for short, the collection of all compact B∞ ⊆ B B∞ B∞ operators on a normed space X. (X) is an ideal of the normed algebra (X) . That B∞ B is, (X) is a subalgebra of (X) such that the product of a compact operator with a B∞ B bounded operator is again compact. We assume that the compact operators act on a complex nonzero Hilbert space H, although the theory for compact operators equally applies (and is usually developed ) for operators on Banach spaces.

29 Theorem 2.1.1 Let T : H H be a linear operator , then the following statements are → equivalent : 1) T (H) ∈ B∞ 2) x * 0 T xn 0. ⇒ →

Theorem 2.1.2 ([Kub12]) If T (X,Y ) and λ C 0 , then ran (T λI) is ∈ B∞ ∈ \{ } − closed.

Theorem 2.1.3 If T (X,Y ) , λ C 0 , and ker (T λI) = 0 , then ∈ B∞ ∈ \{ } − { }

ran (T λI) = H. −

Theorem 2.1.4 (Fredholm Alternative) If T (X,Y ) and λ C 0 , then ∈ B∞ ∈ \{ } ran (T λI) is closed and dim ker (T λI) = dim ker T ∗ λI < . − − − ∞
Theorem 2.1.5 (Fredholm Alternative) Let T (X,Y ) and λ C 0 , then ∈ B∞ ∈ \{ } λ ρ (T ) σp (T ). Equivalently, ∈ ∪

σ (T ) 0 = σp (T ) 0 \{ } \{ }

Corollary 2.1.6 Let T (X,Y ). ∈ B∞ (a) 0 is the only possible accumulation point of σ (T ). (b) If λ σ (T ) 0 , then λ is an isolated point of σ (T ). ∈ \{ } (c) σ (T ) 0 is a discrete subset of C. \{ } (d) σ (T ) is countable.

2.2 Hyponormal operators

In this section we will first examine some general properties of hyponormal operators. Then we continue with a general discussion of a certain growth condition on the resolvent set which obtains for hyponormal operators.

30 2.2.1 Definitions and properties

Definition 2.2.1 An operator T (H) is hyponormal if T ∗T TT ∗, which is equiva- ∈ B ≥ lent to the condition T ∗x T x . An operator T (H) is cohyponormal if its adjoint k k ≤ k k ∈ B is hyponormal . If it is either hyponormal or cohyponormal, then it is called seminormal.

Proposition 2.2.2 Let T (H) , then T is hyponormal operator if and only if T ∗T + ∈ B 2 2λT T ∗+ λ T ∗T > 0, for all λ R. ∈

Proof. Let λ R and x H be given. T is hyponormal operator if and only if ∈ ∈ 2 2 2 2 T ∗x T x T x + 2λ T ∗x + λ T x 0 k k ≤ k k ⇔ k k k k k k ≥ 2 T x, T x + 2λ T ∗x, T ∗x + λ T x, T x 0 ⇔ h i h i h i ≥ 2 T ∗T x, x + 2λ TT ∗x, x + λ T ∗T x, x 0 ⇔ h i h i h i ≥ 2 T ∗T + 2λT T ∗ + λ T ∗T x, x 0 ⇔ ≥ 2 T ∗T + 2λT T ∗ + λ T ∗T >
0 ⇔

Remark 2.2.3 If T (H) is hyponormal, then (T λI) is hyponormal for every ∈ B − λ C. ∈

1 Proposition 2.2.4 Let T (H), and λ C, if T is hyponormal and (T λI)− exists, ∈ B ∈ − 1 then (T λI)− is hyponormal. −

Proof. Since hyponormality is preserved under translation, we may assume λ = 0. Thus

T ∗T TT ∗ 0 and hence − ≥

1 1 1 1 0 T − (T ∗T TT ∗) T ∗− = T − T ∗TT ∗− I. ≤ − −

1 Now since A I implies A− I we have ≥ ≤

1 1 I T ∗T − T ∗− T 0 − ≥

31 and hence 1 1 1 1 1 1 1 1 T ∗− T − T − T ∗− = T ∗− I T ∗T − T ∗− T T − 0 − − ≥ which completes the proof.

Lemma 2.2.5 Let T (H), and λ C , if ker (T λI) ker T ∗ λI , then ∈ B ∈ − ⊆ − (a) ker (T λI) ker (T νI) whenever ν = λ, and
− ⊥ − 6 (b) ker (T λI) reduces T . −

Proof. (a) Let x ker (T λI) and y ker (T νI). Thus T x = λx and T y = νy. if ∈ − ∈ − ker (T λI) ker T ∗ λI , then x ker T ∗ λI , and so T ∗x = λx. Then − ⊆ − ∈ −

νy, x = T y, x = y, T ∗x = y, λx = λy, x h i h i h i h i

and hence (λ ν) y, x = 0 − h i which implies that y, x = 0 whenever ν = λ. h i 6 (b) If x ker (T λI) ker T ∗ λI , then T x = λx and T ∗x = λx. Thus ker (T λI) ∈ − ⊆ − − is T ∗-invariant. But ker (T λI) is T -invariant.
Therefore ker (T λI) reduce T. − −

Corollary 2.2.6 if T (H) is hyponormal, then ∈ B (a) ker (T λI) ker (T νI) whenever ν = λ, and − ⊥ − 6 (b) ker (T λI) reduce T. −

Theorem 2.2.7 If λγ γ Γ is a (nonempty) family of distinct complex numbers (where { } ∈ Γ is nonempty index set), and if T (H) is hyponormal, then the topological sum ∈ B

= ker (T λγI) M − γ Γ ! X∈ reduces T , and the restriction of T to it, T ( ), is normal. |M ∈ B M

32 Proposition 2.2.8 ([Kub12]) T (H) is hyponormal if and only if T ∗x T x ∈ B k k ≤ k k for every x H. Moreover, the following assertions are pairwise equivalent ∈ (1) T is normal.

(2) T n is normal for every positive integer n N. ∈ n n (3) T ∗ x = T x for every x H and every n N. k k k k ∈ ∈

2.2.2 Some conditions implying normality or self-adjointness

In this subsection, we will give some important classical results related to hyponormal operators.

Theorem 2.2.9 ([Sta65]) If T is hyponormal and σ (T ) is an arc, then T is normal.

Corollary 2.2.10 If T is hyponormal and σ (T ) is real, then T is self-adjoint.

Corollary 2.2.11 If T is hyponormal and σ (T ) lies on the unit circle, then T is unitary.

Definition 2.2.12 An operator T is quasi-normal if (T ∗T ) T = T (T ∗T ). An operator T on a Hilbert space H is subnormal if there exists a Hilbert space K, K H, and a ⊇ normal operator S defined on K with T x = Sx for x H. ∈

Remark 2.2.13 One has the following inclusion relation for classes of operators:

Normal Quasi-normal Subnormal Hyponormal. ⊂ ⊂ ⊂

Theorem 2.2.14 If T is quasi-normal and σ (T ) has no interior, then T is normal.

Theorem 2.2.15 Let T be hyponormal with λ ρ (T ). Then ∈

1 1 (T λI)− − ≤ d (λ, σ (T ))

or, equivalently, (T λI) d (λ, σ (T )) , where d (λ, σ (T )) = min λ w : w σ (T ) k − k ≥ {| − | ∈ }

33 Proof. Let λ ρ (T ) , x H and, x = 1, then ∈ ∈ k k 1 1 1 (T λI)− x (T λI)− = max w : w σ (T λI)− − ≤ − | | ∈ − 1 = min w : w σ(T λI)
{| | ∈ − } 1 = min w λ : w σ(T ) {| − | ∈ } 1 = d(λ,σ(T ))

It will be convenient to refer to the conclusion of the above theorem by stating that T satisfies condition G1 ; i.e.the resolvent of T has exactly first order rate of growth with respect to the spectrum of T.

Theorem 2.2.16 ([Nie62]) If T satisfies condition G1 and σ (T ) is real, then T is self- adjoint.

Theorem 2.2.17 ([Don63]) If T satisfies condition G1 and σ (T ) lies on the unit circle, then T is unitary

Theorem 2.2.18 ([Sta65]) If T satisfies condition G1 and σ (T ) is a finite set of points, then T is normal.

However, if T is compact and satisfies condition G1 , T need not be normal. We will sketch a simple example to illustrate this. The operator

0 1 T1 = 0 0   does not satisfies G1 .

We will now define an operator T2 in such a manner that T = T1 T2 does satisfy ⊕ condition G1 and moreover is compact. Let fi ∞ be an orthogonal basis for H2. We { }i=1 now set T2fi = aifi where the ai’sare complex numbers placed on circles concentric to the origin with suffi cient density to ensure that

2 min λ ai λ : for each, 0 < λ < 1 . i | − | ≤ | | | |  34 . This can clearly be done with zero as the only limit point of the ai’s. The operator

T = T1 T2 defined on H1 H2 is completely continuous and satisfies condition G1 by ⊕ ⊕ construction but it is obviously not normal. This example also illustrates that if T satisfies condition G1, and M is a reducing subspace of T then T M may not satisfy | condition G1.

Theorem 2.2.19 ([Halm50]) Let T (H) ∈ B

2 T is hyponormal ; T is hyponormal .

S. Berberian has asked whether an operator must be subnormal if all its powers are hyponormal. In fact J. G. Stampfeli ([Sta65]) gives a negative answer to that question. + Let fi i=∞ be an orthonormal basis for H and define { } −∞

fi+1, i 0 T fi = ≤ .  2f , i > 0  i+1

k k Then T fi = bi,kfi+k where bi,k bi+1,k , so T is hyponormal for k = 1, 2,.... Since | | ≤ | |

2 T f0 = T ∗f0 but T ∗T f0 = T f0 , k k k k 6

we must conclude that T is not subnormal.

Theorem 2.2.20 ([Sta62], Theorem 5) Let T be a hyponormal operator with T n = B, where n is a positive integer and B is a normal operator; then T is normal.

2.2.3 p-Hyponormal operators

The semi-hyponormal operator was first introduced by Professor D. Xia. He also provides an example of a semi-hyponormal operator which is not hyponormal. In this section we shall study p-Hyponormal operators for p > 0.

35 Definition 2.2.21 An operator T (H) is said to be p-hyponormal if ∈ B p p (TT ∗) (T ∗T ) ≤ for a positive number p.

1 Remark 2.2.22 If p = 1, T is hyponormal and if p = 2 T is semi-hyponormal.

The following inequality is called Löwner-Heinz’sinequality.

Proposition 2.2.23 ([Löw34], [Hei51]) Let A, B (H) satisfy 0 B A and ∈ B ≤ ≤ 0 < p < 1. Then Bp Ap. ≤ By Löwner-Heinz’sinequality, every p-hyponormal operator is q-hyponormal if 0 < p < q. There exists a q-hyponormal operator which is not p-hyponormal if 0 < q < p.

Theorem 2.2.24 (Furuta’sinequality [Fur87]) If A B 0, then the inequalities ≥ ≥ (p+2r) r p r 1 (B A B ) q B q ≥ and (p+2r) r p r 1 A q (A B A ) q ≥ hold for p, r 0, q 1 with (1 + 2r)q p + 2r. ≥ ≥ ≥ Theorem 2.2.25 (Hansen’sinequality [Han80]) If A 0 and B 1, then ≥ k k ≤

p p (B∗AB) B∗A B ≥ for 0 p 1. ≤ ≤ Theorem 2.2.26 ([AW99]) Let 0 p 1. Let T be a p-hyponormal operator. The ≤ ≤ inequalities p p n n n P P n n n T ∗ T (T ∗T ) (TT ∗) T T ∗ ≥ ≥ ≥ hold for all positive integer n.

36 Proof. Let T = U T be the polar decomposition of T . For each positive integer n, let p | | p n n n n n n An = T ∗ T and Bn = T T ∗ We will use induction to establish the inequalities

An A1 B1 Bn. (2.1) ≥ ≥ ≥

The inequalities (2.1) clearly hold for n = 1. Assume (2.1) hold for n = k. The induction hypothesis and the assumption that T is p-hyponormal imply

U ∗AnU U ∗A1U A1. ≥ ≥ p k k p Let Ck = U ∗Ak U . Hansen’sinequality implies  

Ck U ∗AkU A1. ≥ ≥

Thus p k+1 k k+1 Ak+1 = T ∗ T p  k k k+1 = T ∗ T ∗ T T p k k+1   p   = T U ∗A U T | | k | |  p 1 k 1 k+1 2p p 2p = A1 Ck A1   A1 ≥ by Furuta’sinequality. On the other hand, the induction hypothesis implies

Bk B1 A1. ≤ ≤

37 Thus p k+1 k+1 k+1 Bk+1 = T T ∗ p k k+1  p  = TBk T ∗   p k k+1 p = U T B T U ∗ | | k | |  p k k+1 p = U T B T U ∗ | | k | |   p 1 k 1 k+1 2p p 2p = U A1 Bk A1 U ∗   UA1U ∗ ≤ = B1 where the inequality follows from Furuta’sinequality. Therefore,

Ak+1 A1 B1 Bk+1 ≥ ≥ ≥

and hence, by induction, inequalities (2.1) holds for n 1. ≥

Corollary 2.2.27 ([AW99]) Let 0 p 1. If the operator T is p-hyponormal, then T n ≤ ≤ p is n -hyponormal.
Concrete examples of non-hyponormal p-hyponormal operators are hard to come by. In [Xia80], Xia gave an example of a singular integral operator which is semi-hyponormal but not hyponormal. The above Corollary allows us to give another example of a semi- hyponormal operator which is not hyponormal. Let A be the operator in Halmos’sbook, thus, A is hyponormal but A2 is not hyponormal. By the above Corollary , A2 is semi-

2n 1 hyponormal. Moreover, A is 2n -hyponormal.
Proposition 2.2.28 Let 0 p 1. If T is p-hyponormal and T n is normal, then T is ≤ ≤ normal.

38 2.3 Normaloid operators

Definition 2.3.1 An operator T (H) is normaloid if r (T ) = T , where ∈ B k k 1 r (T ) = sup λ : λ σ (T ) = lim T n n n {| | ∈ } →∞ k k is the spectral radius of T.

Proposition 2.3.2 r (T ) = T if and only if T n = T n . k k k k k k Theorem 2.3.3 Every hyponormal operator is normaloid.

Proof. Let T (H) be a hyponormal operator on a Hilbert space H. ∈ B n 2 n+1 n 1 Claim 1 T T T − for every positive integer n. k k ≤ k k k k First note that, for any operator T (H) , ∈ B

n 2 n n n n 1 n n 1 T x = (T x, T x) = T ∗T x, T − x T ∗T x T − x k k ≤ k k
for each n 1 and every x H. Now if T is hyponormal, then ≥ ∈

n n 1 n+1 n 1 n+1 n 1 2 T ∗T x T − x T x T − x T T − x k k ≤ ≤ k k and hence for each n 1 ≥

n 2 n+1 n 1 2 T x T T − x k k ≤ k k which ensures the claimed result, thus completing the proof of Claim 1. Claim 2 T n = T n for every n 1. k k k k ≥ The above result holds trivially if T = 0 and it also holds trivially for n = 1. Let T = 0 6 and suppose the above result holds for some integer n 1. By Claim 1 we get ≥

2n n 2 n 2 n+1 n 1 n+1 n 1 T = ( T ) = T T T − T T − . k k k k k k ≤ ≤ k k

Therefore, as T n T n , and since T = 0 , k k ≤ k k 6

n+1 2n n 1 1 n+1 n+1 T = T T − − T T . k k k k k k ≤ ≤ k k

39 Hence T n+1 = T n+1 . Then the claimed result holds for n + 1 whenever it holds for k k k k n, which concludes the proof of Claim 2 by induction. Therefore T n = T n for every integer n 1 by Claim 2, and so T is normaloid. k k k k ≥ n n Since T ∗ = T for each n 1, it follows that r (T ∗) = r (T ) . Thus an operator T k k k k ≥ is normaloid if and only if its adjoint T ∗ is normaloid, and so every seminormal operator is normaloid.

Proposition 2.3.4 An operator T is normaloid if and only if

T = sup T x, x . k k x =1 {|h i|} k k Definition 2.3.5 For a compact convex subset X of the plane, a point λ X is bare if ∈ there is a circle through λ such that no points of X lie outside this circle.

Theorem 2.3.6 ([SY65]) Let T be an operator such that (T λI) is normaloid for − every complex number λ , then we have

W (T ) = co σ (T ) .

Where co σ (T ) denotes the convex hull of σ (T ) .

To prove the theorem stated above we need the following lemma.

Lemma 2.3.7 Let T be an operator and λ W (T ) a bare point of W (T ), then there ∈ exists a complex number λ0 satisfying

λ λ0 = sup µ λ0 : µ W (T ) . | − | | − | ∈ n o Lemma 2.3.8 Let C be a nonempty compact convex subset of the plane, and let S be the collection of all of its bare points. Then C is the closed convex hull of S.

For convenience we state the following known result as a lemma.

40 Lemma 2.3.9 ([Orl63]) For an operator T , and λ W (T ) , and λ = T imply ∈ k k k k λ σ (T ) . ∈ Proof of Theorem. It is suffi cient to show that each bare point of W (T ) belongs to

σ (T ) ( Lemma 2.3.9 ). Let λ be a bare point of W (T ) , there is a λ0 satisfying

λ λ0 = sup µ λ0 : µ W (T ) | − | | − | ∈ n o by lemma 2.3.8 . Thus, by the hypothesis on T and the fact W (T λ0) = W (T ) λ0, − − we have

T λ0I = λ λ0 . k − k | − |

Since λ λ0 W (T λ0), λ λ0 σ (T λ0I) by the above Lemma and so we have − ∈ − − ∈ − λ σ (T ) . Hence the proof is completed. ∈ In [Ber62], S. K. Berberian conjectured that the closure of the numerical range of a hyponormal operator coincides with the convex hull of its spectrum. According to Theorem 2.3.6, we can give an affi rmative answer to his conjecture.

Corollary 2.3.10 For a hyponormal operator T, W (T ) = co σ (T ) .

Corollary 2.3.11 If an operator T (H) is compact and normaloid, then σp (T ) = ∈ B 6 ∅ and there exists λ σp (T ) such that λ = T . ∈ k k k k Theorem 2.3.12 ([Kub12]) Every compact hyponormal operator is normal.

Proof. Suppose T (H) is a compact hyponormal operator on a nonzero complex ∈ B Hilbert space H. The above corollary says that σp (T ) = . Consider the subspace 6 ∅

= ker (T λI) M  −  λ σp(T ) ∈X   of Theorem 2.2.7 with λγ γ Γ = σp (T ). Observe that { } ∈

σp (T ) = . |M⊥ ∅

41 Indeed, if there is a λ σp (T ), then there exists ∈ |M⊥

0 = x ⊥ such that λx = T x = T x 6 ∈ M |M⊥ , and so x ker (T λI) , which is a contradiction. Moreover, recall that T is ∈ − ⊆ M |M⊥ compact and hyponormal . Thus, if ⊥ = 0 , then Corollary 2.3.11 says that M 6 { }

σp (T ) = |M⊥ 6 ∅

which is another contradiction. Therefore, ⊥ = 0 so that = , and hence M { } M H

T = T H = T | |M is normal according to Theorem 2.2.7 .

2.4 Paranormal operators

In this section we discuss a class of paranormal operators. In [Ist66] this is named an operator of class (N) . We show that this class includes hyponormal operators and is included in the class of normaloid operators, also we will give a generalization of Theorem 2.3.12.

2.4.1 Definitions and properties

Definition 2.4.1 An operator T (H) is paranormal if T 2x T x 2 for every ∈ B k k ≥ k k unit vector x in H.

Proposition 2.4.2 Every hyponormal operator is paranormal.

Proof. In fact,

2 2 T x = T x, T x = T ∗T x, x T ∗T x T x . k k h i h i ≤ k k ≤

42 Theorem 2.4.3 Let T (H) . If T is paranormal then ∈ B 1) T is normaloid.

1 2) T − is also paranormal if T is invertible.

Lemma 2.4.4 Let T be a paranormal operator, then

k+1 2 k 2 2 T x T x T x (Pk) ≥ for a positive integer k 1, and every unit vector x in H. ≥ Proof. For the case k = 1

T 2x 2 = T 2x T 2x T 2x T x 2 ≥ k k

and (P1) is clear. Now suppose that (Pk ) is valid for k and we assume T x = 0, then k k 6 2 k+2 2 2 k+1 T x T x = T x T T x k k k k2 2 k T x 2 T x T x T T x T T x ≥ k k k k k k k+1 2 T 3x T x T x ≥ k k k+1 2 2 T x T x ≥ k k by of Lemma (??) and (P ). So (P ) is valid and the proof is complete by the mathe- k k+1 matical induction.

Theorem 2.4.5 ([Fur67]) If T is a paranormal operator, then T n is paranormal for every integer n 1. ≥ Proof. It is suffi cient to show that if T and T k is paranormal, then T k+1 is paranormal too. We may assume T 2x = 0 , then k k 6 2 2(k+1) 2 2 2 2k T 2x T x = T x T T x k k k k 2 T 2x 2 T k T 2x T x ≥ k k 2 k k T k+2x k T 2x k ≥ k k 2 T k+1x T 2x k Tk2xk k ≥ k k = T k+1x 2

43 k+1 by (Pk+1) of Lemma (2.4.4). So T is paranormal. There exists a paranormal operator which is not hyponormal. That is, the class of hy- ponormal operators is properly included in the class of paranormal operators. In [Halm50] Halmos gives an example of hyponormal operator T such that T 2 is not hyponormal. By Theorem 2.4.5, this T is paranormal. Hence we get an example of non-hyponormal, paranormal operator. In Theorem 2.3.12 we prove that every compact hyponormal operator is necessarily normal. The following Theorem is a slight generalization of it.

p1 q1 pm qm Theorem 2.4.6 ([ISY66]) Let T be a paranormal operator such that T ∗ T T ∗ T ··· is compact for some non-negative integers p1, q1, . . . pm, qm . Then T is necessarily a nor- mal operator.

2.4.2 k-paranormal operators

Definition 2.4.7 An operator T is k-paranormal, if T satisfies

T k+1x T x k+1 ≥ k k for any x H with x = 1. ∈ k k Proposition 2.4.8 If T is paranormal then T is k-paranormal.

Theorem 2.4.9 ([FHN67]) If a paranormal operator T has a compact power T k, then T is compact. However, this is not true for normaloid operators in general.

Proof. Let us suppose that

xα 0 ( weakly), xα 1. k k → k k ≤ Since T is (k 1)-paranormal, then − k k T xα k T xα k k T xα , ≥ xα ≥ k k k k

44 k which tells us that T xα converges strongly to 0, since T xα 0 by the compactness of → T . Therefore, T is compact.

To prove the remainder half of the theorem, let us put H = `2 . Define an operator T by 1 0 0 0 0 0 ···  0 0 0 0 0 0  ···    0 1 0 0 0 0   ···   T =  0 0 0 0 0 0   ···    0 0 0 1 0 0   ···    0 0 0 0 0 0   ···     ····················· with respect to the orthonormal basis 

1 0 0 0 1 0       e1 = 0 , e2 = 0 , e3 = 1 ,...             0 0 0       . . . . . .             Then, wen can easily deduce

e1 (i = 1)

T ei =  e (i = 2j) j = 1, 2,...  i+1   0 (i = 2j + 1)  Hence  T = 1 and T k = P (k 2) k k ≥ where P is the projection belonging to the subspace spanned by the scalar multiples of e1. Therefore, T k = T k = 1 k k

45 for all k, which shows that T is a normaloid. Since T k = P for k 2, T k is compact for k 2, whereas T is not compact since the ≥ ≥ range of T contains an infinite orthonormal set ei ; i = 1, 3, 5,... . The second half of { } the theorem is now proved.

2.5 Convexoid operators

In this section we study the class of convexoid operators, also we show the relation between normaloid operators and convexoid operators.

Definition 2.5.1 An operator T is called to be convexoid if

W (T ) = co σ (T ) where co σ (T ) denotes the convex hull of σ (T ) .

Definition 2.5.2 An operator T is said to be spectraloid if

w (T ) = r (T ) or equivalently n n w (T ) = w (T ) (n N∗) ∈ where w(T ) and r(T ) mean the numerical radius and the spectral radius of T respec- tively as follows :

w (T ) = sup λ ; λ W (T ) . {| | ∈ } r (T ) = sup λ ; λ σ (T ) . {| | ∈ } The following theorem gives a characterization of convexoids operators

Theorem 2.5.3 ([FN71]) An operator T is convexoid if and only if (T λI) is spec- − traloid for every complex λ.

46 2.5.1 Examples

It is known that there exist convexoid operators which are not normaloid and vice versa and the classes of normaloids and convexoids are both contained in the class of spec- traloids, and every hyponormal operator is convexoid.

Example 2.5.4 A normaloid operator need not be convexoid. Let H = C3 with the Euclidean norm given by

2 2 2 f = (f1, f2, f3) = f1 + f2 + f3 . k k k k | | | | | | Let 0 1 0 T = 0 0 0 .   0 0 1   2   Then T f = (f2, 0, f3) and T = 1. On the other hand k k

T f, f = f2f1 + f3f3 h i and consequently,

w (T ) = sup f2f1 + f3f3 = 1 f =1 k k  by taking f = (0, 0, 1) . Hence T is normaloid. It can be verified easily that σ (T ) = 0 1 { } ∪ { } and W (T ) is the closed convex set spanned by the disc λ : λ 1 and one point 1. | | ≤ 2 Hence T is not convexoid. 

Example 2.5.5 A convexoid operator need not be normaloid. Let x1, x2,... be an { } orthonormal base for H = `2. Define zn = x2n+1, n = 0, 1, 2,... and z n = x2n, n = ∼ 0, 1, 2,.... Every x in H can be written as

∞ x = αkzk. k= X−∞

47 Let now define the operator S on H by

1 ∞ Sx = α z 2 k k+1 k= X−∞

∞ where x = αkzk. we can check easily that k= −∞ P 1 W (T ) = λ C : λ . ∈ | | ≤ 2   Let us define the operator

0 0 2 L = on C . 0 1   The operator defined on H C2 by ⊕

T (f, g) = (Lf, Sg)

yields 1 W (T ) = λ C : λ = co σ (T ) ∈ | | ≤ 2   T is not normaloid since 1 T = 1 w (T ) = . k k 2

Example 2.5.6 An example of non-convexoid, non-paranormal, normaloid operator.

Let T be an infinite matrix of the form

1 0 0 0 ···  0 M 0 0  ··· 0 0   T =  0 0 M 0  where M = .  ··· 1 0    0 0 0 M   ···       ···············  

48 Then it is clear that T is normaloid , non-paranormal because

1 0 0 0 ···  0 0 0 0  ··· 2   T =  0 0 0 0   ···    0 0 0 0   ···     ···············   and T n = T n = 1. However the relation T 2x T x 2 does not hold for the unit k k k k k k ≥ k k vectors e2(0, 1, 0, 0, ), e4(0, 0, 0, 1, 0, 0, ) etc. T is non-convexoid. In fact W (T ) is the ··· ··· closed convex, set spanned by the disc λ : λ 1 and one point 1, σ (T ) = 0 1 , so | | ≤ 2 { }∪{ } the convex hull of σ (T ) is the closed unit interval [0, 1] , and this unit interval is properly included in W (T ).

Example 2.5.7 An example (T. Ando) of non-hyponormal, paranormal convexoid oper- ator . T. Ando has given the following concrete example as follows : when H is a complex ∞ Hilbert space, K denotes the infinite direct sum of copies of H, i.e. K = Hk (Hk ∼= H) . k=1 M

Given two bounded positive operators A and B on H, the infinite matrix TA,B,n is defined on K, which assigns to a vector

x = (x1, x2, ) the vector y = (y1, y2, ) ··· ···

49 such that, y1 = 0, yj = Axj 1 (1 < j n) and yj = Bxj 1 (n < j) , that is , − ≤ − 0 A 0     A.       A.       ..  TA,B,n =   .    ..       A 0       B 0       B.       ..     T. Ando shows that this operator TA,B,n is paranormal if and only if

AB2A 2λA2 + λ2 0 (λ > 0) − ≥ and that it is hyponormal if and only if B2 A. He observed the operator ≥ 1 1 1 1 2 T = TA,B,n with A = C 2 ,B = C− 2 DC− 2   where 1 1 1 2 C = and D = , 1 2 2 8 then T is paranormal , but the tensor product T Tis not paranormal. He shows that ⊗ this paranormal operator T is convexoid and non-paranormal T T is also convexoid. ⊗

Example 2.5.8 A normaloid operator need not be convexoid. In H = C3, consider the operator 1 0 0 T = 0 0 0 .   0 1 0    

50 We have, as in example 2.5.4, that σ (T ) = 0, 1 and W (T ) = co σ (T ) ,S , where { } { } 1 S = λ C : λ . ∈ | | ≤ 2   Example 2.5.9 A slight modification of the above example produces a spectraloid operator that is not normaloid. In H = C3, let the operator

1 0 0 T = 0 0 0 .   0 2 0     We have T = 2 and w (T ) = r (T ) = 1. k k

Since a convexoid operator is not always a normaloid by the above examples.

2.6 Class A operators

In this section We shall introduce a new class “class A” given by an operator inequal- ity which includes the class of log-hyponormal operators and is included in the class of paranormal operators.

Definition 2.6.1 An operator T (H) is called a log-hyponormal operator if T is ∈ B invertible and

log (TT ∗) log (T ∗T ) . ≤

Remark 2.6.2 Since log :]0, ] R is operator monotone, for 0 < p < 1, every ∞ → invertible p-hyponormal operator T , is log-hyponormal.

Definition 2.6.3 ([FIY98]) An operator T belongs to class A if

T 2 T 2 . ≥ | |

51 We would like to remark that class "A " is named after the "‘absolute" values of two operators T 2 and T . We call an operator T class A operator briefly if T belongs to | | | | class A. We obtain the following results on class A operators.

Theorem 2.6.4 1) Every log-hyponormal operator is class A operator. 2) Every class A operator is paranormal operator.

The following theorems and lemma play an important role in the proof of the above theorem.

Theorem 2.6.5 ([FFK93][Fur92]) Let A and B be positive invertible operators. Then the following properties are mutually equivalent 1) log A log B. ≥ p p 1 p p 2 2) A A 2 B A 2 for all p 0. ≥ ≥ r r r r p r+p 3) A A 2 B A 2
for all p 0 and r 0. ≥ ≥ ≥
Theorem 2.6.6 (Hölder-McCarthy inequality [McC67]) Let A be a positive oper- ator. Then the following inequalities hold for all x in H :

r r 2(1 r) 1) A x, x Ax, x x − for 0 < r 1. h i ≤ h i k k ≤ r r 2(1 r) 2) A x, x Ax, x x − for r 1. h i ≥ h i k k ≥

Lemma 2.6.7 ([Fur95]) Let A and B be invertible operators. Then

λ (λ 1) (BAA∗B∗) = BA (A∗B∗BA) − A∗B∗ holds for any real number λ .

Proof of Theorem 2.6.4. 1) Suppose that T is log-hyponormal. T is log-hyponormal iff 2 2 log T log T ∗ . (2.2) | | ≥ | |

52 By the equivalence between (1) and (2) of Theorem 2.6.5, (2.2) is equivalent to

1 2p p 2p p 2 T T T ∗ T for all p 0. (2.3) | | ≥ | | | | | | ≥
Put p = 1 in (2.3), then we have

1 2 2 2 T T T ∗ T (2.4) | | ≥ | | | | | |

2
By Lemma 2.6.7 and T ∗ = TT ∗,(2.4) holds iff | | 1 2 2 2 T T T T ∗ T T − T ∗ T | | ≥ | | | | | |
iff 1 2 2 T ∗ T T T ∗T | | ≥ so that
T 2 T 2 ≥ | |

that is, T is class A. 2) Suppose that T is class A , i.e.,

T 2 T 2 . ≥ | |

Then for every unit vector x in H ,

2 2 2 2 T x = (T )∗ T x, x k k 2 = T 2 x, x | | DT 2 x, x 2E by (2) of Theorem 2.6.6 ≥ h| | i 2 T 2 x, x ≥ | | 4 = T x k k Hence we have T 2x T x 2 for every unit vector x in H ≥ k k

so T is paranormal. Whence the proof of Theorem 2.6.4 is complete.

53 2.6.1 Quasi-class A operators

In this section we introduce quasi-class A operators, denoted , satisfying QA 2 2 T ∗ T T T ∗ T T and we prove basic structural properties of these operators. The | | ≥ | | quasi-class A operators were introduced , and their properties were studied in [JK06].

Definition 2.6.8 ([JK06]) An operator T (H) is quasi-class A if ∈ B

2 2 T ∗ T T T ∗ T T. ≥ | |

We denote the set of quasi-class A operators by To be shown in the next example, QA the class of quasi-class A operators properly contains classes of class A operators.

Example 2.6.9 ([JK06]) First, we consider finite dimensional Hilbert space operators.

Let H = C2 and let 0 0 T = . 1 0 Then by simple calculations we see that T is not paranormal with the unit vector (1, 0) and even not normaloid but quasi-class A. There exists an example that T is not paranormal but quasi-class A and normaloid; if

1 0 0 T = 0 0 0   0 1 0     then T is not paranormal but quasi-class A and normaloid. Now we consider unilateral weighted shift operators as an infinite dimensional Hilbert space operator. Recall that given a bounded sequence of positive numbers α : α0, α1,... (called weights), the unilateral 2 weighted shift Wα associated with α is the operator on H = ` defined by

Wαen = αnen+1 for all n 0, ≥

54 2 where en ∞ is the canonical orthonormal basis for ` . We easily see that Wα can be { }n=0 never normal, and so in general it is used to giving some easy examples of non-normal

operators. It is well known that Wα is hyponormal if and only if α is monotonically in-

creasing. Also, straightforward calculations show that Wα is class A if and only if α is monotonically increasing. It is meaningless to use this characterization for distinguishing some gaps between hyponormal operators and class A operators. However, for oper- QA ators, Wα has a very useful characterization. Indeed, simple calculations show that Wα belongs to if and only if QA 0   α0 0   Wα =  α1 0       α2 0     .. ..   . .     1 where α0 is arbitrary and α1 α2 α3 . So if Wα has weights α0 = 2 and αi = ≤ ≤ ≤ · · · i (i 1),then Wα is quasi-class A but not normaloid because Wα = 2 = 1 = r (Wα). ≥ k k 6 Theorem 2.6.10 ([JK06]) Let T and T not have a dense range. Then ∈ QA AB T = on H = ran (T ) + ker (T ∗)⊥ , 0 0    where A = T is the restriction of T to ran (T ) , and A is class A operator. |ran (T ) Moreover, σ (T ) = σ (A) 0 . ∪ { } Theorem 2.6.11 ([JK06]) Let T and its invariant subspace. Then the re- ∈ QA M striction T of T to is also a operator. |M M QA Theorem 2.6.12 ([JK06]) Let T and (T λI) x = 0 for some λ = 0,then ∈ QA − 6

(T λI)∗ x = 0. −

55 Chapter 3

Similarities involving bounded operators

The purpose of this chapter is finding any conditions implying that a bounded operator T is self-adjoint or unitary.

3.1 Introduction

In this section, we talk about similarity in C∗-algebra , by giving a classical result of A Berberian.

Definition 3.1.1 A unitary element u of C∗-algebra is said to be cramped if its spec- A trum is contained in some semicircle of the unit circle

iθ σ (u) e : θ0 < θ < θ0 + π . ⊂  Beck and C.R.Putnam showed that if T is a bounded operator which is unitarily equivalent to its adjoint T ∗, via cramped unitary operator U, necessarily T is self-adjoint. The proof in [BP56] utilizes the spectral resolution of U. The purpose of the next theorem is to give a proof in which the spectral resolution is replaced by an application of the Cayley transform.

56 Theorem 3.1.2 ([Ber62]) Let be a C∗-algebra, if u is a cramped unitary element of A , and z is an element of such that A A

uzu∗ = z∗

then z is self-adjoint.

Corollary 3.1.3 ([BP56]) Let U, T (H) , if u is a cramped unitary such that ∈ B

UTU ∗ = T ∗

then T ∗ = T.

Theorem 3.1.4 ([McC64]) Let φ be a linear transformation on a algebra with involu- tion such that ϕ (x) is self-adjoint whenever x is, and such that 1 is not in the point − spectrum of ϕ. Then ϕ (z) = z∗ implies that z = z∗.

It perhaps not immediately apparent that Theorem 3.1.4 implies Theorem 3.1.2,. To see this consider a Banach algebra with unit and with involution *. For r, s in , A A denoted by Lr,Ls the linear operators on defined by A

Lr (z) = rz and Ls (z) = zs

the operators Lr,Ls commute and the spectrum of LrLs is contained in λµ : λ σ (Lr) , µ σ (Ls) . { ∈ ∈ } This in turn is contained in λµ : λ σ (r) , µ σ (s) . Now let u given and let { ∈ ∈ }

φ (z) = LuLu∗ z = uzu∗. If z is self-adjoint then

φ (z)∗ = uz∗u∗ = uzu∗ = φ (z) .

If u is unitary, then the spectrum of φ is contained in λµ : λ, µ σ (u) , and if u is { ∈ } cramped , this set can not contain 1. −

57 3.2 Operators similar to their adjoints

3.2.1 Conditions implying self-adjointness of operators

In this section we try to answer the following question : Question suppose that T is a bounded operator and S is an invertible operator for which 0 / W (S) and ST = T ∗S, then when does it follow that necessary T is self-adjoint!? ∈

Definition 3.2.1 Let A and B be a bounded operators. We said that A is similar to B iff SA = BS

for some bounded invertible operator S.

Theorem 3.2.2 ([She66]) Let T be a bounded hyponormal operator. if S is any bounded operator for which 0 / W (S), then ∈

ST = T ∗S T = T ∗. ⇒

For proving this theorem, we need certain results which we formulate in the form of lemmas.

Lemma 3.2.3 ([Ber61]) Let T be a bounded hyponormal operator and let λ1 ,λ2 ∈ σapp (T ) , such that λ1 = λ2. if xn and yn are the sequences of unit vectors of H such 6 that (T λ1) xn 0 and (T λ2) yn 0, then xn, yn 0. k − k → k − k → h i →

Lemma 3.2.4 ([Ber65]) If T is a bounded hyponormal operator, then σ (T ∗) = σapp (T ∗) .

Lemma 3.2.5 ([Sta65]) If T is a bounded hyponormal operator such that σ (T ) is a set of real numbers, then T is self adjoint.

Lemma 3.2.6 If an operator A is similar to an operator B, then A is bounded below iff

B is bounded below. In other words if A and B are similar, then σapp (A) = σapp (B) .

58 PROOFOFTHETHEOREM. Since 0 / W (S),T is invertible. Hence ∈ 1 T = S− T ∗S and it follows from Lemmas 3.2.4 and 3.2.6 that

σ (T ) = σ (T ∗) = σapp (T ∗) = σapp (T )

Now, it is suffi cient, by virtue of Lemma 3.2.5, to prove that σ (T ) is real. Suppose that

there exists a λ σ (T ) such that λ = λ. Since λ σ (T ) = σapp (T ) , there exists a ∈ 6 ∈ sequence xn of unit vectors such that T ∗ λ xn (T λ) xn 0. − ≤ k − k → Since 0 / W (S), the relation
∈ 1 1 T ∗ λ xn = STS− λ xn = S T λ S− xn 0 − − − →

1
1
1 1 implies that T λ S− xn 0. Hence xn,S − xn = SS− xn,S − xn . Put yn = − → h i h i 1 S− xn 1 , then y = 1 and Sy , y 0 i.e. 0 W (S), a contradiction. This complete S xn n
n n k − k k k h i → ∈ the proof of the theorem.

Corollary 3.2.7 Let T be a bounded seminormal operator. If for any bounded operator S, for which 0 / W (S), then ∈

ST = T ∗S T = T ∗. ⇒

1 Theorem 3.2.8 ([Wil69]) If T is any operator such that S− TS = T ∗, where 0 / ∈ W (S), then the spectrum of T is real.

Proof. It is enough to show that the boundary of σ (T ) lies on the real axis. Since this is a subset of the approximate point spectrum of T , it suffi ces to show that if x is a n n N { } ∈ sequence of unit vectors such that (T λ) xn, then λ is real. This latter assertion follows − from the inequality

1 1 1 λ λ S− xn, xn = (T ∗ λ) S− xn, xn T ∗ λ S− xn, xn − h i h − i − − 1 1
(T ∗ λ) S− xn + S− (T
λ) xn ≤ k − k k k k − k 1 1 = S− (T λ) xn + S− (T λ) xn k − k k k k − k 1 2 S− (T λ) xn ≤ k k k − k

59 and the fact that 0 / W (S) implies 0 / W (S 1). ∈ ∈ − To recover Theorem 3.2.2, from Theorem 3.2.8, we need only observe that if T is hyponormal, then W (T ) is the convex hull of σ (T ) .

1 Corollary 3.2.9 Let T (H) be a convexoid operator, if S− TS = T ∗, where ∈ B 0 / W (S), then T = T ∗. ∈

Theorem 3.2.10 If T or T ∗ is p-hyponormal operator and S is an operator for which

p > 0, 0 / W (S) and ST = T ∗S, then T is self-adjoint. ∈

Proof. Suppose that T or T ∗ is p-hyponormal. According to Theorem 3.2.8 we get

σ (T ∗) = σ (T ) R. ⊂

Thus m2 (σ (T )) = m2 (σ (T ∗)) = 0, for the planer Lebesgue measure m2. Now apply

Putnam’s inequality for p-hyponormal operators to T or T ∗ (depending upon which is p-hyponormal) to get

p p p 2p 1 (T ∗T ) (TT ∗) r − drdθ = 0. k − k ≤ π σZZ(T )

It follows that T is normal. Since σ (T ∗) = σ (T ) R here, T must be self-adjoint. ⊂

Lemma 3.2.11 ([JKTU08]) Let T = U T be the polar decomposition of a class A | | operator T . Then T U T is semi-hyponormal and | | | |

σ ( T U T ) = r2eiθ : reiθ σ (T ) . | | | | ∈  Lemma 3.2.12 ([JKTU08]) For any operator T (H) ∈ B

1 1 2 2 2 2 2 2 T ∗ T T T T ∗ T T T ∗ | | ≥ | | ⇐⇒ | | ≥ | |

The next Theorem was mentioned in ([JKTU08]).

60 Theorem 3.2.13 Let T be a class A operator. If S is an arbitrary operator for which

0 / W (S) and ST = T ∗S, then T is self-adjoint. ∈

Proof. Since ST = T ∗S and 0 / W (S), then σ (T ) R by Theorem3.2.8. Let T = U T ∈ ⊂ | | be the polar decomposition of T . Then T U T is semi-hyponormal and | | | | σ ( T U T ) = r2eiθ : reiθ σ (T ) by Lemma 3.2.11 because T is class A operator. On | | | | ∈ the other hand, since T is class A operator, the following inequalities hold by Lemma 3.2.12 : 1 1 2 2 2 2 2 2 2 T = T ∗ T T T T ∗ T T T ∗ | | ≥ | | ⇐⇒ | | ≥ | |

Using the M. Ito and T. Yamazaki’sresult, we have

1 2 2 2 T T T ∗ T . | | ≥ | | | | | | Therefore, we have

1 1 2 2 2 2 2 2 T T T ∗ T T T T ∗ T | | − | | ≤ | | − | | | | | | 1 1 2 2 = (( T T )∗ (
T T )) (( T T )(
T T )∗) | | | | − | | | |

Since σ (T ) is real, σ ( T T ) is also real. Thus m2σ ( T T ) = 0, where m 2 is the planer | | | | Lebesgue measure. Applying the Putnam’sinequality for semi-hyponormal operators:

1 1 1 (T ∗T ) 2 (TT ∗) 2 drdθ = 0 − ≤ 2π σZZ(T )

we have T 2 T 22 because T T = T U T is semi-hyponormal . | | − | | | | | | | | Now let AB T = on ran (T ) + ker (T ∗) 0 0  be a 2 2 matrix representation ofT , and let P be the orthogonal projection onto × ran (T ). Then since T ∗(T ∗T TT ∗)T = 0, we have P (T ∗T TT ∗)P = 0. Therefore, − − A∗A AA∗ = BB∗ and hence A is hyponormal. Let −

S1 S2 S =   S3 S4  

61 then since ST = T ∗S and 0 / W (S), we have 0 / W (S1) and S1T = T ∗S1. Since A is ∈ ∈ AB hyponormal, Ais self-adjoint and hence B = 0. Hence T = is self-adjoint. 0 0  The following is an extension of Theorem 3.2.13 and I. H. Kim ([Kim06, Corrolary 5]) to the quasi-class A operators.

Theorem 3.2.14 ([JKTU08]) If T is a quasi-class A operator and S is an arbitrary

operator for which 0 / W (S) and ST = T ∗S, then T is self-adjoint. ∈ Theorem 3.2.15 Let S,T be two bounded operators satisfying:

1 S− T ∗S = T,S∗ST = TS∗S and 0 / W (S) ∈ then T is self-adjoint.

The following Lemma plays an important role in the proof of the above Theorem.

Lemma 3.2.16 ([Ber64]) Let T be a bounded operator for which 0 / W (T ). Then T is 1 ∈ invertible and the unitary operator T (T ∗T )− 2 is cramped.

Proof of Thoerem 3.2.15. Since 0 / W (S), S is invertible. So, let S = UP be ∈ 1 1 its polar decomposition. Remember that P = (S∗S) 2 is positive and U = S(S∗S)− 2 is unitary. By Lemma 3.2.16 , U is even cramped.

Since S∗ST = TS∗S, we have

P 2T = TP 2 or PT = TP.

Hence we may write 1 S− T ∗S = T 1 P − U ∗T ∗UP = T ⇐⇒ 1 U ∗T ∗U = PTP − ⇐⇒ 1 U ∗T ∗U = TPP − ⇐⇒ U ∗T ∗U = T ⇐⇒ T ∗ = UTU ∗. ⇐⇒

62 As U is cramped, Theorem 3.1.2 applies and yields the self-adjointness of T , establishing the result.

3.2.2 Operators similar to self-adjoint ones

Let T be a bounded linear operator on a Hilbert space. In this section a suffi cient con-

1 ditions are given in order that S− TS is self-adjoint for some bounded invertible linear operator S . Theorem 3.2.8, fails if the operator S is merely required to be invertible. Even normality of both S and T does not help, as the following example shows :

2 Example 3.2.17 Let T be the bilateral shift (T en = en+1) on the span ` of the ortho- normal set en ∞ and let S be the self-adjoint unitary defined by Sen = e n. Then { }−∞ −

1 1 S− TS = T − = T ∗ but the spectrum of T is not real.

Here is the promised generalization :

1 Theorem 3.2.18 If S− TS = T ∗ where 0 / W (S), then T is similar to a self-adjoint ∈ operator.

Proof. Since 0 / W (S) is convex and does not contain 0, we can separate 0 from W (S) ∈ by a half-plane. If necessary, we may replace S by eiθS for suitably chosen θ to insure that this half-plane is Re z  for some  > 0. Then, if ≥ 1 A = (S + S∗) 2

the numerical range of A lies on the real axis to the right of , hence A is positive and invertible. Since

TA = AT ∗

63 1 1 it follows that A− 2 TA 2 is self-adjoint.

The proof of Theorem 3.2.18, shows that if TS = ST ∗ where S is positive and invertible, then T is similar to a self-adjoint operator. Both assumptions on S are essential here.

1 Example 3.2.19 The condition S− TS = T ∗ do not imply that T is normal. It suffi ces to take T = SB where S is positive, B is self-adjoint, and S and B do not commute.

Proposition 3.2.20 Let T (H) , if T is similar to a self-adjoint operator, then T is ∈ B similar to T ∗ and the similarity can be implemented by an S with 0 / W (S). ∈

1 Proof. If R− TR is self-adjoint, then

1 (RR∗)− T (RR∗) = T ∗

and 0 / W (RR ) because RR∗ is positive and invertible. ∈ ∗

3.3 Operators with inverses similar to their adjoints

Let T be a bounded linear operator on a Hilbert space. In this section we give a necessary

1 and suffi cient conditions are given in order that S− TS is unitary for some bounded invertible linear operator S. Also we discuss the case when T is left invertible and similar to their adjoint.

3.3.1 Operators similar to unitary ones

2 2 1 1 If P is a positive invertible operator and if TP = P T ∗ then P − TP = PT ∗P − is 1 2 2 1 1 1 self-adjoint. Similarly the condition T − P = P T ∗ implies that P − T − P = PT ∗P − is unitary. Hence T is similar to a self-adjoint operator (to a unitary operator) if and only

1 if T and T ∗ are conjugate (T − and T ∗ are conjugate) by means of a positive invertible operator.

64 Theorem 3.3.1 ([SM73]) Let J (H), such that J (X∗) = J (X)∗ for all X ∈ B ∈ (H) , then J (S) = 0 for some S such that 0 / W (S) if and only if J (A) = 0 for some B ∈ positive invertible A.

We have the following important corollaries :

1 Corollary 3.3.2 ([Wil69]) If S− TS = T ∗ where 0 / W (S), then T is similar to a ∈ self-adjoint operator.

1 1 Corollary 3.3.3 ([SM73]) Let T be an invertible operator, such that S− T − S = T ∗ where 0 / W (S), then T is similar to a unitary operator. ∈ For the proof of these corollaries it suffi ces to take

J (X) = i (TX XT ∗) and J (X) = TXT ∗ T, respectively. − − Definition 3.3.4 Let T (H) , we said that T is power bounded if ∈ B

n sup T : n N is finite. {k k ∈ } Theorem 3.3.5 ([Cas83]) Let T,S (H) , such that S is invertible. The following ∈ B assertions are equivalent : (1) T is similar to a unitary operator. (2) T and S are power bounded. 1 (3) T is power bounded,(I λS)− exists for λ < 1 and − | | 1 sup (1 λ ) (I λS)− : λ < 1 − | | − | |  is finite. (4) For each x in H the expressions

n 1 k 2 sup (n + 1)− T x : n N ∈ ( k=0 ) X

65 and n 1 k 2 sup (n + 1)− T ∗ x : n N ∈ ( k=0 ) X 1 are finite, (I λS)− exists for λ < 1 and − | |

1 sup (1 λ ) (I λS)− : λ < 1 − | | − | |  is finite. (5) For each x in H the expressions

n 1 k 2 sup (n + 1)− T ∗ x : n N ∈ ( k=0 ) X and n 1 k 2 sup (n + 1)− S x : n N ∈ ( k=0 ) X are finite. 1 1 (6) For λ < 1 the inverses (I λT )− and (I λS)− exist and for every x and y in H | | − − the expression

+π 2 iθ 1 iθ 1 sup 1 r 1 re− T − 1 re S − x, y dθ : 0 r < 1  − − − ≤   Zπ D E 
−

is finite  

1 Theorem 3.3.6 If T is an operator such that T ∗ = U ∗T − U, where U is a cramped unitary operator, then T is unitary.

1 Proof. From T ∗ = U ∗T − U we have

1 UT ∗ = T − U (3.1)

1 Now by taking the inverses, we get UT ∗− = TU. Again by taking the adjoints, we have 1 UT − = T ∗U, and hence 2 2 T ∗U = U T ∗.

66 It follows by an argument similar to that of W. A. Beck and C. R. Putnam [BP56] that

UT ∗ = T ∗U (3.2)

1 Hence from (3.1) and (3.2), T ∗U = T − U which implies that T is unitary. The following assertion appears as Corollary 3 in [SM73] ( ) Let S,T (H) with T invertible, if ∗ ∈ B

1 1 S− T − S = T ∗ (3.3)

where 0 / W (S), then ∈ T is normaloid T is unitary. ⇒ Deprima constructs a suitable counterexample to the above assertion. First note that by Corollary 3.3.3, (3.3) holds iff T is similar to a unitary. Consequently (3.3) implies that r(T ) = 1 so that T is then normaloid iff T = 1. Thus assume H is separable and let k k U0 be the bilateral shift defined by

U0ek = ek+1, k Z, ∈ where ek Z is an orthonormal basis for H. U0 is unitary and { ∈ }

σ (U0) = λ; λ = 1 . { | | }

Let Q be the positive operator defined by

Qek = αkek+1, k Z, ∈

with 1 αk = α−k = 2 for k > 0 and α0 = 1. − Set 1 T0 = Q− U0Q

67 then

ek+1 for k / 0, 1 T0ek = ∈ { − }  1 e otherwise  2 k+1

Clearly T0 = 1, but T0 is not unitary. k k  Since in the above example σ (T0) = σ (U0) and T0 = 1, we see that W (S) = k k λ; λ 1 so that T0 is also convexoid. Hence ( ) is also false when we replace the { k k ≤ } ∗ assumption T is normaloid by the assumption that T is convexoid.

Theorem 3.3.7 ([DeP74]) Let T,S (H) with T invertible. If ∈ B 1 1 (1) S− T − S = T ∗ with 0 / W (S), ∈ (2) T is either convexoid or normaloid,

1 (3) T − is either convexoid or normaloid, then T is unitary.

Remark 3.3.8 If one of the conditions (2) and (3) is a normaloid condition, the appli- cation of Corollary ?? may be replaced by Corollary ?? .

Corollary 3.3.9 Let T,S (H) with T invertible. If (1) holds and if T is hyponormal, ∈ B then T is unitary.

1 Corollary 3.3.10 If T,S (H) with T invertible. If T − S = ST ∗ with 0 / W (S), ∈ B ∈ and T is normal, then T is unitary.

Theorem 3.3.11 Let S,T be two bounded operators satisfying:

1 1 S− T − S = T ∗,S∗ST = TS∗S and 0 / W (S), ∈ then T is unitary.

1 Proof. Let S = UP where U is unitary and P is positive (where P = (S∗S) 2 ). We then have 1 1 S∗ST = TS∗S = T − S∗S = S∗ST − ⇒

68 hence 2 1 1 2 1 1 P T − = T − P so that PT − = T − P.

Therefore, 1 1 S− T − S = T ∗ 1 1 P − U ∗T ∗UP = T − ⇐⇒ 1 1 U ∗T ∗U = PT − P − ⇐⇒ 1 1 U ∗T ∗U = T − PP − ⇐⇒ 1 U ∗T ∗U = T − ⇐⇒ 1 T ∗ = UT − U ∗. ⇐⇒ 1 Since 0 / W (S), U is cramped so that Theorem 3.3.6 applies and gives us T ∗ = T − , ∈ completing the proof.

3.3.2 Operators with left inverses similar to their adjoints

In this section we discuss the case when T is similar to an isometry.

Theorem 3.3.12 ([Pat73]) If T is a left invertible operator with a left inverse T1 and 1 if there exist an operator S such that S− T1S = T ∗ and 0 / W (S), then T is similar to ∈ an isometry.

Proof. .Since 0 / W (S), 0 can be separated from W (S) by a half plane. If necessary, ∈ we may replace S by Seiθ for suitable θ, so that this half plane lies strictly on the right of

1 1 2 the imaginary axis. Let A = 2 (S + S∗); then A is positive and invertible. Let A denote 1 the positive square root of A. Then A 2 is invertible. Now

1 T1AT1∗ = 2 (T1ST1∗ + T1S∗T1∗) 1 = 2 (ST ∗T1∗ + TTS∗) 1 = 2 (S + S∗) = A

69 Thus

T1AT1∗ = A. (3.4)

1 1 Let B = A 2 T1∗A− 2 , then

1 1 ∗ 1 1 B∗B = A 2 T1∗A− 2 A 2 T1∗A− 2  1   1  = A− 2 T1AT1∗A 2 = I

by using (3.4). Since

1 1 1 1 1 − 1 B = A 2 S∗− TS∗A− 2 = S∗A− 2 T S∗A− 2 ,     T is similar to an isometry. The preceding Theorem may not be true if S is merely invertible, and it does not satisfy the condition 0 / W (S). This can be seen by the following example. ∈

Example 3.3.13 Let H be a two-dimensional Hilbert space and T be an operator on H with the matrix √2 1 .  1 √2 Now if S is the invertible operator with the matrix

1 0 0 1 −   1 1 then T ∗ = S− T − S. However, T cannot be similar to an isometry as its spectrum does not lie in the unit disc.

The following is the converse to Theorem 3.3.12.

Theorem 3.3.14 If T is similar to an isometry, then T has a left inverse T1 satisfying 1 S− T1S = T ∗ for some operator S with 0 / W (S). ∈

70 1 Proof. Our hypothesis implies the existence of an invertible operator R such that R− TR 1 is an isometry. If we put S = RR∗, then 0 / W (S) and T1 = ST ∗S− is a left inverse of ∈ T . Our next result shows that a suitable restriction on T in Theorem 3.3.12 guarantees T to be an isometry.

Theorem 3.3.15 Let T be a left invertible normaloid operator satisfying the hypothesis of theorem 3.3.12. If T1 is also normaloid then T is an isometry.

Proof. By Theorem 3.3.12, T and T1 are similar to an isometry and coisometry respec- tively. Consequently, r(T ) = 1 and r(T1) = 1. Since T and T1 are normaloid, T = 1 k k and T1 = 1. Then for any x in H, k k

x = T1T x T x x . k k k k ≤ k k ≤ k k

Thus T x = x for every x in H, which shows that T is an isometry. k k k k If T1 is not normaloid in theorem 3.3.15, then T may not be an isometry. We mention the following example to illustrate this point.

Example 3.3.16 Let ei ∞ be an orthonormal basis for H. Define T as follows : { }i=1 1 T e1 = e2 and T en = en+1 for n = 1. 2 6

Then T is a left invertible operator with left inverse T1 defined as follows :

T e1 = 0  T en = T e2 = 2e1   T en = en 1 for all n = 1 and n = 2 − 6 6   Clearly T is normaloid. Since T1 = 2 and r(T1) = 1, T1 is not normaloid. Also it can k k be seen that T is not an isometry. However, if S is a diagonal operator with diagonal

1 1, 1, 1,... , then S− T1S = T ∗ with 0 / W (S). {− } ∈

71 Theorem 3.3.17 ([Pat73]) Let T be a left invertible operator with the polar decompo-

sition UP , where U is an isometry and P is positive and invertible. Let T1 be the left 1 inverse of T with the polar decomposition P − U ∗. If, for a cramped unitary operator V ,

T ∗ = V ∗T1V , then T is an isometry.

The next Theorem was proved by S.M.Patel in 1973.

Theorem 3.3.18 Let T be a left invertible operator with a left inverse T1. If there exists 1 p an operator S such that T ∗ = S− T S, 0 / W (S) and p is a nonnegative integer, then 1 ∈ σ (T ) lies in the unit disc.

Although Theorem 3.3.18 is an extension of Theorem 3.3.12, it does not generalize it, for if p = 1, the conclusion of the theorem cannot say whether T is similar to an isometry. However, we have

Theorem 3.3.19 Let T be a left invertible operator with a left inverse T1. If there exists 1 p a self-adjoint operator S such that T ∗ = S− T S where 0 / W (S) and p is a nonnegative 1 ∈ integer, then T is similar to an isometry.

q 1 p Corollary 3.3.20 Let T,S (H). If T ∗ = S− T S where 0 / W (S) and p, q are ∈ B ∈ integers, then σ (T p) is real where p = q and σ (T ) lies in the unit circle whose p + q = 0. 6

Corollary 3.3.21 If T is nilpotent operator such that T p+1 = 0 then T does not satisfy

1 p T ∗ = S− T S where 0 / W (S). ∈

The following Theorem is a natural generalization of the result of De Prima in [DeP74].

1 q 1 p Theorem 3.3.22 If T and T − are spectraloid and T ∗ = S− T S where 0 / W (S) and ∈ p = q, then T is unitary. 6

q 1 p Theorem 3.3.23 If T satisfies the condition T ∗ = S− T S for p = q where S is an | | 6 | | invertible self-adjoint operator, then σ (T ) lies in the unit circle.

72 Proof. It is suffi cient to prove the theorem for positive integers p and q. Since

q 1 p 1 q p T ∗ = S− T S and S− T ∗ S = T , we have

1 p2 pq 1 q2 pq S− T ∗ S = T and S− T ∗ S = T

i.e. p2 q2 p2 q2 T ∗ = T ∗ or T − = I.

p2 q2 By spectral mapping theorem, λ − = 1 for each λ σ (T ) . This gives us that σ (T ) is ∈ a finite set and lies in the unit circle.

3.4 Similarities involving normal operators

The purpose of this section is to discuss Embry’sresult.

Theorem 3.4.1 ([Emb70]) If H and K are commuting normal operators and AH = KA, where 0 is not in the numerical range of A, then H = K.

Proof. Let h and k be the spectral resolutions of H and K respectively. Since AH = KA, then Ah (α) = k (α) for each complex Borel set α by [Put54]. This last equation together with the fact that h (α) and k (α) are commuting projections implies that

p (α) Ap (α)∗ = q (α) Aq (α)∗ = 0 (3.5) for each Borel set α, where

p (α) = (I h (α)) Ah (α) − (3.6) q (α) = h (α) A (I h (α)) − since 0 / W (A), equation (3.5) implies that p (α) = q (α) = 0. Thus by (3.6) ∈ Ah (α) = h (α) A

73 for each Borel set α and consequently, AH = HA. Finally, HA = KA and since 0 / ∈ W (A), H = K. The following two examples show that if H and K are normal and AH = KA, then H and K may differ if 0 / W (A) or if H and K do not commute, even if A is unitary. ∈ Example 3.4.2 If

1 0 0 1 2 0 K = ,A = , and H = 0 2 1 0 0 1       then H and K are normal, commute and AH = KA, but H = K. 6 Example 3.4.3 If

1 1 1 0 1 i K = ,A = , and H = 1 2 0 i  i 2 −       then H and K are normal, AH = KA and 0 / W (A), but H = K. ∈ 6

Corollary 3.4.4 ([Emb66]) If AA∗ and A∗A commute and 0 / W (A), then A is nor- ∈ mal.

Proof. Let H = A∗A, K = AA∗ and note that AH = KA, so that Theorem 3.4.1 is applicable.

Corollary 3.4.5 If 0 / W (A) and there exist real numbers r and s such that r2 + s2 = 0 ∈ 6 and A commutes with rAA∗ + sA∗A, then A is normal.

Several special cases of Corollary 3.4.5 are known. If A is quasi normal and 0 / W (A), ∈ then A is normal [Bro53] . If A commutes with AA∗ A∗A, then A is normal [Put54]. − This last follows from Corollary 3.4.5 by applying the corollary to A λI,which commutes − with

(A λI)(A λI)∗ (A λI)∗ (A λI) − − − − − for λ / W (A) . ∈

74 Corollary 3.4.6 ([Put57]) If A2 is normal and 0 / W (A), then A is normal. ∈

Theorem 3.4.7 ([Emb68]) Let A (H) . If σ (A) σ ( A) = , then A and A2 ∈ B ∩ − ∅ commute with exactly the same operators.

Let D be the set of all bounded operators A for which either 0 / W (A) or σ (A) ∈ ∩ σ ( A) = . − ∅

Corollary 3.4.8 If A D and AE = EA, where either A or E is normal, then E = 0. ∈ −

Proof. If σ (A) σ ( A) = , then by Theorem 3.4.7 AE = EA since A2E = EA2. ∩ − ∅ Therefore E = 0. Assume now that 0 / W (A). If E is normal, we apply Theorem 3.4.1 ∈ and have E = E or E = 0. If A is normal, then A∗E = EA∗ , and thus − −

A (E E∗) = (E E∗) A − − −

Since E E∗ is normal,E = E∗ by Theorem 3.4.1. Consequently,E is normal and a second − application of Theorem 3.4.1 yields E = E = 0. −

Corollary 3.4.9 If A is a normal element of D, then A and A2 commute with exactly the same operators.

Proof. Assume that A2E = EA2 and let H = AE EA. Then −

AH = HA −

and by Corollary 3.4.8, H = 0.

Corollary 3.4.10 If AE = E∗A and AE∗ = EA, where A D, then E is self-adjoint. ∈

Corollary 3.4.11 If AE = E∗A, where A D and either A is unitary or E is normal, ∈ then E is self-adjoint.

75 Proof. If E is normal, then AE∗ = EA, if A is unitary, then

EA∗ = A∗E∗ and consequently,

AE∗ = EA.

Thus in either case Corollary 3.4.10 may be applied. Corollary 3.4.11 includes a slight improvement of a result of J. P.Williams. In [Wil69]

Williams proved that σ (E) is real if AE = E∗A, where 0 is not in the closure of W (A). Thus if E is normal, E is self-adjoint. In particular, Williams noted that if E is normal and AE = E∗A, where A is a cramped unitary operator, then E is self adjoint. More generally, in [BP56] W. A. Beck and C. R. Putnam and in [Ber62] S. K. Berberian proved this same result without the hypothesis that A is normal. Finally, in [McC64] C. A.

McCarthy obtained a generalization from which it follows that if AE = E∗A, A unitary and σ (A) σ ( A) = , then E is self-adjoint. All of these results are included in Corollary ∩ − ∅ 3.4.11.

Corollary 3.4.12 Let AH = KA and A∗H = KA∗, where A D. If A is unitary or H ∈ and K are normal, then H = K.

Corollary 3.4.13 Let H,K (H) , such that HH∗ = KK∗ and H∗H = K∗K. If ∈ B there exists an element A of D such that AH = KA and A∗H = KA∗, then H = K.

3.5 Quasi-similarity of operators

Two operators A : H H and B : K K are quasi-similar provided there exist −→ −→ quasi-invertible operators X : H K and Y : K H such that XA = BX and −→ −→ YB = AY ..

Theorem 3.5.1 If A : H H is invertible, B : K K is hyponormal, and −→ −→ X : H K has dense range and satisfies XA = BX, then B is invertible. −→

76 Theorem 3.5.2 ([Cla75]) Quasi-similar hyponormal operators have equal spectra.

Proof. If A and B are quasi-similar hyponormal operators, then, for any complex number λ, A λ and B λ are also quasi-similar and hyponormal, so by Theorem 3.5.1 they are − − both invertible or both non invertible. Thus the spectrum of A is the same as that of B. (Note that the injectivity of the interwining maps was never used.) Define the natural map (or the natural quotient map) π : (H) (H) / (H) B −→ B B∞ by π (T ) = S (H): S = T + K for some K (H) = T + (H) . { ∈ B ∈ B∞ } B∞

Definition 3.5.3 The essential spectrum (or the Calkin spectrum) σe (T ) of T (H) ∈ B is the spectrum of π (T ) in unital Banach algebra (H) / (H), B B∞

σe (T ) = σ (π (T )) .

S. Clary ([Cla75]) proved that quasi-similar hyponormal operators have equal spec- tra and he asked whether quasi-similar hyponormal operators also have essential spectra. Later L. R. Williams [Wil80]) showed that two quasi-similar quasi-normal operators and under certain conditions two quasi-similar hyponormal operators have equal essential spec- tra, B. C. Gupta ([Gup85]) showed that quasi-similar k-quasihyponormal operators have equal essential spectra and L. Yang ([Yan93])proved that quasi-similar M-hyponormal operators have equal essential spectra, and R. Yingbin and Y. Zikun ([YZ99]) showed that quasi-similar p-hyponormal operators have also equal spectra and essential spectra. Very recently, I. H. Jeon, J. I. Lee and A. Uchiyama ([JLU03]) showed that quasi-similar injective p-quasihyponormal operators have equal spectra and essential spectra and A. H. Kim and I. H. Kim ([KK06]) showed that quasi-similar (p, k)-quasihyponormal operators have equal spectra and essential spectra.

Theorem 3.5.4 If S,T (H) are quasi-similar quasi-class A operators, then ∈ B σ (S) = σ (T ) and σe (S) = σe (T ).

77 Chapter 4

Similarities involving unbounded operators

In this chapter, we develop basic concepts and results about general closed operators on Hilbert space, also we discuss some conditions implying self-adjointness of unbounded operators. At the end of the chapter we deal with similarity of unbounded operators.

4.1 Preliminaries

Definition 4.1.1 A linear operator from a Hilbert space H1 into H2 is a linear mapping

T of a linear subspace of H1 , called the domain of T and denoted by D (T ), into H2.

Let S and T be two linear operators from H1 into H2. By definition we have S = T if and only if D (S) = D (T ) and S(x) = T (x) for all x D(S) = D(T ). We shall say that T ∈ is an extension of S or that S is a restriction of T and write S T , when D (S) D (T ) ⊆ ⊆ and S(x) = T (x) for all x D(S). ∈

78 The complex multiple αT for α C, α = 0, and the sum S +T are the linear operators ∈ 6 from H1 into H2 defined by

D (αT ) = D (T ) , (αT ) x = αT x for x D (T ) . ∈ D (S + T ) = D (S) D (T ) , (S + T ) x = Sx + T x for x D (S + T ) . ∩ ∈

If R is a linear operator from H2 into a Hilbert space H3, then the product RT is the linear operator from H1 into H3 given by

D (RT ) = x D (T ): T x D (R) , (RT ) x = RT x or x D (RT ) . { ∈ ∈ } ∈

It is easily checked that the sum and product of linear operators are associative and that the two distributivity laws

(S + T ) Q = ST + TQ and R (S + T ) = RS + RT ⊇ hold, where Q is a linear operator from a Hilbert space H0 into H1.

Definition 4.1.2 The graph (T ) of an operator T from H1 into H2 is the set G

(T ) = (x, T x): x D (T ) . G { ∈ }

Obviously, the relation S T is equivalent to the inclusion (S) (T ) . ⊆ G ⊆ G

Proposition 4.1.3 Let T be a linear operator on H such that D(T ) is dense in H. If T x, x = 0 for all x D(T ), then T x = 0 for all x D(T ). h i ∈ ∈

Definition 4.1.4 An operator T is called closed if its graph (T ) is a closed subset of G the Hilbert space H H, and T is called closable (or preclosed) if there exists a closed × linear operator S such that T S. ⊆

Remark 4.1.5 The operator T is called the closure of the closable operator T .

79 4.1.1 Adjoint Operators

Let (H1, ., . ) and (H2, ., . ) be Hilbert spaces. Let T be a linear operator from H1 into h i1 h i2 H2 such that the domain D(T ) is dense in H1 . Set

D(T ∗) = y H2 : There exist u H1 such that T x, y = x, u for x D(T ) . { ∈ ∈ h i2 h i1 ∈ }

Since D(T ) is dense in H1 , the vector u H1 satisfying T x, y = x, u for all ∈ h i2 h i1 x D(T ) is uniquely determined by y. Therefore, setting T ∗y = u, we obtain a well- ∈ defined mapping T ∗ from H2 into H1. It is easily seen that T ∗ is linear.

Definition 4.1.6 The linear operator T ∗ is called the adjoint operator of T .

Let T be a densely defined linear operator on H. Then T is called symmetric if T T ∗. ⊆ Further, we say that T is self-adjoint if T = T ∗ and that T is essentially self-adjoint if its closure T is self-adjoint. We now begin to develop basic properties of adjoint operators.

Proposition 4.1.7 ([Con90]) Let T : H1 H2 and S : H2 H3 be linear operators −→ → such that D(ST ) is dense in H1, then

1) If D(S) is dense in H2, then (ST )∗ T ∗S∗. ⊇ 2) If S is bounded and D(S) = H2, then (ST )∗ = T ∗S∗. 1 3) If T is invertible with inverse T − in (H), then (ST )∗ = T ∗S∗. B

Theorem 4.1.8 ([Con90]) Let T be a densely defined linear operator from H1 into H2.

1) T is closable if and only if D(T ∗) is dense in H2.

2) If T is closable, then (T )∗ = T ∗, and setting T ∗∗ := (T ∗)∗, we have

T = T ∗∗.

3) T is closed if and only if T = T ∗∗.

80 4) Suppose that ker (T ) = 0 and ran (T ) is dense in H2, then T ∗ is invertible and { }

1 1 (T ∗)− = T − ∗ .

1 5) Suppose that T is closable and ker (T ) = 0 . Then the inverse T − of T is closable if { } and only if ker T = 0 . If this holds, then { }
1 1 T − = (T − ).

1 6) If T is invertible, then T is closed if and only if T − is closed.

Definition 4.1.9 Let T be a closed linear operator on a Hilbert space H. A complex number λ belongs to the resolvent set ρ (T ) of T if the operator T λI has a bounded − 1 everywhere on H defined inverse (T λI)− . −

The set σ (T ) = C ρ (T ) is called the spectrum of the operator T. \

Definition 4.1.10 σp (T ) := λ C : ker (T λI) = 0 is the point spectrum of T. { ∈ − { }}

Remark 4.1.11 Formally, the preceding definition could be also used to define the spec- trum for a not necessarily closed operator T. But if λ ρ (T ), then the bounded everywhere ∈ 1 defined operator (T λI)− is closed, so is its inverse T λI by Theorem 4.1.8, and hence − − T . Therefore, if T is not closed, we would always have that ρ (T ) = and σ (T ) = C ∅ according to Definition 4.1.9, so the notion of spectrum becomes trivial.

By Definition 4.1.9, a complex number λ is in ρ (T ) if and only if there is an operator B (H) such that ∈ B B (T λI) T I and TB (T λI) = I. − ⊆ −

Proposition 4.1.12 ρ (T ) is an open subset, and σ (T ) is a closed subset of C.

81 4.1.2 Self-adjoint Operators

Self-adjointness is the most important notion on unbounded operators . The main results about self-adjoint operators are the spectral theorem and the corresponding functional calculus based on it. In this section, T denotes a densely defined linear operator on a Hilbert space H.

Definition 4.1.13 We say that T is normal if D(T ) = D(T ∗) and

T x = T ∗x for all x D(T ). k k k k ∈

It is clear that T is normal if and only if T ∗ is normal. Moreover, we have

ker (T ) = ker (T ∗) for any normal operator T .

Proposition 4.1.14 T is normal if and only if T is closed and T ∗T = TT ∗.

Definition 4.1.15 The operator T is called symmetric if

T x, y = x, T y for all x, y D(T ). h i h i ∈ Proposition 4.1.16 ([Con90]) T is symmetric if and only if T x, x is real for all x h i ∈ D(T ).

Definition 4.1.17 A densely defined symmetric operator T on a Hilbert spaceH is called

self-adjoint if T = T ∗ and essentially self-adjoint, briefly e.s.a., if T is self-adjoint, or

equivalently, if T = T ∗.

A self-adjoint operator T is symmetric and closed, since T ∗ is always closed.

Proposition 4.1.18 If T is a symmetric operator on H such that ran(T ) = H, then T

1 is self-adjoint, and its inverse T − is a bounded self-adjoint operator on H.

Corollary 4.1.19 A closed symmetric linear operator T on H is self-adjoint if and only

if σ (T ) R. ⊆

82 Theorem 4.1.20 (Fuglede-Putnam) Let S B(H). Let T1 and T2 be an unbounded ∈ normal operators and

ST1 T2S ⊆ then

ST ∗ T ∗S. 1 ⊆ 2

Corollary 4.1.21 If S is bounded and T1 and T2 are unbounded and normal, then

ST1 = T2S = ST ∗ = T ∗S. ⇒ 1 2

Theorem 4.1.22 (Fuglede-Putnam-Mortad) Let S be a closed operator with domain

D(S). Let T1 and T2 be two unbounded normal operators with domains D(T1) and D(T2) respectively. If D(T2) D(ST2), then ⊂

ST1 T2S = ST ∗ T ∗S. ⊂ ⇒ 1 ⊂ 2

Theorem 4.1.23 Let T and S be two normal operators. Assume that S is bounded. If ST = TS, then ST (and so TS) is normal.

Proof. Since ST = TS, by Corollary 4.1.21 we have ST ∗ = T ∗S. Then we have

(ST )∗ ST = T ∗S∗ST = T ∗S∗TS T ∗TS∗S ⊂ and

ST (ST )∗ = STT ∗S∗ = TST ∗S∗ = TT ∗SS∗ = T ∗TS∗S.

Whence

(ST )∗ ST ST (ST )∗ . ⊂ But ST is closed for it equals TS which is closed since is closed and S is bounded.

Therefore, ST (ST )∗ and (ST )∗ ST are both self-adjoint and hence ST is normal.

83 Theorem 4.1.24 Let T be a bounded and invertible operator. Let S be unbounded and

closed. Assume further that D(S) D(STS). Then ST and TS are normal iff STT ∗ = ⊂ T ∗TS and S∗ST TSS∗. ⊂ Consider next the following example:

Example 4.1.25 Let T and S be the two operators defined by

ix x2 ix T f(x) = e f(x) and Sf(x) = e − f(x)

on their respective domains

2 2 x2 2 D (T ) = L (R) and D (S) = f L (R): e f(x) L (R) . ∈ ∈ n o Then T is unitary (so STT ∗ = T ∗TS is verified) and S is normal. Moreover, we can easily check that:

2 2x2 2 D (S∗ST ) = f L (R): e f(x) L (R) ∈ ∈ n o and 2 2x2 2 D (TSS∗) = D (SS∗) = f L (R): e f(x) L (R) ∈ ∈ n o too. Since

S∗ST f(x) = TSS∗f(x), f D (S∗ST ) = D (TSS∗) , ∀ ∈

we have S∗ST = TSS∗. We also see that both TS and ST are normal on their equal domains 2 x2 2 D (TS) = D (ST ) = f L (R): e f(x) L (R) ∈ ∈ n 2 o since they are the multiplication operator by the function ex . Nonetheless we have

2 2x2 2 D (STS) = f L (R): e f(x) L (R) ∈ ∈ n o 3 x2 3 x2 and so D (S) " D (STS) as, for instance, e− 2 D (S) but e− 2 / D (STS) . ∈ ∈ This example suggests that replacing "bounded and invertible" by "unitary" might allow us to drop the condition D (S) D (STS) there. This is in fact the case and we have ⊂

84 Theorem 4.1.26 Let T be a unitary operator. Let S be unbounded and closed. Then ST

and TS are normal iff S∗ST TSS∗. ⊂ Proof. The proof of suffi ciency is as before. Note that with T assumed unitary, the first condition of Theorem 4.1.24 is automatically satisfied.

Let us suppose that ST and TS are normal and let us check that S∗ST TSS∗. In ⊂ fact, since TS is normal, we have

(TS)∗ TS = S∗T ∗TS = S∗S = TS (TS)∗ = TSS∗T ∗.

Hence SS∗T ∗ = T ∗S∗S. Accordingly by taking adjoints,

TSS∗ = S∗ST

establishing the result. We now turn to the case of two unbounded normal operators. We have

Theorem 4.1.27 Let T be an unbounded invertible normal operator. Let S be an un- bounded normal operator. If ST = TS, T ∗S ST ∗ and S∗T TS∗, then ST is normal. ⊂ ⊂ The same method of proof yields

Theorem 4.1.28 Let T be an unbounded invertible normal operator. Let S be an un- bounded normal operator. If ST TS, T ∗S ST ∗ and S∗T TS∗, then ST is normal. ⊂ ⊂ ⊂ Corollary 4.1.29 Let T and S be two unbounded invertible normal operators with do- mains D(T ) and D(S) respectively. If ST = TS and D(T ),D(S) D(ST ), then ST ⊂ (and TS) is normal.

Proof. Note first that the closedness of ST is clear. Now we have

ST TS = ST ∗ T ∗S = S∗T TS∗. ⊂ ⇒ ⊂ ⇒ ⊂ Similarly we have

TS ST = TS∗ S∗T = T ∗S ST ∗ ⊂ ⇒ ⊂ ⇒ ⊂ So we came back to the setting of Theorem 4.1.24.

85 4.2 Quasi-similarity of unbounded operators

In this section, we generalized some notions of bounded operators to unbounded operators on Hilbert space, and we extend some familiar results on quasi-similar bounded operators to unbounded operators.

Definition 4.2.1 (Quasi-similarity) Let T and S be densely defined linear operators in Hilbert spaces H1 and H2, respectively. If there exist quasi-invertible operators XTS from H1 into H2 and XST from H2 into H1 such that

XTST SXTS and XST S TXST ⊂ ⊂ then we say that T is quasi-similar to S and is denoted by T S. ∼

Remark 4.2.2 if T S, then S∗ T ∗. ∼ ∼

Definition 4.2.3 A densely defined operator T is said to be hyponormal if:

1) D (T ) D (T ∗) . ⊆ 2) T ∗x T x for all x D (T ) . k k ≤ k k ∈

a densely defined linear operator T in a Hilbert space H is said to be subnormal if there exist a Hilbert space K containing H as a closed subspace and a normal operator N in K such that

D (T ) D (N) and T x = Nx for all x D (T ) . ⊆ ∈

Proposition 4.2.4 A subnormal operator is hyponormal.

Proof. Suppose T is subnormal operator in a Hilbert space H with normal extension N in K, and let us take x D (T ) , for y D (T ) , one has ∈ ∈

T x, y = Nx, y = x, N ∗y h iH h iK h iK

86 hence y D (T ∗) .Moreover if P is the orthogonal projection of K onto H, then the above ∈ equalities implies that

T ∗y = PN ∗y for all y D (T ∗) . ∈ It follows that T x = Nx k k k k = N ∗x k k PN ∗x ≥ k k = T ∗x k k for all x D (T ) , thus T is hyponormal. ∈ The following Theorem is an unbounded version of Clary’sresult [Cla75] on unbounded operators.

Theorem 4.2.5 ([OS89]) Let T be a closed hyponormal operator in a Hilbert space H and let S be a closed densely defined operator in Hilbert space K. If there exists a linear bounded transformation X with dense range from K to H such that XS TX, then the ⊂ spectrum of T is contained in that of S.

Corollary 4.2.6 Quasi-similar closed hyponormal operators have equal spectra.

The next Theorem is a generalization of Douglas’ result in [Dou69] to unbounded operators.

Theorem 4.2.7 ([OS89]) Let T and S be normal operators. If T and S are quasi- similar, then T and S are unitarily equivalent.

T +T ∗ S+S∗ Proof. Let T1(resp. S1) be the closure of 2 ( resp. 2 ); the real part of T ( resp

T T ∗ S S∗ . S) and Let T2(resp. S2) be the closure of −2i ( resp. −2i ); the imaginary part of T ( resp. S). It then follows from the spectral theory for a normal operator (ex. Theorem

7.32 in [Wei80]) that all Ti,Si (i = 1, 2) are self-adjoint operators satisfying

T1T2 = T2T1 and S1S2 = S2S1

87 and

T = T1 + iT2,S = S1 + iS2.

Let X = XTS be a quasi-invertible transformation with XT SX. It then follows from ⊂ the Fuglede-Putnam theorem that XT ∗ S∗X, and so ⊂

XT1 S1X,XT2 S2X. ⊂ ⊂

Since both of (Ti iI) and (Si iI) are (every where defined) boundedly invertible and − −

X (Ti iI) (Si iI) X − ⊂ −

for i = 1, 2, it follows that

1 1 X (Ti iI)− = (Si iI)− X − − for i = 1, 2. Let X = UP be the polar decomposition of X. Clearly U is unitary. Applying the Proof 1 1 of Lemma 4.1 in [Dou69], for bounded normal operators (Ti iI)− and (Si iI)− , it − − easily seen that they are unitarily equivalent with the common intertwinig operator U, namely 1 1 U (Ti iI)− = (Si iI)− U − − for i = 1, 2. Hence, by noticing that U is unitary, one has

UT = U (T1 + iT2)

= S1U + iT2U = SU.

This completes the proof of the Theorem.

Theorem 4.2.8 ([OS89]) Let T be a closed densely defined symmetric operator, if T is

quasi-similar to its adjoint T ∗, then T is self-adjoint.

88 Proof. Let X be a quasi-invertible operator such that

XT ∗ TX. ⊂

For y D (T ∗) with T ∗y = iy, one has X (y) D (T ) and ∈ ∈

iX (y) = XT ∗ (y) = TX (y) .

Since the point spectrum of a symmetric operator is real, it follows that X (y) = 0, by the injectiveness of X, y = 0. Analogously,

(ran (T iI))⊥ = ker (T ∗ + iI) = 0 . − { } hence the deficiency indices of T are (0, 0) . This means that T is self-adjoint.

Remark 4.2.9 In general the above Theorem fails if the closed operator T is merely normal, and so T is unitarily equivalent to T ∗.

Corollary 4.2.10 Let T be a densely defined, closed symmetric operator. If T is quasi- similar to a self-adjoint operator S, then T is also self-adjoint, and moreover T and S are unitarily equivalent.

Proof. Since T S, then ∼ S T ∗ and T T ∗ ∼ ∼ according to Theorem 4.2.8, T is self-adjoint. By Theorem 4.2.7 T and S are unitarily equivalent.

Theorem 4.2.11 ([OS89]) Let T be a closed subnormal operator and S be a self-adjoint operator in a Hilbert space H. If there is a positive, quasi-invertible operator X in H such that XS TX ⊂ then, T = S.

89 Theorem 4.2.12 Let T be a closed subnormal operator in a Hilbert space H, If T is quasi-similar to a self-adjoint operator S, then T is also self-adjoint, and moreover T and S are unitarily equivalent.

Proof. Let X be a quasi-invertible operator such that

XS TX. ⊂ Let X = UP be the polar decomposition of X. Then U is unitary and P is positive. Since

U ∗TUP = U ∗TX U ∗XS = PS ⊃ and U ∗TU is subnormal, It follows by Theorem 4.2.11, that U ∗TU = S.

4.3 Similarities involving unbounded normal opera- tors

In this section we prove and disprove some generalizations of a result about some simi- larities involving normal operators due to M. R.Embry in 1970. Some interesting conse- quences are also given. If N and M are unbounded normal operators, and AN MA, then ⊂

APBR (N) = PBS (M) A

where PBR (N) and PBS (M) are the spectral projections of N and M respectively and where

BR (N) = z C : z R and BS (M) = z C : z S { ∈ | | ≤ } { ∈ | | ≤ } are two closed balls in C where R and S are two positive numbers. The numerical range of an operator A, defined on a Hilbert space H, is denoted by W (A) and is defined as

W (A) = Ax, x : x D (A) , x = 1 . {h i ∈ k k }

90 We adopt the following definition of commutativity of two unbounded normal operators: Two normal operators are said to commute if their associated spectral projections do. As introduced by Devinatz-Nussbaum in [DN57], we say that the unbounded operators N,H and K have the property P if they are normal and if

N = HK = KH.

Devinatz-Nussbaum proved (in the same paper) the following result

Theorem 4.3.1 If N,H and K have the property P , then H and K commute.

Proposition 4.3.2 Assume that H and K are two bounded self-adjoint operators such that 0 / W (K). If HK is normal, then it is self-adjoint. ∈

Proof. Let N = HK, Since N is normal and

KN = KHK = N ∗K then HK = (HK)∗ . Now we give the generalization of Embry Theorem to unbounded H and K.

Theorem 4.3.3 ([Mor10]) Assume N,H and K are unbounded operators having the property P . Also assume that D (H) D (K). Assume further that A is a bounded ⊂ operator for which 0 / W (A) and such that AH KA. Then H = K. ∈ ⊂

Proof. Let PBR (H) and PBS (K) be the spectral projections of H and K by respectively.

Then PBR (H) and PBS (K) are two bounded normal operators. The property P (more precisely Theorem 4.3.1) then guarantees that they are commuting operators.

Now since AH KA and since ran (PB (H)) D (H) (by the spectral theorem), we ⊂ R ⊂ immediately see that

AHPBR (H) = KAPBR (H) .

91 Whence (by the remark below the Fuglede-Putnam theorem in the introduction)

AHPBR (H) = KAPBR (H) = KPBS (K) A.

Therefore we are in a bounded setting and Embry’sTheorem then applies and implies that

HPB (H) x = KPB (K) x for all x D (H) . R S ∈ Sending both R and S to infinity (in the strong operator topology) gives us

Hx = Kx for all x D (H)( D (K)) . ∈ ⊂ Whence H K. Now since normal operators are maximally normal, H = K. The proof ⊂ is complete. An interesting application of the previous theorem is the following.

Corollary 4.3.4 Assume that A is a bounded operator such that 0 / W (A). If H is an ∈ unbounded normal operator such that AH H∗A, then H is self-adjoint. ⊂

Proof. Obvious since HH∗ = H∗H as H is normal and also since HH∗ is self-adjoint.

Remark 4.3.5 The necessity of 0 / W (A) was justified in [Emb70] by a counterexample. ∈ Now we give an example which shows that Property P cannot be completely eliminated.

Take A = I (the identity operator on the whole Hilbert space). Now take any non- closed symmetric operator H and hence it is neither self-adjoint nor normal (e.g., take

H such that Hf (x) = if 0 (x) on D (H) = C0∞ (R)). Then Property P is not fulfilled, − AH H∗A, 0 / W (A) but H is not self-adjoint. ⊂ ∈ Now we give an analog of Proposition 4.3.2 for unbounded H (this is also akin to a result obtained in [Mor03]). We have

Proposition 4.3.6 Assume that H and K are two self-adjoint operators such that H is unbounded and 0 / W (K). Assume further that K is bounded. If HK is normal, then it ∈ is self-adjoint.

92 The case where all operators are unbounded fails to be true in general even if A is assumed to be self-adjoint and even if " "is replaced by "= "in the assumption ⊂ AH = KA. We have

Theorem 4.3.7 Let A, H and K be unbounded operators. Assume that N,H and K have the property P . Also assume that A is self-adjoint. Then AH = KA and 0 / W (A) do ∈ not necessarily imply that H = K.

Proof. We give a counterexample. Consider the following operators A and H defined by

Af (x) = (1 + x ) f (x) and Hf (x) = i (1 + x ) f 0 (x) | | − | | on their respective domains

2 2 D (A) = f L (R) : (1 + x ) f L (R) ∈ | | ∈  and 2 2 D (H) = f L (R) : (1 + x ) f 0 L (R) . ∈ | | ∈  In order to find H∗ , the adjoint of H, some technical work is required. One has to do it

first for f C0∞ (R∗) , the space of smooth functions with compact support away from ∈ the origin. Then one has to mimic the arguments used in [Mor03] for slightly different operators. One finds the following

H∗f (x) = f (x) i (1 + x ) f 0 (x) ± − | | on 2 2 D (H∗) = f L (R) : (1 + x ) f 0 L (R) . ∈ | | ∈ Now simple calculations yield 

2 AH∗f (x) = HAf (x) = i (1 + x ) f (x) i (1 + x ) f 0 (x) ± | | − | | for every f in

2 2 2 D (AH∗) = D (HA) = f L (R) : (1 + x ) f, (1 + x ) f 0 L (R) . ∈ | | | | ∈  93 This shows that AH∗ = HA. Now since H is normal (see [Mor09]), then so is H∗

and besides, HH∗ = H∗H (and HH∗ is self-adjoint) and hence Property P is verified. Obviously A is self-adjoint on D (A) and 0 / W (A). ∈ As one can see, all these assumptions are not suffi cient to make H = H∗ .

4.4 Unbounded operators similar to their adjoints

Theorem 4.4.1 ([DM17]) Let S be a bounded operator on a C-Hilbert space H such that 0 / W (S). Let T be an unbounded and closed hyponormal operator with domain ∈ D(T ) H. If ST ∗ TS, then T is self-adjoint. ⊂ ⊂ To prove the theorem stated above we need the following results.

Lemma 4.4.2 Let T be a densely defined and closed operator in a Hilbert space H, with domain D(T ) H. Assume that for some k > 0, ⊂ T x k x for all x D(T ). k k ≥ k k ∈ Then ran(T ) is closed.

Proposition 4.4.3 Let T be an unbounded, closed and hyponormal operator in some Hilbert space H. Then W (T ) co σ(T ), where co σ(T ) denotes the the convex hull of ⊂ σ(T ).

Proof. The proof is divided into three claims:

Claim 1: σa (T ∗) = σ (T ∗) . By definition, σa (T ∗) σ (T ∗) . To To show the reverse ⊆ inclusion, let λ / σa (T ∗). Then there exists a positive number k ∈

T ∗x λx k x for all x D(T ∗). k − k ≥ k k ∈

Hence T ∗ λI is clearly injective. Besides, and by Lemma 4.4.2, ran(T ∗ λI) is closed − − as T ∗ λI is closed for T ∗ is so. Now, since T is hyponormal, so is T λI which means − −

94 that such that

T x λx T ∗x λx k x for all x D(T ) D(T ∗). − ≥ k − k ≥ k k ∈ ⊂

Whence T λI is also one-to-one so that −

ran (T ∗ λI)⊥ = ker T λI = 0 or ran (T λI) = H. − − { } ∗ −
Thus T ∗ λI is onto since we already observed that its range was closed. Therefore, − λ / σ (T ∗). ∈ Claim 2: σ(T ) R. Let λ σ (T ∗) = σa (T ∗). Then for some xn D(T ∗) such that ⊂ ∈ ∈ xn = 1 we have T ∗xn λxn 0 as n . Since ST ∗ TS and xn D(T ∗), k k k − k −→ −→ ∞ ⊂ ∈ we have ST ∗xn = T Sxn so that we may write the following

1 0 λ λ Sxn, xn = ST ∗S− λ + λ T Sxn, xn ≤ − h i − −
S ( T ∗ λ) xn, xn + λ
T xn, x n ≤ |h − i| − S T ∗xn λxn + xn, (λ
T ∗) xn ≤ k k k − k |h − i| S T ∗xn λxn + S T ∗xn λxn ≤ k k k − k k k k − k = 2 S T ∗xn λxn 0. k k k − k −→

(where in the second inequality we used the fact that xn D(T ∗) and Sxn D(T ) both ∈ ∈ coming from ST ∗ TS). However, the condition 0 / W (S) forces us to have λ = λ ⊂ ∈ Accordingly,σ(T ∗) R or just σ(T ) R (remember that σ(T ∗) = λ : λ σ(T ) ). ⊂ ⊂ ∈ Claim 3: T = T ∗. Since σ(T ) R, Proposition 4.4.3 implies that W (T ) R , which ⊂ ⊂ clearly implies that T x, x R for all x D(T ), which, in its turn, means that T is h i ∈ ∈ symmetric. Hence T is quasi-similar to T ∗ via S and I, so that Theorem 4.2.8 applies and gives the self-adjointness of T . This completes the proof.

The condition ST ∗ TS in the foregoing theorem is not purely conventional, i.e. we ⊂ may not obtain the desired result by merely assuming instead that ST T ∗S, even with ⊂ a slightly stronger condition (i.e. symmetricity in lieu of hyponormality). This is seen in the following proposition

95 Proposition 4.4.4 There exist a bounded operator S such that 0 / W (S), and an un- ∈ bounded and closed hyponormal T such that ST T ∗S whereas T = T ∗. ⊂ 6 Proof. Consider any unbounded, densely defined, closed and symmetric operator T which is not self-adjoint. Let S = I, i.e. the identity operator on the Hilbert space. Then S is bounded and 0 / W (S). Also, T is closed and hyponormal. Finally, it is plain that ∈

T = ST T ∗ = T ∗S. ⊂

Remark 4.4.5 We have not insisted on the explicitness of the example T in the previous proof. This was not too important. Besides, there are many of them. For instance, the interested reader may just consult Exercise 4 on page 316 of [Con90].

Theorem 4.4.6 Let S be a bounded operator such that 0 / W (S). Let T be an unbounded ∈ hyponormal and invertible operator. If ST T ∗S, then T is self-adjoint. ⊂ Proof. Since T is invertible with an everywhere defined bounded inverse, we have

1 1 ST T ∗S = ST − = T − ∗ S. ⊂ ⇒ 1
Since T is hyponormal, the bounded T − too is hyponormal (see [Jan94]). Hence by 1 [She66], T − is self-adjoint. Hence T is self-adjoint. The condition of invertibility in the foregoing theorem may not simply be dispensed with. This is illustrated by the following example:

Example 4.4.7 Let A be an unbounded operator defined on a Hilbert space H, with do-

main D(A) H. Set T = A A, then T is not closed and hence it is surely not − self-adjoint. Finally, let S = I the identity operator on H. Now we verify that the re- maining conditions (except for invertibility) of the theorem are fulfilled.

1) T is hyponormal for T ∗ = 0 with D(T ∗) = H D(A) = D(T ) so that ⊃

T x = T ∗x for all x D(T ). k k k k ∈

96 2) Since S = I, obviously 0 / W (S). Moreover, ∈

T = 0D(A) T ∗ = 0H so that ST T ∗S ⊂ ⊂

Last but not least, we have a very nice and important result which generalizes Berberian Theorem to unbounded operators.

Theorem 4.4.8 ([DM17]) Let U be a cramped unitary operator. Let T be a closed operator such that UT = T ∗U. Then T is self-adjoint.

Proof. First we prove that U 2T = TU 2. Since U is bounded and invertible, we have

(UT )∗ = T ∗U ∗ and (T ∗U)∗ = U ∗T

(by Proposition 4.1.7). Hence T ∗U ∗ = U ∗T . We may then write

2 2 U TU ∗ = U (UTU ∗) U ∗

= UT ∗U ∗

= UU ∗T ∗

= T ∗,

2 2 2 2 2 2 2 giving U T = TU or TU ∗ = U ∗ T or U T ∗ = T ∗U .

Next, we prove that TU = UT ∗. We have

TU = U ∗T ∗UU

= U ∗UUT ∗

= UT ∗.

Hence also U ∗T ∗ = TU ∗. The penultimate step in the proof is to prove that T is normal. To this end, set

1 S = 2 (U + U ∗). Following [Wil69], S > 0.

Then we show that STT ∗ T ∗TS. We have ⊂

UTT ∗ = T ∗UT ∗ = T ∗TU

97 and

U ∗TT ∗ = T ∗U ∗T ∗ = T ∗TU ∗.

Hence 1 STT ∗ = 2 (U + U ∗) TT ∗ 1 1 = 2 UTT ∗ + 2 U ∗TT ∗ 1 1 = 2 T ∗TU + 2 T ∗TU ∗

TT ∗S ⊂ So according to Corollary 5.1 in [Sto01], TT ∗ = TT ∗, and remembering that T is taken to be closed, we immediately conclude that T is normal. Accordingly, and by Corollary 4.3.4 ,

UT = T ∗U as 0 / W (U), establishing the result. ∈

Remark 4.4.9 Evidently, a hypothesis like UT T ∗U would not yield the desired result. ⊂ For example, take T to be any symmetric and closed unbounded operator T which is not self-adjoint. Let U = I be the identity operator on the given Hilbert space. Then clearly

UT T ∗U while T = T ∗. ⊂ 6

Remark 4.4.10 Going back to the previous proof, we observe that this proof may well be applied to bounded operators. Hence we have just given a new proof of Berberian Theorem which bypasses the Cayley transform.

Corollary 4.4.11 Let S be a bounded operator and T be an unbounded closed operator

1 satisfying: S− T ∗S = T,S∗ST = TS∗S and 0 / W (S). Then T is self-adjoint. ∈

98 Bibliography

[Amb45] W. AMBROSE, Structure theorems for a special class of Banach algebras. Trans. Amer. Math. Soc. 57 (1945), 364—386.

[And72] T.ANDO, Operators with a norm condition, Acta Sci. Math. Szeged, 33 (1972), 169-178.

[AW99] A. ALUTHGE and D. WANG, Powers of p -hyponormal operators, J. of Inequal. Appl., 3 (1999 ), 279—284.

[BD73] F. F. BONSALL and J. DUNCAN, Complete Normed Algebras. Ergebnisse der Math- ematik und ihrer Grenzgebiete 80, Springer Verlag, 1973.

[Ber61] S. K. BERBERIAN, Introduction to Hilbert space, Oxford Univ. Press, New York, 1961.

[Ber62] – – – – – – —, A note on operators unitarily equivalent to their adjoints, J.London Math. Soc. 37 (1962), 403-404.

[Ber64] – – – – – – —, The numerical range of a normal operator, Duke Math. J., 31, (1964) 479-483.

[Ber65] – – – – – —, A note on hyponormal operators, Pacific J. Math. 12 (1962), 1871- 1875.

[BP56] W. A. BECK and C. R. PUTNAM, A note on normal operators and their adjoints, J.London Math. Soc. 31 (1956), 213-216.

99 [Bro53] A. BROWN, On a class of operators, Proc. Amer. Math. Soc. 4 (1953), 723-728.

[Cas83] J.A. VAN CASTEREN, Operators similar to unitary or selfadjoint ones, Pacific Math. J. 104(1983), 241—255.

[CI95] M. CHO¯ and M. ITOH, Putnam’s inequality for p-hyponormal operators, Proc. Amer. Math. Soc., 123 (1995) 2435—2440.

[Cla75] S. CLARY, Equality of spectra of quasisimilar hyponormal operators Proc. Amer. Math. Soc. 53 (1975), 88—90.

[Con90] J. B. CONWAY, A Course in Functional Analysis, Springer, 1990 (2nd edition).

[DeP74] C. R. DEPRIMA, Remarks on "Operators with inverses similar to their adjoints, Proc. Amer. Math. Soc., 43 (1974) 478-480.

[DM17] S. DEHIMI, M. H. MORTAD, Bounded and unbounded operators similar to their adjoints, Bull. Korean Math. So c. 54 (2017), No. 1, pp. 215—223

[DN57] A. DEVINATZ, A. E. NUSSBAUM, On the Permutability of Normal Operators, Ann. of Math. (2), 65 (1957), 144—152.

[Don63] W. F. DONOGHUE, On a problem of Nieminen, Inst. Hautes Études Sei. Publ. Math, no. 16 (1963),31-33. MR 27 #2864.

[Dou69] R.G. DOUGLAS , On the operator equations S∗XT = X and related topics , Acta Sci . Math . , 30(1969),19- 2.

[Emb66] M. R. EMBRY, Conditions implying normality in Hilbert space, Pacific. J. Math.18 (1966), 457-460.

[Emb68] – – – – – , Conditions implying normality in Hilbert space, Pacific. J. Math. 18 (1966), 457-460.

100 [Emb70] – – – – – , Similarities Involving Normal Operators on Hilbert Space, Pacific J. Math., 35 (1970), 331—336.

[FFK93] M.FUJII, T.FURUTA and E.KAMEI, Furuta’s inequality and its application to Ando’s theorem, Linear Algebra Appl., 179 (1993), 161-169.

[FHN67] T. FURUTA, M. HORIE and R. NAKAMOTO : A remark on a class of operators. Proc. Japan Acad., 43, 607-609 (1967).

[FIY98] T. FURUTA, M. ITO and T. YAMAZAKI, A subclass of paranormal operators in- cluding class of log-hyponormal and several related classes, Sci. Math., 1 (1998), 389—403.

[FN71] T. FURUTA, R. NAKAMOTO, On the numerical range of an operator, Proc. Japan Acad. 47 (1971), 279—284.

[Fur67] T. FURUTA, On the class of paranormal operators, Proc. Japan Acad. 43 (1967) 594—598.

1 (p+2r) [Fur87] – – – – , A > B > 0 assures (BrApBr) q B q for r 0, p 0, q 1 ≥ ≥ ≥ ≥ with (1 + 2r) q p + 2r, Proc. Amer. Math. Soc., 101 (1987), 85-88. ≥ [Fur92] – – – – , Applications of order preserving operator inequalities, Oper. Theory Adv. Appl., 59 (1992), 180-190.

[Fur95] – – – – , Extension of the Furuta inequality and Ando-Hiai log-majorization, Linear Algebra Appl., 219 (1995), 139-155.

[Gup85] B. C. GUPTA, Quasisimilarity and k-quasihyponormal operators Math. Today 3 (1985), 49—54.

[Gel41] I. M. GELFAND, Normierte Ringe. Rec. Math. (Mat. Sbornik) 51 (1941), 3—24.

101 [Halm50] P. HALMOS, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950), 125-134.

[Hal67] – – – – – – , A Hilbert space problem book, Van Nostrand, Princeton, N.J., 1967.

[Han80] F. HANSEN, An operator inequality, Math. Ann., 246 (1980), 249-250.

[Hau19] F. HAUSDORFF, Der Wertvorrat einer Bilinearform, Math. Z. 3 (1919), 314—316.

[Hei51] E. HEINZ, Beiträge zur Störungstheorie der Spektralzerlegung, Math. Ann., 123 (1951), 415-438.

[Ist66] V. ISTRA˘ ¸TESCU, On some hyponormal operators, Pacific J. Math. 22 (3) (1967) 413—417.

[ISY66] V. ISTRA˘ ¸TESCU, T. SAITS, and T. YOSHINO: On a class of operators. TShoku. Math. Journ., 18, 410-413 (1966).

[Jan94] J. JANAS, On Unbounded Hyponormal Operators, Ark. Mat., 112/1 (1994) 75-82.

2 2 [JK06] I. H. JEON, I. H. KIM, On operators satisfying T ∗ T T T ∗ T T.. Linear Alg. | | ≥ | | Appl. 418 (2006), 854—862.

[JKTU08] I. H. JEON, I. H. KIM, K. TANAHASHI and A. UCHIYAMA, Conditions implying self— adjointness of operators, Integral Equations Operator Theory, 61(2008), 549— 557.

[JLU03] I. H. JEON, J. I. LEE and A. UCHIYAMA, On p-quasihyponormal operators and quasisimilarity Mathematical Inequalities and Applications 6 (2003), 309—315.

[Kat68] T. KATO, Smooth operators and commutators, Studia Math. 31 (1968), 535-546.

[Kim06] I. H. KIM, The Fuglede-Putnam theorem for (p,k)-quasihyponormal operators J. Inequalities and Applications 9 (2006), 397—403.

102 [KK06] A. H. KIM and I. H. KIM, Essential spectra of quasisimilar (p,k)-quasihyponormal operators J. Inequalities and Applications 9 (2006), 1—7.

[Kub12] C. S. KUBRUSLY, Spectral theory of operators on Hilbert spaces. New York, NY, USA: Birkh¨ auser, 2012.

[Löw34] K. LÖWNER, Über monotone Matrixfunktionen, Math. Z., 38 (1934), 177-216.

[McC64] C. A. MCCARTHY, On a theorem of Beck and Putnam, J. London Math. Soc. 39 (1964), 288-290.

[McC67] – – – – – – —, cp, Israel J. Math., 5 (1967), 249-271.

[Mor03] M. H. MORTAD, An Application of the Putnam-Fuglede Theorem to Normal Prod- ucts of Selfadjoint Operators, Proc. Amer. Math. Soc., 131/10 (2003), 3135— 3141.

[Mor09] – – – – – , On Some Product of Two Unbounded Self-adjoint Operators, Inte- gral Equations Operator Theory, 64/3 (2009), 399—408.

[Mor10] – – – – – , On a Beck-Putnam-Rehder Theorem, Bull. Belg. Math. Soc. Simon Stevin, 17/4, (2010) 737-740.

[Na07] N.A.SOMIRY, On Powers of p-hyponormal Operators, The Islamic University of Gaza, 2007.

[Nag36] M. NAGUMO, Einige analytische Untersuchungen in linearen metrischen Ringen. Japan J. Math. 13 (1936), 61—80.

[Nie62] T. NIEMINEN, A condition for the self-adjointness of a linear operator, Ann. Acad. Sei.Fenn. Ser. A I No. 316 (1962),3-5.

[NF70] B. Sz.-NAGY and C. FOIA¸S, Analyse harmonique des opérateurs de l” espace de Hilbert, Masson, Paris; Akad. Kiado’, Budapest, 1967; English rev. transl.,

103 NorthHolland, Amsterdam; American Elsevier, New York; Akad. Kiado, Bu- dapest, 1970.

[Orl63] G. H. ORLAND, On a class of operators, Proc. Amer. Math. Soc, 15(1964), 75-79.

[OS89] S. ÔTA, K. SCHMÜDGEN, On Some Classes of Unbounded Operators, Integral Equa- tions Operator Theory, 12/2 (1989) 211-226.

[Pat73] S. M. PATEL, Operators with left inverses similar to their adjoints, Proc. Amer. Math. Sot. 41(1):127-131 (1973).

[Pet08] S. PETRAKIS, Introduction to Banach algebras and the Gelfand-Naimark Theo- rems. University of thessalonki.

[Pin68] J. D. PINCUS, Commutators and systems of singular integral equations, I,Acta Math. 121 (1968), 219-249.

[Put54] C. R. PUTNAM, On the spectra of commutators, Proc. Amer. Math. Soc. 5 (1954), 929-931.

[Put57] – – – – – —,On square roots of normal operators, Proc. Amer. Math. Soc. 8 (1957)768-769.

[Put63] – – – – – —, On the structure of semi-normal operators, Bull. Amer. Math. Soc. (N.S.) 69 (1963), 818-819.

[Put70] – – – – – —, An inequality for the area of hyponormal spectra, Math Z. 116 (1970), 323-330.

[Rus68] M. ROSENBLUM, A spectral theorem for self adjoint singular operators, Amer. J. Math. 88(1966), 314-328.

[Sch61] M. SCHREIBER, On the spectrum of a contraction, Proc. Amer. Math. Soc., 12(1961), 709-713.

104 [She66] I. H. SHETH, On hyponormal operators, Proc. Amer. Math. Soc., 17 (1966) 998- 1000.

[SM73] U. N. SINGH and K. MANGLA, Operators with inverses similar to their adjoints, Proc. Amer. Math. Soc. 38 (1973),258-260.

[Sta62] J.G. STAMPFLI, Hyponormal operators, Pacific J. Math., 12 (1962), 1453-1458.

[Sta65] – – – – – —, Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117(1965), 469—476. MR 33:4686.

[Sta67] – – – – – —, Minimal range theorems for operators with their spectra, Pacific J. Math. 23 (1967), 601-612. MR 37 #4655.

[Sto01] J. STOCHEL, An asymmetric Putnam-Fuglede theorem for unbounded operators, Proc. Amer. Math. Soc., 129/8 (2001) 2261-2271.

[SY65] T. SAITO and T. YOSHINO, On a conjecture of Berberian. Tohoku. Math. Journ., 17, 147-149(1965).

[Toe18] O. TOEPLITZ, Das algebraische Analogon zu einem Satz von Fejer, Math. Z. 2 (1918), 187—197.

[Wei80] J. WEIDMANN , Linear operators in Hilbert spaces, Springer-Verlag, Berlin Heidelberg-New York (1980).

[Wil69] J. P. WILLIAMS, Operators Similar to Their Adjoints, Proc. Amer. Math. Soc., 20 , (1969) 121-123.

[Wil70] – – – – – —, Finite operators. Proc. Amer. Math. Soc., 21, 129-136 (1970).

[Wil80] L. R. WILLIAMS, Equality of essential spectra of certain quasisimilar seminormal operators Proc. Amer. Math. Soc. 78(2) (1980), 203—209.

105 [Xia63] D. XIa, On non-normal operators, I, Chinese J. Math. 3 (1963), 232-246.

[Xia80] – – —, On the non-normal operators-semihyponormal operators, Sci. Sininca, 23 (1980), 700-713.

[Yan93] L. YANG, Quasi-similarity of hyponormal and subdecomposable operators J. Func- tional Analysis 112 (1993), 204—217.

[YZ99] R. YINGBIN and Y. ZIKUN, Spectral structure and subdecomposability of p- hyponormal operators Proc. Amer. Math. Soc. 128 (1999), 2069—2074.

106