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1973 On Reductive Operators. Thomas P. Wiggen Louisiana State University and Agricultural & Mechanical College

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Recommended Citation Wiggen, Thomas P., "On Reductive Operators." (1973). LSU Historical Dissertations and Theses. 2440. https://digitalcommons.lsu.edu/gradschool_disstheses/2440

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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 WIGGEN, Thomas P., 1945- ON REDUCTIVE OPERATORS. The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1973 Mathematics

University Microfilms, A XEROX Company, Ann Arbor, Michigan ON REDUCTIVE OPERATORS

A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial, fulfillment of the requirements for the degree of Doctor of Philosophy in ' The Department of Mathematics

by Thomas P. Wiggen B.S., University of North Dakota, 1967 M.S., University of North Dakota, 1969 May, 1973 ACKNOWLEDGMENT

The author is deeply indebted to the late Dr. Pasquale Porcelli for his guidance and encourage­ ment while the research for this paper was being done. The author also wishes to express his gratitude to Dr. Ernest Griffin for his help and advice in the writing of this dissertation and to Dr. John Dyer for many discussions on the material contained herein. TABLE OP CONTENTS

CHAPTER PAGE ACKNOWLEDGMENT ...... i i ABSTRACT...... iv INTRODUCTION ...... 1 I THE REDUCTIVE OPERATOR CONJECTURE...... 10 I I INVERSES OF REDUCTIVE OPERATORS ...... 28 I I I POWERS OP REDUCTIVE OPERATORS...... 38 IV RESOLVENT TECHNIQUES ...... ^5 BIBLIOGRAPHY ...... 55 VITA ...... 59

i l l ABSTRACT

Reductive operators are the principal concern of this dissertation. The most important problem concerning reductive operators is that of determining whether or not every reductive operator is normal. In Chapter I we show that a reductive operator A is normal if f(A) is normal for some analytic function f on {z: |z| < ||A||) . It is also shown that reductive quasinormal operators are normal. Concerning the spectrum of a reductive operator A , we prove that a (a) = 7r(A) , the approximate point spectrum of A . An example is given to show that the set of re­ ductive operators on an infinite-dimensional Hilbert space is not closed in the strong operator topology. Chapter II is devoted to the study of inverses of reductive operators. We show that if A is reductive and i f 0 is in the unbounded component of the resolvent set p(A) , then A -1 is reductive. This result is used to demonstrate that the bilateral shift is a direct sum of

i v V

two reductive operators. We also show that a reductive operator A is normal if and only if A © A is reduc­ tive, thus obtaining a new equivalence to the invariant subspace conjecture. In Chapter III we investigate powers (positive and negative) of reductive operators. With various restric­ tions on the spectrum of an operator A , we show that A is reductive if An is reductive and that An is re­ ductive if A is reductive. Chapter IV is chiefly concerned with quasinilpotent operators and their invariant subspaces. This departs only slightly from the topic of reductive operators, be­ cause any transitive operator is reductive. We show that a quasinilpotent operator A has a proper invariant sub­ space if and only if, for some non-zero vectors x and y , the analytic function \ "* ((\-A) ”'I'x,y) on C\ {0} has a pole at \ = 0 . INTRODUCTION

This section introduces most of the notation and conventions used in the remainder of the text. It also mentions most of the background knowledge which w ill be required of the reader. The symbols Z , Z+ , R and C will denote the sets of integers, positive integers, real numbers and com­ plex numbers respectively. If A. e C , then X will denote the conjugate of X . A Hilbert space H will always be a complex Hilbert space. The inner product of two vectors x and y in H will be denoted by (x:,y) and \|x|| will denote the norm of the vector x . A unit vector is a vector x such th a t ||x|J = 1 . By an operator A on a Hilbert space H , we will mean a continuous linear transformation of H into itself. The adjoint of an operator A on H will be denoted by A , so (Ax,y) = (x,A y) for all vectors x and y in

1 2

11 . The symbol I represents the identity operator on H , that is Ix = x for all vectors x e H . For con­ venience we will use (\-A) to represent the operator (\*I-A) . If A is an operator on H and if there exists . an operator B on H such that BA = AB = I , then we say that A is invertible and that B is the inverse of A , written B = A~^ . The symbol UAH will denote the operator norm of the operator A on H , that is, ||A|| = sup {||Ax|| : x e H , ||x|| = 1} . The symbol B(H) de­ signates the set of all operators on H . We will assume that the reader is familiar with the uniform, strong operator and weak operator topologies on B(H) . The uniform topology on B(H) is the topology ge­ nerated by all sets of the form N(A$e) = {b e B(H): ||B-A||. < e} where e > 0 . The stro n g o p erato r topology on B(H) i s the topology generated by all sets of the form

N(A;x1, • • *,xn; e) = {B e B(H) s|| (B-A)x±|| < e , i= l, 2, • • • ,n} where e > 0 , n e z+ and xi****>xn are arbitrary vec­ tors in H . The weak operator topology on B(H) is the topology generated by all sets of the form

N(A; Xx, ••*,xn; yx, ••*,yn; e) = (b e b (H): | ((B-A)xjL,yi ) | < e , i = 1,2,•••,n} where e > 0 , n e Z+ and • • *,xn ,y 1, ••*,yn are arbitrary vectors in H . The strong operator topology is stronger than the weak operator topology and weaker than the uniform topology. Further discussion on these can be found in [ 5], [ 6]*

[7 ] j [12] or [ 16]. A linear manifold Vi of H is a subset of H which is closed under the operations of vector addition and scalar multiplication. A subspace of H is a closed linear manifold of H . If jf is a subset of H , V [x:x e X) will denote the linear manifold generated by X and V(x:x e x } w ill denote the subspace generated by X . The symbol X will denote the closure of the set X • Also, for any subset X o f H , X x will denote the orthogonal complement of X , th a t i s , X*~ = (y e H:(x,y) = 0 for all x e X 3 . X x is always a subspace of H . The subspaces (0} and H are called the trivial sub­ spaces of H and all others are called proper subspaces. A subspace Vi of H is invariant for an operator A if [Ax:x e Vi) c Vi . We say that Vi reduces A if Vi is * invariant for both A and A or, equivalently, if both Vi and Tn1' are invariant for A . The restriction of an operator A to an invariant subspace Vi is denoted by

An operator A is Hermitian if A = A , normal if *X* *X* AA = A A , isometric if ||Ax|| = ||x|| for all x € H , •X* •X* and unitary if AA = A A = I . If A is defined and isometric on a dense subspace of H and if the range of A is dense in H , then A has a unique extension to a unitary operator on H . A projection is an operator * 2 A which satisfies A = A = A . A is said to be quasi- * normal if A commutes with A A , and reductive if every invariant subspace for A reduces A . A transitive operator is an operator which has no proper invariant sub- spaces. Clearly every operator on a one-dimensional Hilbert space H is transitive - and these are the only known examples. Operators which are not transitive are called intransitive. A compact operator is an operator which maps any bounded sequence of vectors to a set with a convergent subsequence. Every operator on a finite - dimensional Hilbert space is compact. A normal part of an operator A is the restriction of A to a reducing subspace on which it is normal. An operator is said to be normal-free if it has no normal p a r t. We l e t 71(A) = (x e H: Ax = 0} , the null-space of A , and we let /?(A) = {Ax:x e H) , the range of A . The re la tio n s h ip s 71(A) = /?(A ) A and /?(A) = 7l(A ) a re valid for any operator A on H . 5

The spectrum o f A , o(A) , is {X e C: (x-A) is not invertible) . The resolvent of A , p(A) , is the complement of a (A) . The operator-valued function ^ -» (X-A)’"’1' defined on p (A) is called the resolvent function of the operator A . For any pair of vectors x and y in H , the function X •* ((X -A )_1x ,y ) is an analytic function on P (A) , hence we say the resolvent function is analytic on p(A) . Following Dunford and Schwartz ,([7]* P. 580 ) we de­ fin e 0p (A) = (X € o(A): 7*(X-A) £ (0)} ,

or (A) = {X e 0(A): fl(X-A) = (0) and /?"(x-A) ^ H) , and ac (A) = (X 6 a(A): 7?(x-A) = (0) and /?(X-A) = H) .

The sets a (A) , a (A) and a (A) are called, respectively, the P • x C point spectrum, residual spectrum and continuous spectrum o f A . C learly cr (A) , a (A) and a (A) are disjoint p r w < •x* sets whose union is a (A) . Also a„(A) c a (A ) because Jr ^ (A )x = 7{{A ) . An eigenvalue of A is a number X e 0^ (a) , that is a number x such that (X-A)x = 0 for some unit vector x e H , and this vector is called an eigenvector of Acorresponding to the eigenvalue X . A number X e a (A) is called an approximate eigenvalue of A if there exists a sequence (*n3 vectors in H such th a t ||xn || = 1 fo r each n and ||(X-A)xn || 0 . The set of all approximate eigenvalues of A will be de­ n o te d by tt(A) and will be called the approximate point spectrum of A . An operator A is said to be nilpotent if An = 0

i , f o r some n e Z , and quasinilpotent if a (A) = (0) . A vector x £ H is called a cyclic vector for an o p e ra to r A on H i f V(Anx :n e Z+ or n=0) = H . The commutant of an operator A on H is (A) * = = (B e B(H): BA = AB and BA* = A*B} . B. Fuglede [9] has shown that if A is normal and . BA = AB , then also BA = A B . Thus for a normal operator A on H we have {A)‘ = (B e B(H): BA = AB) . A subalgebra Ot o f B(H) is called a *-subalgebra of B(H) (or simply a *-algebra) i f A e Ot implies A * e Ot . An element A of a *-sub- a lg e b ra Ot of B(H) is called a positive element of Ot i f A = B*B fo r some B e o t . For any operator A , (A)' is a *-algebra and is closed in the weak operator topology. A continuous linear functional on an algebra Ot is a continuous linear mapping from Ot into C . A positive continuous linear functional f on a *-algebra o t s a t i s ­ fies the additional condition that f(A) > 0 whenever A is a positive element of Ot • I f O t is a commutative uniformly closed *-subalgebra of B(H) containing I and :i.r j? is a maximal ideal in (JL , then the factor ring O t/J is isometrically isomorphic to C . Thus each maximal ideal J determines a continuous linear func­ tio n a l on o t . The space M of all maximal ideals of Ot is a closed subset ofthe unit ball of the dual of Ot and is thus a compact Hausdorff space in the relative ■weak topology. Moreover, 01 is isometrically isomorphic to the algebra C(M) of all continuous complex-valued functions on M . The isometric isomorphism from Ot* to C(M) is called the Gelfand transform and the symbol ^(m) denotes the image of an operator A e O t. Proofs of the facts stated above and further information on this topic can be found in Naimark [1 6 ] or Dunford and Schwartz [7], Vol. II Another non-trivial result we w ill assume is the spectral theorem for normal operators. A good exposition o f th is may be found in Halmos [1 0 ]. The theorem s ta te s that every normal operator A on H has a representa- tion of the form where, for each Borel C s e t A c C , E(A) is a p ro je c tio n in B(H) , E(C) = I , 0 0 0 0 M E( u A ) = £ E(A ) whenever C^v,)v,_i i s a d is jo in t se- n=l n n=l n x quence of Borel subsets of C , and E(A^)E(Ag) = 0 whenever A-^ fl Ag = 0 . Moreover, if BA = AB then I»:(A) l.i BE(A) for every Borel subset A of C . In par­ ticular, each projection E(A) commutes with A itself. If the only values of E(A) are I and 0 then A is a multiple of I . The family of projections {E(a ) 3 is called the spectral family of the normal operator A . If f is a complex-valued function which is defined and analytic on a neighborhood V, o f a (A) fo r some operator A e B(H) we will simply say that f is ana­ lytic on o(A) . The set F^ will be the set of all functions analytic on a (A) (where the neighborhood depends on the function). If f e F^ we can find a rec­ tifiable Jordan curve r in 14 which contains a (A) in its interior and define an operator f(A) in B(H) by the formula

f(A) = ST I T where dx denotes Lebesgue measure on the complex plane. (Note that r is a positive distance from a(A) since T and a(A) are disjoint compact sets.) Also, the operator f(A) so defined is independent of the parti­ cular curve r chosen because the resolvent function \ -* (\-A ) -1 is analytic on p(A) and the Cauchy integral theorem is valid. The correspondence from F^ into B(H) d efin e d above has th e follow ing p ro p e rtie s : 9

(1) if f(X) h 1 then f(A) = I ; ( 2) if f(\) = \ then f(A) = A; (3) if f(X) = a^f^X) + a 2±'2 (X) where f-^fg e PA > al ,a2 6 0 ' then f G FA and f (A) = = c^f^A) + a2f 2 (A) i (4) i f f(X-) = f 1 (X )f2 (X) where fl,f2 € FA ’ then f 6 FA ^ f (A ) = f l ( A) f 2^A) 5 and ( 5) if the sequence (fn(x )3 converges uniformly to f ( \ ) in some neighborhood M of a(A) and f e f a for each n , then f e F^ and the sequence (f (A)'} converges in the uniform topology to f(A) . In parti­ cular, if f is a polynomial in X , then f(A) is the corresponding polynomial in A . This procedure of constructing analytic functions of an operator will be referred to as the operational calculus. Some good sources of information on the operational calculus are Dunford and Schwartz [ 7] , Lorch [15] and Naimark [16]. Finally, the reader may find it helpful to have at least a vague idea of the concept of a direct integral. For this the reader is referred to Naimark [16], Dixmier

[5], or Pedersen [17]. CHAPTER I THE REDUCTIVE OPERATOR CONJECTURE

Reductive operators, as mentioned in the introduc­ tion, are operators for which every invariant subspace i s a reducing subspace. The name ' ‘red u c tiv e o p e ra to r ' 1 is of recent origin and was suggested by Paul Halmos. In earlier literature these operators have traveled under the names of ''completely normal operators*', as in Dyer and Porcelli [ 8]3 ''operators with property (P ) ' 1 as in Wermer [2 3 ]; or simply as 11 operators for which every invariant subspace is reducing'1, as in Sarason [20], The first proposition below is very elementary and its proof will be omitted.

Proposition 1.1. The following are equivalent for A e B(H): (a) A is reductive. •x* (b) A is reductive. (c) If a subspace % is invariant for A , so is .

10 (<1) A subspaec' 7I\ is invariant for A iff it is inva­ riant for both of the Hermitian parts of A . (e) A.-A is reductive for some A e C . (f) A-A is reductive for all A e c . (g) AA is reductive for some non-zero A e C . (h) AA is reductive for all A e C . In particular, we note that the set of reductive operators is closed under the operations of translation, scalar multiplication and the taking of adjoints.

Proposition 1.2. If A e B(H) and the sub space %. re - •X* <}f duces A , then A ^ = (A|^) .

Proof. If x,y e % , then (A*|^x,y) = (A*x,y) = (x,Ay) = = (x,A|^y) . Since x and yare arbitrary in ^ , we have (A|r)* = A*|^ .

Proposition 1.3. If A is reductive, then crr(A) = 0 .

Proof. Suppose a € or (A) . Then ^(x-A) = (0) and /?(A-A) is not dense in H . Hence {0) 0 /?(A-A)x = ^ •X* - ) and so there is a vector x / 0 in A) ) It follows that A*x = \ x and 7f[ = V{x) is invariant for A . By Proposition 1.1(b), A is reductive so is •X* •X’ — » also invariant for A .Also, (A|^) = A l? 7| " = = (A*I|^)* so A = A *1 on % . Thus we have A e CTp(A) > 12

a contradiction because ar(A) and °p(A) are disjoint. We thus conclude that ar (A) = 0 . Hermitian operators are clearly reductive because A = A • f t . Normal operators need not be reductive as is shown by th e fo llo w in g example:

Example 1.4. Let H be a separable infinite-dimensional Hilbert space and let (en:n e z3 be an orthonormal basis for H . Let U be the operator on H defined by i*en = e n+1 for n e Z . The adjoint ty* o f U is the operator defined by H • f t en = en_-j_ for n e z , and % i s •ft ft called a bilateral shift on H . We have KU = V. l( = I so V. is normal (unitary, in fact). However, V. is not reductive because every subspace of the form V{e^, ejc+ii * ’ * 3

• f t is invariant for U but not for U We saw that our example above was, in fact, a unitary operator. It is interesting to note that this is, in a sense, the only example of a unitary operator which is not reductive. John Wermer has shown in [23] that a unitary operator is reductive iff it does not have a reducing sub- space on which it acts like a bilateral shift. The statement that every reductive operator is normal is an open question, which we shall refer to as the re­ d u c tiv e o p e ra to r co n je c tu re . T. Ando has shown in [1] th a t a compact operator is reductive iff it is normal. Another open q u estio n , more w idely known than th e reductive operator conjecture, is the statement that every operator on a Hilbert space of dimension greater than 1 has a proper invariant subspace. This we shall refer to as the invariant subspace conjecture. In th e e a rly I930»s J . von Neumann proved (but did not publish) the result that every compact operator on a Hilbert space of dimension greater than 1 has a proper invariant subspace. This was later extended to arbitrary Banach spaces by N. Aronszajn and K. T. Smith [2], More recently it has been shown that polynomially compact oper­ ators have proper invariant subspaces (cf. Bernstein and Robinson [3], Halmos [10]). An easy generalization of this is the fact that every analytically compact operator has a proper invariant subspace. Some other results in this direction can be found in Chernoff and Feldman [4]. John Dyer and Pasquale Porcelli have proven the fol­ lowing remarkable result [ 8 ]:

Theorem 1.5. The following two statements are equivalent: (a) The reductive operator conjecture is true. (b) The invariant subspace conjecture is true.

It is easy to show that (a) =* (b) . Suppose that (b) is false. Then we have a transitive operator A on a Hilbert space II of dimension greater than 1 . Clearly A is reductive because its only invariant sub­ spaces are (0) and H . However, A is not normal because normal operators are not transitive (by the spec­ tral theorem for normal operators, for example). Hence (a) is false. The proof of the converse is more difficult and re­ quires some knowledge of direct integral theory. Some good references on this subject are Pedersen [17] and Dixmier [5]. Before indicating the proof that (b) => (a) in Theorem 1.5, we w ill pause to develop some necessary ma­ chinery. First we note that, in attacking the reductive operator conjecture, it suffices to consider only reductive operators A with a cyclic vector . This simplifica- tion is possible because V[A ?0:n e Z U [0}) reduces A and A is normal iff its restriction to each such subspace is normal. Also, by Ando's result, we may assume that V[An50:n e Z+ U [0)) is infinite-dimensional. Thus we assume for the remainder of this proof that H is a se­ parable infinite-dimensional Hilbert space and that A is a reductive operator on H with a cyclic vector such th a t ||Sq || =1 . Let E be a maximal commutative *-sub- algebra of [A}' . Let = [B§0:B e fcj , a subspace 15

of H . If E50 4 H , we can find a vector §^e(E§0)x such that ||§1|| - 1 . It is easy to show that the sub­ spaces E5q and are orthogonal. If E §0 © E§^ ^ H , we can choose a vector 52g(E5q©E§1)x and continue in this manner. Since H is separable we can thus write 00 ______H = 2 © E§. . We w ill d efin e i =0 1 t n + H = { E B.%.: B. e E , n e z u {0)) , a linear mani- i =0 1 fold which is dense in H . For each vector we can define a continuous positive linear functional f^ on the algebra E by f^(B) = (B?^,§^) . Since the algebra E is isometrically isomorphic to C(M) , the algebra of continuous functions on the maximal ideal space M of E , each functional f^ may be considered as a functional on C(M) . As a result, each functional f^ induces a regu­ lar Borel measure p^ on M and we have f . (13) = f $(m)dp.(m) for each B e E , where $(m) de- 1 J M notes the Gelfand transform of B . Because Is a cyclic vector for A , it can be shown that the support of the measure pQ is M and that each p^ is absolutely continuous with respect to pQ . Also, the vectors

i = l , 2,••*, can be chosen so that dp^ - XM, dpQ where I XM is the characteristic function of a pQ-measurable set 16

contained in M . Next we define a linear map from E§i into C(Mi ) by

L2(Mi^pi) and for B e E we have ||^BS^||2 =

= (^B §i ,r(I_B5i ) = (^(m) lM^*^(m) |M^) = J m i*(m)^(m)d^i (m) =

J B*B(m)dpi (m) = f^(B*B) = (B§i ,B§i ) = ||B?i ||2 . Thus Mi each map ^ has a unique extension to a unitary operator

^ which maps E§^ onto L^M^I-l ) . Using the symbol 11 = " to mean 1'is unitarily equivalent to*1 , we CO p have H = Z © L (M.,p.) via the unitary operator i=0 U = Z © U. . Let Hn = V{§.:i e .+ u (0)) and for i=0 u i each m e M , let H(m) = V{Si#.m e , a subspace of the « separable Hilbert space HQ . For a vector r| e H (de- n fined previously) of the form n = Z B.,5. where B. e E , i=0 1 we can define a vector-valued function Ti(m) on M by the equation T](m) = Z B . (m)XM (m)g. . Then r)(m) e H(m) fo r i=0 1 iVli 1 each m e M and furthermore 12 = j 0'lBi5i“2 = = Jo =

= Z J |£(m )l2XM (m)dn(m) = J Z [|£(m )|2XM (m)]dp(m) = i=0 M wi M i=0 “i J ||,n(m)||2dp(m) . Thus the map v ’:r| -♦ (Ti(m))m £ M maps a r® deni’.i) r.ubsot of II onto a donut1 subset of Il(m)dp(m) M in an isometric fashion and so

p0 (1.1) H ^ H(m)dP(m) M i via the unitary map V which extends V and is defined i on all of H . For vectors r\ e H , the vector-valued fu n ctio n ( Tl(m ) 3 meM def,ine^ everywhere on M . This is not the case for all vectors in H . If T) e H is , i arbitrary, we can find a sequence (r| } of vectors in H converging to r| such that the sequence of functions

Crin(m) )meM converSes pointwise a.e. (p) on M . This allows us to define Ti(m) a.e. (p) on M and this is the best we can do. Our next task is to represent an operator B on H as an operator on the direct integral of the Hilbert spa­ ces (H(m))meM . In general, we cannot represent all oper ators in B(H) in such a fashion, but we are able to re- i present every operator In E in this way. In particular * the reductive operators A and A have such a represen­ ta tio n . i Let € H be as before and let p e H . Let

B e e ' . Then p(m) is defined everywhere on M and (B|.)(m) is defined a.e. (p) on M . We let 00 = (me M: (B§i)(m) is not defined} and N = U . I/or m e M\N we define f(r|(in)) = (ri(m), (B^) (m)). Then f

In ;i linear functional on the dense linear manifold h' n H(m) of H(m) and |f(r)(m ))| | |h(m )||.|| (B§1) (m)|| , so f is bounded. As a result of this, f extends uniquely to a bounded linear functional F defined on all of H(m) with ||F|| = ||f|| < ||(B§i )(m)|| . By the Riesz representation theorem, there is a unique vector Cj_(m) in H(m) such that F(§(m)) = (§(m),£^(m)) for all %(m) e H(m) . We w ill d efin e an o p erato r B(m) on H(m) by letting B(m)§i(m) = ^(m) for i=0,l,••• , and ex­ tending linearly. (Note that the vectors ^(m) 3j^_o contain a basis for H(m) .) For m e N we will define B(m) to be the zero operator on H(m) . The operators B(m) so defined can be shown to have the following pro­ perties: (a) B(m)5(m) = (B§)(m) a.e. (p) for § e H ;

(b) B(m)ri(m) = (Bn)(m) fo r a l lm e M , t] e h ' ; (c) |JB(m)|| < ||B|| fo r a l l m e M. We w rite

(1 . 2 ) B = B(m)dp(m) where J M\K

[ f B(m)dp(m)][ g(m)dp(m)] = f B (m )§ (m)dp(m) J M d M M

If B^ and B2 are any two operators in E we can show that (B^+ B2)(m) = B1(m) + B2(m) a.e. (|j) , B^Bg(m) = B1 (m)B2 (m) a .e . (n) , and th a t (B^)(m) = •X* = (B^(m)) a.e. (|i) . As a result we have

(1.3) B-J+ B2 = J^[B1(m) + B2(m)]dn(m)

(1.4) B B2 = Jr® B1 (m) • B2 (m)dp(m) and

(1.5) B* = f [B1(m)]*dn(m) . JM We now return to the proof of Theorem 1.5. Let A be a reductive operator on the separable infinite-dimen­ sional Hilbert space H and let be a cyclic vector for A with ||§0|| = 1 . According to the above construe tion, we have a representation.

«© (1 .6 ) A = A(m)dH(m) M where for each m , A(m) is an operator on the Hilbert space H(m) as defined above. Dyer and Porcelli showed that the ''coordinate operators'' A(m) are transitive a.e. (p.) . if we assume that the invariant subspace con Jecture is true, then each operator A(m) must be an operator on a 1-dimensional Hilbert space H(m) . Due to the fact that operators on a 1-dimensional Hilbert space commute, we have r%© # AA = A(m)A (m)dp(m) M

p© * A (m)A(m)dp(m)

* = A A and so A is normal. This shows that (b) * (a) and com­ pletes the proof of Theorem 1.5.

Some progress towards a solution of the reductive operator conjecture is given by the following result:

Proposition 1.6. If A is reductive and if f(A) is normal for some non-constant analytic function f on { z :|z | < ||Aj|} , then A is normal.

Proof. Suppose that f is as described above and f(z) = 00 = 2 a zpis the power series expansion of f about the

p=° 0 0 point z = 0 . Then 2 anA converges in the uniform p=0 p 0 0 p topology and so f(A) = 2 a AF . According to (1.6) we r® p = ° p have A = A(m)dp(m) where the operators A(m) are M n rj transitive a.e. (p) . If we let ^(A) = 2 a AF , then P=0 p p© 1'n(A)(m) = fn(A(m)) a.e. (p) and fn(A) = J fn (A)(m)dp(m)

p© f (A(m))dp(m) . Hence for almost all m we have M n 21

fn(A(m)) - i’n(A)(m) for all n and {fn(A(m))) con­ verges in the uniform topology to f(A(m)) . (Note that f(A(m)) i s defined because ||A(m)|| < ||A|| .) I t follow s that f(A)(m) = f(A(m)) a.e. (p) and that c0 f(A) = f(A(m))dp(m) . Since f(A) is normal, f(A(m)) M is normal a.e. (p) . If f(A(m)) is normal, then the spectral family of f(A(m)) commutes with A(m) because A(m) commutes with f(A(m)) . As a result, f(A(m)) must be a multiple of the identity operator I on H(m) when­ ever f(A(m)) is normal, say f(A(m)) = • Im . If we let g(z) = f(z) - ctjH , then g(A(m)) = 0 so A(m) is analytically compact. This implies that A(m) must be an operator on a 1-dimensional Hilbert space H(m) , be­ cause A(m) is transitive. This happens a.e. (p) on M and it follows that A is normal. As an application of Proposition 1.6 we obtain the following result:

Corollary 1.7. If A is reductive and if An commutes . *X* Yfl -U w ith (A ) fo r some m yn e z , then A is normal.

Proof. If An commutes with (A*)m , then Anm is normal.

Our next result states that every operator in B(H) has a decomposition into normal and normal-free parts. 22

Proposition 1.6. Every operator A in B(H) can be expressed as the direct sum of a normal operator and a normal-free operator. This decomposition is unique.

P ro o f. The s e t o f a l l reducing subspaces fo r A on -which A is normal is partially ordered by inclusion, and Zorn1s lemma gives us a maximal such sub space H-^ . Hence A|^

is normal and Al^x is normal-free and we have the desired ,H1 decomposition. The uniqueness is clear. Alternatively, we can define the subspace by H-^ = {W(A,A*) (AA*-A*A)x: x e H, W(A,A ) is a word in A ■x- and A ) . It is easy to show that reduces A ,

A |jj is normal, and if K reduces A and A |k is normal, then K c H-^ .

We can now restrict our attention to reductive normal- free operators in studying the reductive operator problem.

Proposition 1.9. If A is reductive normal-free on H , then a(A) = a (A) , that is, A has no eigenvalues, c

Proof. If \ e a (A) , then there is a non-zero vector J r X 6 h such that (\-A)x = 0 . This means V{x} reduces A and the restriction of A to this one-dimensional sub­ space is certainly normal, a contradiction. Hence ap(A) = = 0 . We saw earlier that or(A) = 0 for any reductive operator A so a(A) = o (A) . \0

Proposition 1.10. If A is reductive normal-free on H and i f 771 is an invariant sub space for A , then (Ax:x e Wi) is dense in 7H .

Proof. Since A is reductive, reduces A and A|^ is again reductive normal-free. We may, therefore, with­ out loss of generality, assume that % = H . Now, e ith e r A is invertible in which case A is onto, or else 0 e a (A) which again means A has dense range.

Our next result is an extension of Proposition 1.9.

Proposition 1.11. If p is any non-constant polynomial and if A is reductive normal-free on H , then

a (P (A)) = <*c(p(A)) •

Proof. It suffices to show that neither [p(A)] nor [p(A)] can have an eigenvalue. Suppose \ is an eigen value of p(A) . Factor the polynomial p(A) - \ in to n linear factors II (A-a.) (assuming without loss of ge- i= l 1 nerality that p is monic). If X is a eigenvalue of p(A) there is a vector x ^ 0 in H such that n i II (A-a. )x = 0 . As a result one of the a. s must be i= l 1 an eigenvalue of A , a contradiction of Proposition 1.9 •H* Thus p(A) has no eigenvalues. Since A is also re­ ductive normal-free, we can show in a similar manner that [p(A)] has no eigenvalues and the result follows.

Recall that an operator A is said to be quasinormal * if A commutes with A A .

Proposition 1.12. If A is quasinormal and reductive on H , then A is normal.

Proof. It suffices to show that A has no normal-free part. We will assume that A is normal-free, (because the normal-free part would also be quasinormal and reduc­ tive), and arrive at a contradiction. Since A is quasi- norm al, we have 0 = A(A A) - (A A)A = (AA -A A)A . The operator A has dense range by Proposition 1.10, so ,y, AA and A A agree on a dense subset of H . This means AA = A A , which cannot be because A is normal-free.

The full power of a reductive operator is not used in proving the above result. As a matter of fact the fol­ lowing statement is true:

Proposition 1.13. An operator A on H isnormal iff A is quasinormal and 7({A ) reduces A .

Proof. If A is normal it is certainly quasinormal. Also 7l (A ) is clearly invariant for A . If x e ?j(A ) then we need to show that Ax e 71 (A*) . But fo r such an x we have A (Ax) = A(A x) = 0 , hence 7?(A ) reduces

A • Conversely suppose that A is quasinormal and that

/ 9?(A ) reduces A . Then a |^ ( a *) 'the zero operator which is clearly normal, and we only need to consider• A(?Ka*)X . On 7l {A ) , A is still quasinormal and has dense range, so the proof of Proposition 1.12 shows that

A |^(a *)X is normal. Thus A is normal. It is well-known that an operator A in B(H) is invertible iff it satisfies the following two conditions:

(1.7) /?(A) is dense in H and (1.8) ||Ax|| > c||x|| for some c > 0 for all x e H .

Proposition 1.14. If A is reductive on H , then A is invertible iff (1.8) holds.

Proof. It suffices to show that (1.8) implies (1.7) for reductive operators. Suppose (1.8) holds and that ■X* ' ? e /?(A) . Then § is orthogonal to AA § and so (S,AA*§) = 0 . Thus (A*§,A*§) = 0 which implies •X — A § = 0 . Now, V{§) reduces A so we also have A? = 0 Since (1.8) holds, we have 0 = ||A?|| 2 CP|| ^d so ? = 0 It follows that A has dense range, i.e., (1.7) holds. We might remark here that Proposition 1.14 is also true if ''reductive'* is replaced by ' ‘normal**.

Proposition 1.15. If A is reductive on H and X s o(A) , then \ is an approximate eigenvalue for A , (that is, ^(A) = ir(A)) .

Proof. The number X is in a(A) iff (1.8) fails for the operator X-A . Thus for each n we can find a unit vector xn such that j|(X-A)xn|| < ^ . This means that X is an approximate eigenvalue for A .

Proposition 1.16. The set of reductive operators on H is not closed in the strong operator topology on B(H) .

Proof. Let H be a separable infinite-dimensional Hil­ bert space and let (e^:i £ Z) be an orthonormal basis for H . Define a sequence of operators on H by

i+i lf -n ^ 1 = < e -n if i=n (e± if |i| > n

•ft For each n , = I , and the sequence {^) converges to the bilateral shift in the strong operator Pn+l * * „ 2n topology. For each n , = I = so Kn = ^ • As a result, every invariant subspace for is * invariant for ty and V is reductive. The bilateral shift, as we saw earlier, is not reductive. It is natural to ask what the SOT (strong operator topology) closure of the set of reductive operators is. I t is known th a t the SOT clo su re o f th e s e t of normal operators is the set of subnormal operators (operators with normal extensions, cf. [12]) . If the reductive operator conjecture is true, the SOT closure of the set of reductive operators would necessarily be contained in the set of subnormal operators. If one can show that the set of reductive operators is SOT dense in B(H) , this would show the reductive operator conjecture to be false, because B(H) contains operators which are not subnormal. CHAPTER I I INVERSES OF REDUCTIVE OPERATORS

R. G. Douglas has asked whether or not the inverse of an invertible intransitive operator is intransitive (cf. [12]). This question is clearly equivalent to the question of whether or not the inverse of an invertible . transitive operator is transitive. A seemingly more ge­ neral question is indicated by the following:

Proposition 2.1. If the inverse of each invertible re­ ductive operator is reductive, then the inverse of each invertible transitive operator is transitive.

Proof. Suppose A is transitive and A"1 exists. Then A is reductive so A-1 is reductive by our hypothesis. Any invariant subspace for A ^ reduces A ^ , and hence reduces A as well. Since A has no proper reducing sub­ spaces, A"1 has no proper invariant subspaces and A ^ is transitive.

28 The converse of Proposition 2.1 has not been settled. According to (1.6), we can express a reductive operator A as a direct integral of a family (A(m)}meM 0:f transi­ tive operators. If A is invertible, then the 11 coordi­ nate operators'' A(m) are invertible for almost all m e M . If the inverse of a transitive operator is transi­ tive, then A”^ , being the direct integral of the family

{A"’'L(m) 3mejy[ 3 is a direct integral of a family of transi­ tive operators. The problem is that we do not know that a direct integral of a family of transitive operators is reductive. Indeed, we do not even know when the direct sum of two transitive operators is reductive. (However, we shall soon see that if transitive operators exist on Hilbert spaces of dimension greater than 1, it is not true in general.) The following results suggest that this problem is non-trivial.

Proposition 2.2 (Sarason). If A is a normal operator on H , then the weakly closed algebra generated by A is a *-algebra iff A is reductive.

Proof. The reader is referred to [20],

Proposition 2.3. If A is reductive and normal on H , then A © A is reductive and normal on H © H . Proo L’. Let W(A) denote the weakly-closed algebra ge­ nerated by A . Let T e W(A©A) and let ^ © Xg and yx © y2 be unit vectors in H © H . Let e > 0 be arbi­ trary. There exists a polynomial p such that

|((T-p(A©A))y1 © y2Jxi © *2) | < e/3 . Then |((T - p(A © A ))x1 © Xg,y1 © y2)| < e/3 also, where p denotes the polynomial whose coefficients are the conju­ gates of the coefficients of p . By Proposition 2.2, W(A) is a *-algebra, hence W(A) is strongly closed

(cf. [5], P. 43, [6], p. 333, [16], p. 448). Thus, there

— ■X’ exists a polynomial q such that || (p (A ) - q(A))xJ| < e/3 fo r i= l, 2 . Hence |((T*-q(A©A) )x1©x2, y.j© yg) | £ 1 |((T*- p(A*© A*))x1© x2 , y2© y2)| +

+ |((P(A*) - q(A))x1,y1)| + |((p(A*) - q(A))x2,y2)| <

< s/3 + s/31| y || + e/31|y2|| < e . The same c a lc u la tio n s work for any finite sequence [x^n^© x^n^, y^1^© ‘K’ and so T e W(A©A) . The operator A © A is clearly normal, (because A is), and W(A©A) is a *-algebra, so by Proposition 2.2 A © A is reductive.

Proposition 2.4. If A is any operator on H , and if A © A is reductive, then A is normal and reductive.

Proof. If A © A is reductive, then A must be reduc­ tive, because it is the restriction of A © A to the reducing subspace H © (0) . Consider the subspace of H © H defined by = (x © Ax: x e H] . This subspace is invariant for A © A , so by our hypothesis it is •X* also invariant for A © A . Let x e H be arbitrary. •X* -X* -X* •X* Then (A © A ) (x©Ax) = A x © A Ax e so by the defini- "X* *x* tio n o f % we have A Ax = AA x . Hence A is normal.

Corollary 2.5. The following statements are equivalent: (a) If A is reductive, then A © A is reductive. (b) If A is reductive, then A is normal (i.e., the reductive operator conjecture is true).

Proof. Suppose (a) is true and A is a reductive oper­ ator on H . Then A © A is reductive on H © H , and by Proposition 2 A A is normal. Conversely, suppose (b) is true and A is a reductive operator on H . Then A is both reductive and normal,so A © A is reductive on H © H by Proposition 2.3. Getting back to the problem of the inverse of an in­ vertible reductive operator, let us first recall the re­ solvent function X -* (x-A)”'1’ for X e p(A) . For any two vectors x,y e H , the mapping X -> ((x-A)_1x,y) is an analytic function on the open set p(A) .

Proposition 2.6. If the sub space of H is invariant for the operator A on H and if x is in the unbounded component of p(A) , then 771 is invariant for (X-A)-1 .

Proof. For |x| > ||A|| we have (X-A)- = £ X-p Ap . P=0 It follows that (X-A)- is invariant on 771 whenever |X | > ||A|| . Choose v e c to rs x e 771 Y e 771^ . Then the analytic function X -> ((X-A)_1x,y) is zero for | x | > ||A|j , hence it is zero everywhere on the unbounded component of P (A) . Since x e 7H and y e Tlf are a r ­ bitrary, the result follows.

Proposition 2.7. If 0 is in the unbounded component of P (A) , then A and A-1 have e x a ctly th e same in v a ria n t su b sp aces.

Proof. Since 0 is in the unbounded component of p(A) , we can find a curve T which contains a (A) in its in­ terior such that 0 is exterior to r j and such that p lies entirely in the unbounded component of p(A) . The fu n ctio n X ■* X-1 is analytic on r and its interior so, according to the operational calculus, we have A"1 = J x"1(X-A)“‘1dx . If 771 is an invariant sub- r space for A , it is also invariant for (X-A)"*1 for each X e p . As a result 771 is invariant for A-1 . Since 0 is also in the unbounded component of p(A"*1) , the rest of the proposition follows by symmetry. Corollary 2.6. Lf 0 Is in the unbounded component of p(A) , then (a) is transitive if A is transitive. (b) A"1 is intransitive if A is intransitive. (c) A~'L is reductive if A is reductive.

Proof. Statements (a) and (b) are immediate by Propo­ sition 2.7. Suppose A is reductive and % is invariant for . Then ^ is invariant for A by Proposition 2.7. Hence % reduces A , and so 7)\ reduces A-1 . This means that A”’*’ is reductive.

Corollary 2.9. If A is reductive, transitive or in­ transitive, respectively, there exists a translation (a+A) of A such th a t (a+A)- ^ e x is ts and is re d u c tiv e , transitive or intransitive, respectively.

Proof. We may assume A ^ 0 . Choose a = 2||A|| > 0 . Then 0 is in the unbounded component of p(a+A) , so we apply Proposition 2.8 to the operator ch -a .

The next result gives a sufficient (but not necessary) condition on a(A) to insure that 0 is in the unbounded component o f P(A) .

Proposition 2.10. If for some real number t , a (A) n • a (A) = 0 , then 0 is in the unbounded component of p(A) .

Proof. Choose 6 > 0 so that dist (a(A), ei^'a(A)) > > 36 , and cover a(A) with a finite collection of balls with radius 6 and centers in a(A) . Call this open cover U . It follows that n e V. ~ 0 by our choice of 6 . Each component of the boundary of V. is a Peano space. If no component of K separates 0 from 00 , there is a path from 0 to °° missing , and hence it also misses a(A) . Thus we may suppose that some component of separates 0 from 00 . Let C^ denote the unbounded component of C \ Uq and let B = bdry (V0) n C^, . Then B is a Peano space and B separates 0 from 00 (cf. [14], Theorem 3.3). Let Xq be a point on B with minimal modulus and let i*fc be a point on B with maximal modulus. Then e B , zL *fc so is path-connected to 0 in C \ e B because zL"fc components of C \ e B are path-connected. Likewise, n 1" iaQ ft e B , so |iQ is path-connected to 00 . Now, i "b e B separates 0 from *» because B does, and B is path-connected so B contains a path from Hq to . Then B must intersect e B , because every path from i t —■ i t —• X0 to |iQ intersects e B . This means ty f) e U ^ 0 , a contradiction. Hence no component of ty separates 0 from , so o(A) does not separate 0 from 00 . 35

Noto that the truth of the invariant subspace con­ jecture will imply that the inverse of an invertible transitive operator is transitive, (because all transi­ tive operators are then operators on one-dimensional Hilbert spaces)-. It was (and still is) hoped that the truth of the reductive operator conjecture would imply that the inverse of an invertible reductive operator is re­ ductive. At the time of this writing,a proof of the above implication has not been achieved. However, the following information may be enlightening: Suppose that A is a normal reductive operator and that A~'L exists. We let = (z e C: re(z) < 0) and A2 = (z e C: re(z) !> 0) . Then A^ and Ag are two dis­ joint Borel subsets of C , and we have two corresponding projections = E(A^) and Eg = E(a 2) i*1 bhe spectral family of A . Furthermore, E-^Eg = 0 = EgE^ because A^ and A2 are disjoint, and E^ + Eg = I because Ai u Ag = C . We l e t ^ = [x e H: E^x = x) and

Ai = AU :for i==1*2 • Then A = Al® A2 on the Hilbert i space H-j© Hg = H . Note that each of the operators A^ , Ag is normal and reductive (because H^ reduces Aj) , and that a(Ai) c n a (A) so that the hypothe­ ses of Corollary 2.8 are satisfied. As a result A ^ and Ag"1" are reductive operators on the Hilbert spaces 11^ iind 112 respectively. This means A-1 = A^1© Ag1 is the direct sum of two reductive normal operators. We know that A^, Ag, Ag1 and A-j© Ag are normal and reductive. Can we prove that A^1© k^" is reductive? The fact that A^© Ag is reductive must not be neglected because the statement is false otherwise. (In other words, it is false that the direct sum of any two reduc­ tive operators is reductive.) For example, let ^ be a bilateral shift and decompose % in the manner described

— 1 * above. We then have = “Uj® tyg and for i=l,2 . Since every invariant subspace for is in- — 1 * variant for ^ , 1=1,2 , we have the result that the non-reductive operator U is the direct sum of two reductive operators. Thus we have:

Proposition 2.11. The direct sum of two reductive oper­ ators need not be reductive.

The same example can be used to demonstrate the fol­ low ing:

Proposition 2.12. The sum of two commuting reductive operators need not be reductive.

Proof. Let % be a bilateral shift and let A-^ , Ag be as before. Then U = + tyg where for ±■-1,2 . Clearly 2 = 0 = ^2^1 so we only need t0 ahow that and l<2 are reductive. Now

^1 = % |H © 0 and U2 = 0 © ^Ih where Hi * H2 are as in the preceding example and W(t(1) = W(t(1|H ) © (0) ,

W(^2) = (0) © . By Proposition 2.2, W ^ ) and W(t(2) are *-algebras, so and ty2 are reductive as d e sire d . CHAPTER I I I POWERS OP REDUCTIVE OPERATORS

In this chapter we will investigate powers of oper­ ators. In particular we will be concerned with finding sufficient conditions on an operator A on H that sub- spaces invariant for An (n^O) are invariant for A . We shall restrict our attention to the case where 0 is in the unbounded component of p(A) . This implies that there is a path S from 0 to 00 which misses a(A) . We may, without loss of generality, assume that S is simple and that S coincides with the negative real

axis in some neighborhood of 00 . For such a path S , we will define Lg(X) to be the branch of the logarithm function on C \ S which is real for large positive values o f \ . The function Lg(X) is then analytic on a (a) , so we can choose a path r which encloses a(A) and which does not intersect S and define the operator

38 39

(3.1) LS(A) = 2^1 LgdXx-A)'1^ r by the operational calculus.

Proposition 3.1. If Lg(A) is the operator defined above, then A = exp (Lg(A)) .

Proof. The function exp Lg(\) is analytic on a (A) , so exp (Lg(A)) = 2^1 J exp (Lg (\)) (\-A)- 1 d\ r = X-Cx-Aj^dX r = a .

Thus it is possible to recover the operator A from the operator Lg(A) .

Proposition 3.2. If a path S from 0 to «> misses a(An ) , n^O , then A = exp Lg(An )) .

Proof. The function exp • Lg(xn)) is analytic on o(A) , so

exp (n LS(An)) = 2^ r J exp (n LS^ n) ) r

2v i rI exp (Lo(\))(\-A)“ ° 1d\ 4 0

- air J r

= a .

Proposition 3.3. If S is a path from 0 to °° missing a (A) , then A and Lg(A) have e x a c tly the same in ­ variant subspaces.

Proof. If is invariant for A , then (X-A)-’*' is invariant on ^ for all X e T because r lies in the unbounded component of P(A) . Thus I»g(A) is invariant

on % . Conversely, if % is invariant for Lg(A) , it is also invariant for exp (Lg(A)) = A .

Corollary 3.4. If S is a path from 0 to 00 m issing o(A) , then (a) A is transitive iff Lg(A) is transitive (b) A is reductive iff Lg(A) is reductive.

Proof. Part (a) is immediate by Proposition 3.3. Suppose A is reductive and 7H is invariant for Lg(A) . Then ^ is invariant for A , so reduces A . The pro­ jection P which maps H onto % commutes w ith A , hence P commutes with (X-A)"^ for each X e T , hence P commutes with Lg(A) . This means that % reduces A . Conversely, If Lg(A) is reductive and 7l[ i s in ­ variant for A , then % reduces Lg(A) by Proposi­ tion 3.3. This means that P commutes with Lg(A) > hence P commutes with exp (Lg(A)) = A and 771 r e ­ duces A .

Proposition 3.5. If n^O and if S is a path from 0 to oo missing a(An) , then every invariant subspace for An is invariant for A .

Proof. By Proposition 3.2 we have A = exp (—• Lg(An)) . I f % is invariant for An , it is also invariant for Lg(An) by Proposition 3.3. Thus it is also invariant for exp (n ' LS

If n > 0 , the converse of Proposition 3.5 is i clearly true. If n < 0 and if S is a path from 0 to oo missing a(A) , then every invariant subspace for A is invariant for A -1 by Proposition 2.7, and so is invariant for An . Thus we have:

Corollary 3.6. (a) If n < 0 and if S and S* are paths from 0 to 00 missing a (A) and cr(An) respec­ tively, then A and An have exactly the same invariant subspaces.

(b) If n > 0 and if S is a path from 0 t o oo m is- sing o(An) , then A and An have exactly the same invariant sub spaces.

Proposition 3.7. If n^O , if S is a path from 0 to <» m issing 0(An ) > and if A is reductive, then An is reductive.

Proof. If An is invariant on % , then so is A by Proposition 3.5. Thus the projection P onto 7!\ com­ mutes with A , hence it commutes with An .It follows that An is reductive.

Proposition 3.8. (a) If n > 0 , if S is a path from 0 to oo missing a(An) , and if An is reductive, then A is reductive. (b) If n < 0 , if S and S* are paths from 0 to oo missing a(A) and o(An) respectively, and if An is reductive, then A is reductive.

Proof of (a). Suppose fn is in v a ria n t fo r A . Then % is also invariant for An , so by our hypothesis, % is invariant for (A*)n = (An)* . The path S’ = e S} misses cr((A ) ) , so 7ft is invariant for A by Proposition 3.5. This proves that A is reductive.

Proof of (b). Suppose 7ft is invariant for A . By Corollary 3.6 (a) is also invariant for An . Since A is reductive, ?/j is invariant for (A ) , and again ^ T1 S is a path from 0 to «> missing a((A ) ) , so 7ft * is invariant for A by Proposition 3.5. Thus in this case we also have shown that A is reductive.

One consequence of Proposition 3.5 is this: If S is a path from 0 to °° missing

Proposition 3.9. If A is a reductive normal-free oper­ ator on H and n^O , then any proper invariant subspace for An must be infinite-dimensional.

Proof. We will suppose that An has a finite-dimensional invariant sub space 7H and show th a t we must have 7H - (0 ) V It First we note that A (flj), the image of 7K under A , i s finite-dimensional for every k e Z . If n > 0 we will l e t 57^ = % + A(tfj) + ••• + An-1(??0 , the linear manifold generated by the manifolds {A^(Wl):0 < k < n-l) . Notice

th a t 7%^ is finite-dimensional and hence 7ftn is closed (i.e., ^ is a subspace). Also ^ is invariant for A because A(An'* 1 ({9J)) = An(5 n) c ^ . Because A is reductive, the restriction of A to 7/^ -is reductive. However, any reductive operator on a finite-dimensional space must be normal (cf. Ando [1]). Thus we must have ^ = {0) and 7K = {0) , because A is normal-free. In the case where n < 0 , we only need to note that An (??i) c im plies 7H c A“n(#|) , and since A“n cannot increase dimension we have = A~n($) . Thus Wj i s i n ­ variant for A-n and -n > 0 . Now the proof for positive powers applies and we conclude that Wl = {0 ) . CHAPTER. IV RESOLVENT TECHNIQUES

This chapter grew out of an attempt to find invariant subspaces for quasinilpotent operators. Recall that the resolvent function X -» (x-A)“'L of an operator A on H is an analytic function defined on the open set p(A) . This means that, for any two v e c to rs x and y in H , the fu n ctio n X ■* ((X-A) *^x,y) is analytic on P(A) . For |X | > ||A|| the equation (X-A)~ = £ X"p“ A is valid, where the series converges p =0 in the uniform topology on B(H) . This gives the Laurent expansion

(4.1) ((X-A)- 1x ,y ) = S (kVx,y)X~v ~1 , |XI > ||A|| . p =0

By ' 1 resolvent techniques11, we w ill mean growth conditions on the resolvent function. These have been investigated by several others previously. The reader is referred to Putnam [19] and Stampfli [21] and to their

45 bibliographies for further information. An operator will be called a (Gn) operator if the inequality

(4.2) || (^-A)”1|| < K/(dist 0,o(A )])n is satisfied for some positive integers K and n for all X e p(A) . If this inequality is satisfied on a set U H P(A) , where H is a neighborhood of a(A) , we will say that A is locally (Gn) . If XQ e a(A) , and if there exists a neighborhood V. o f x 0 such that (4.2) holds in K n P(A) , we w ill say th a t A is (Gn ) a t XQ . Thus a locally (Gn) operator is (Gn) at each point in its spectrum, and conversely, if an operator is (Gn) at each point in its spectrum it is locally (Gn) .

If XQ e ct(A) , and if (^^3^=1 is a sequence in p(A) converging to xQ such that inequality (4.2) holds for each \k , we will say that A is sequentially (Gn) at XQ . A sequentially (Gr ) operator is an operator which is sequentially (Gn) at each point in its spectrum. Clearly, A is sequentially (Gn) at XQ if it is local­ ly (Gn ) X0 • A weaker growth condition on an operator A is given by the inequality

(4.3) |((X-A)- 1x,y)| < K/[dist (X,a(A)]n , whore x,.y II aiKi K,n are positive Integers. An oper­ ator which satisfies (^.3) on p(A) will be called a (Gn,x,y) operator. We can define the notions of locally (Gn,x,y), (Gn,x,y)at \Q e a(A) , sequentially (Gn,x,y) a t \ q e a ( a ) , and sequentially (Gn,x,y) in a manner analogous to the case for (Gn) operators above. It has been shown that every hyponormal operator A (that is, A*A-AA* is positive) is a (G^) operator (cf. [18], p. 58). A (G1) operator is locally (Gn) for n > 1 ; simply let K = (z e P(A): dist (z,a(A)) < 1} Thus we have many examples of operators which satisfy the above restrictions. It is easy to show that any of the above restrictions are invariant under translations and multiplication by non-zero scalars, so if A satisfies one of the above restrictions so do a+A and PA for

P 4 0 . An important and still unsolved problem is the ques­ tion of whether or not every quasinilpotent operator has a proper invariant subspace. In what follows, we will impose a growth condition of the type (^. 2 ) on a quasinil­ potent operator. The result obtained shows that this is a very strong restriction on a quasinilpotent operator. It forces the operator to be nilpotent. Note that for the case of a quasinilpotent operator 48

A we have dist [\,a(A)] - |\| .

Proposition 4.1. If A isquasinilpotent, and if there exists a neighborhood of 0 and an analytic function f(x) on such that ||f (x.) (\-A)”'L|| < K for some con­ stant K and for all 0 4 X € K > then A is nilpotent. Conversely, if A is nilpotent of order N , then ||XN(X-A)-1|| is bounded in a neighborhood of 0 .

Proof. Suppose f(X) is analytic and ||f(X) (x-A)'"'L||

||\N(X-A)_1|| = \\*Z ApxN“p_1|| < N 21 ||Ap|| fo r |x| < 1 p =0 ~' p =0

Thus we see that a quasinilpotent operator A is nilpotent iff it is locally (Gn) for some n . For sequentially (&n) operators our results are not as im­ p re ssiv e .

Proposition 4.2. If A is sequentially (Gn) and nilpotent of order m , then m

. 0 0 , , _ Proof. We have a sequence j Jconverging to 0

HU oh t.hnt. II fX £ TC/lx J n fnv 1= 1. . P.... . S i n c e

A = 0 we have (X.-A)"1*- = . 2. xlp" Ap , so 3 p =0 3 o m_1 « i « m-1 t ||X?( 2 Xj A)|| = || 2 X“"P_1AP|| < K . I f m >n , we J p =0 d p =0 3 — m—2 have||XrJ"mAm~1|| - || 2 X1?"15”^ ! ! < K , b u t th isis im- 3 p =0 3 — possible because

||xn-nAm-l|| . y s p =0

•||AP|| , and the right hand side of the above in­ equality becomes infinite as Xj -> 0 . Hence m < n as claim ed. This means that a sequentially (Gn) nilpotent oper­ ator is locally (G ) . In particular, a sequentially 50

(G^) nilpotent operator must bo the zero operator. A similar result by B. Wadhwa [22] states that any sequen­ t i a l l y (G^) a lg e b ra ic o p erato r i s norm al.

Proposition 4.3. If A is quasinilpotent and sequen­ tially (Gn) , that is, if A is quasinilpotent, \k -♦ 0 and ||xk(\k-A)-1|| is bounded, then ||An(xk-A) _1|| is bounded. In particular, 0 is an approximate eigenvalue fo r A .

Proof. Since A is quasinilpotent we have (Xk-A)-^ = = Z \ t P_1Ap . Let K be a bound for ||Mr(Xir“A)-1|| . p =0 K Then K > «x“ ( 2 xrP-^APjII = || 2 > || E X^'Vll - - p =0 p =0 p=n

- II Z Xv A II • There is no loss of generality in p =0 . V assuming that lxkl < 1 for each k . Then

IjHs1X^“P -1AP|| < n ^ 1 ||AP|| . I f we l e t K* = K + *2 ||AP|| p=0 K - P=0 p=0 we have || Z x£"P"1AP|l-= || ^ xkP" 1An+P|| 5=||An ( B XkP_ 1AP) || = p=n p =0 p =0

= llAn(^k*’A^ — K' as ^e s ^r e d * Next we w ill show th a t 0 e 7r(An ) . Since || (xk“A) ^\\ -* 00 as k -♦ °° we may assume || (Xk“A.)-1|| > k . Then we can find a unit vector xfc such that || (Xk-A) *-1 xk|| > k . Let 51

_ (xk~A> It follows that K ’ || (xk-A)'lxJ

1— ” ll< £ * 0 as k 00 and 0 e 7r(An)

as claimed. Now by Problem 59 of [12], ^(A31) = [7r(A)]n = = (0) so ir(A) = {0} and the proof is complete.

A growth condition more closely related to the pro­ blem of finding invariant subspaces for quasinilpotent oper­ ators is given by the inequality (^.3). We first prove the following result for an arbitrary operator in 33(H) .

Proposition 4.4. If A e B(H) , x and y are non-zero vectors in H , and the function X ■* ((\-A)~^x,y) on p(A) has a meromorphic extension to the entire complex plane, then A has a proper invariant subspace.

P ro o f. Suppose f(X) = ( (X-A) “^x^y) has a meromorphic

extension F(x) to all of C . Let ai'* ',>an be biie distinct poles of F , and let be the order of the 1 n Y, pole at cu for i=l, •**,n . Let G(x) = [ H (X-ct^) ]F(x) . i= l Then G(x) is analytic on all of C . For any positive

integer m , and any curve C in P(A) en clo sin g

a(A) 3 we have 52

(Am[ n (A-c^) ^x.y) = Ju ” n [(X-a1)Yl](x-A)"1x,y)d\ i —1 p i —1

=■ 2^: I = o . r n Y± Since y ^ 0 , th e v e c to r II (A-a^) x is not cyclic i= l 1 for A . If this vector is non-zero, A has a proper n y * invariant subspace, namely V(Am II (A-cl, ) 1x: m e z+ U {0)} . i= l n Y. I f TI (Aa.) x = 0 then one of the numbers cu is an i= l 1 1 eigenvalue for A and again A has a proper invariant subspace.

The next result is a converse to Proposition 4.4.

Proposition 4.3. If A e B(H) has a proper invariant subspace, then there exist non-zero vectors x,y e H such that the restriction of the function f(x) = ((X-A)“^x,y) to the unbounded component of p(A) has a meromorphic extension to all of C .

Proof. Suppose is a proper invariant subspace for A . Choose non-zero vectors x e ^ and y 6 Iff • By Proposition 2.6, = 0 for all X in the unbounded component of P(A) . Thus F(x) = 0 for all X e c is a meromorphic extension to all of C . Thus, in the case where o(A) is a finite set of points, A has a proper invariant subspace iff for some vectors x,y ^ 0 the function X -♦ ((x-A)_1x,y) defined on P(A) has a pole (possibly of order 0) at each point in o(A) . For a quasinilpotent operator A , we have •

Corollary 4.6. If A is quasinilpotent, then A has a proper invariant subspace iff |((X-A)-1x,y)|

Thus we see that a transitive operator on a separable infinite-dimensional Hilbert space (if one exists) must be badly behaved. That is to say, for any pair of non­ zero vectors x,y e H , the function X -» ((x-A) ^x,y) must have an essential singularity at \ = 0 . In light of the bad behavior of transitive quasinil­ potent operators as witnessed above, it is clear that a transitive quasinilpotent operator on a separable infi­ nite-dimensional Hilbert space must be sequentially (GR,x,y) for any integer n > 0 , and for any vectors x,y e H . This is true because the function X -♦ ((x-A)"'1'x,y) has values arbitrarily close to 0 in any neighborhood of 0 . Thus the notion of sequentially (Gn*:x>y) operators w ill not aid the search for invariant subspaces for quasinilpotent operators. In fact, we have the following result:

Proposition 4.7. If the dimension of H is greater than 1 , and if A is a quasinilpotent operator on H , then A is sequentially (GQ,x,y) for some non-zero vectors x and y in H .

Proof. If A has a proper invariant subspace, we choose non-zero vectors x e 7K and y e 7HX . Then for any se­ quence {Xk3 converging to zero we have ((xk-A)""'Lx,y) = 0 If, on the other hand, A has no proper invariant sub­ spaces we choose x and y arbitrarily. Since the func­ tio n \ ((X-A) ’"1x,y) has an essential singularity at 0 , we can choose a sequence {Xk3 converging to zero such that |((Xk-A)_1x,y)| < 1 for each k . BIBLIOGRAPHY

1. Ando, T., ' ‘Note on Invariant Subspaces of a Compact Normal Operator,' ’ Archiv der Mathematik, 14(1963), 337-340.

2. Aronszajn, N. and K. T. Smith, 11 Invariant Subspaces of Completely Continuous Operators,' r Annals of Mathematics, 60(1954), 345-350.

3. Bernstein, A. R., and A. Robinson, ’’Solution of an Invariant Subspace Problem of K. T. Smith and P. R. Halmos,’' Pacific Journal of Mathematics, 1 6 (1966), 421-431.

4. Chernoff, P. R., and J. Feldman, ’’Invariant Subspaces for Analytically Compact Operators,’’ Michigan Mathematics Journal, 17(1970), 23-27.

5. Dixmier, J., Les Algebres d'Operateurs dans l'Espace Hilbertien, Gauthier-Villars, Paris, 1957.

55 56

6 . Dixmier, J Les C -Algebres et leurs Representations, Gauthier-Villars, Paris, 1964.

7. Dunford, N. and J. T. Schwartz, Linear Operators, Parts I and II, Interscience Publishers, Inc., New York, 1957 and 1963.

8 . Dyer, J. and P. Porcelli, '»Concerning the Invariant Subspace Problem,’’ preliminary report, Notices of the American Mathematical Society, 17(1970), abstract number 677-47-4.

9. Fuglede, B., ’’A Commutativity Theorem for Normal O p e ra to rs ,’ 1 P roceedings o f th e N atio n al Academy of Science, 36(1950), 35-40.

10. Halmos, P. R., Introduction to Hilbert Space, Chelsea, New York, 1957.

11. Halmos, P. R., ’’Invariant Subspaces of Polynomially Compact Operators,' ’ Pacific Journal of Mathematics,

16(1966), 433-437.

12. Halmos, P. R., A Hilbert Space Problem Book, Van Nostrand, Inc., Princeton, N. J., 1967.

13. Halmos, P. R., ’’Ten Problems in Hilbert Space,11 Bulletin of the American Mathematical Society,

76(1970), 887-933. 1*1. Hocking, J. G. and G. S. Young, Topology, Addison- Wesley, Reading, Mass., 1961.

15. Lorch, E. R., Spectral Theory, Oxford University Press, New York, 1962.

16. Naimark, M. A ., Normed R ings, Groningen, The N ether­ lands, 1964.

17. Pedersen, E. A., A Decomposition Theory for Rings of Operators, dissertation, LSU(1966),(unpublished).

18. Putnam, C. R., Commutation Properties of Hilbert Space Operators and Related Topics, Springer-Verlag,

New York, 1967.

19. Putnam, C. R., ''Almost Normal Operators, their Spectra, and Invariant Subspaces,' 1 Bulletin of the American Mathematical Society, (to appear).

20. Sarason, D., ' 'Invariant Subspaces and Unstarred Operator Algebras,1 * Pacific Journal of Mathematics,

17(1966), 511-517.

21. Stampfli, J. G., ''A Local Spectral Theory for Oper­ ators III: Resolvents, Spectral Sets and Similarity, Transactions of the American Mathematical Society,

1 6 8 (1972), 133-151. Wadhwa, 13. L., 1 * Operators Satisfying a Sequential Growth Condition,1' preliminary report, Notices of the American Mathematical Society, 19(1972), abstract number 700-B6.

Wermer, J., *1 On Invariant Subspaces of Normal Operators,*f Proceedings of the American Mathema­ tical Society, 3(1952), 270-277. VITA

Thomas P. Wiggen was bom in Minneapolis, Minnesota on June 29* 1945. He moved to Portland, North Dakota at an early age and was graduated from Portland High School in 1963. He entered the University of North Dakota in January, 1964, where he received his B.S. degree in June, 1967* and his M.S. degree in June, 1969. In September, 1969* he e n ro lle d in th e G raduate School o f Louisiana State University. He is presently a candidate for the degree of Doctor of Philosophy in Mathematics.

59 EXAMINATION AND THESIS REPORT

Candidate: Thomas P. Wiggen

Major Field: Mathematics

Title o£ Thesis: On Reductive Operators

Approved: g 7r ,£ Major Professor and Qmairman

. ______Dean of the Graduate School

EXAMINING COMMITTEE:

Z u l

Date of Examination:

A pril 2k, 1973