NEW ZEALAND JOURNAL OF MATHEMATICS Volume 23 (1994), 11-18

L O M O N O S O V ’ S T H E O R E M A N D E S S E N T IA L L Y N O R M A L O P E R A T O R S

S c o t t W . B r o w n

(Received August 1993)

Abstract. An invariant subspace theorem due to Lomonsov is generalized. This generalized theorem is then used to study essentially normal operators. It is shown that if T is a bounded essentially normal operator on a Hilbert space, then for some nonzero x and y in the Hilbert space and some Borel measure on the spectrum of T, it holds that f pdfi = {p (T )x ,y ) for all polynomials p.

The invariant subspace problem for Hilbert space operators remains unsolved. Lomonosov [3] proved a very strong invariant subspace result for compact opera­tors on a Banach space. This author [1] and others have proved invariant subspace results for certain kinds of operators on a Hilbert space. Many of these Hilbert space proofs depended on an understanding of the algebra generated by the par­ticular operator involved. In the following, Lomonosov’s theorem is generalized using the same techniques as those given in [3]. Independently, Lomonosov [4] has developed generalizations that include those given here, but the techniques do not follow so directly from [3]. In this note, these generalizations are used to examine the algebras generated by certain Hilbert space operators. Specifically, if a given operator and the algebra that it generates are well understood, then some (very mild) statements can be made about the algebra generated by a compact perturbation of the given operator.

Let H be a complex, infinite dimensional, separable Hilbert space, and let B(H) denote the space of bounded operators on H. Let )C(H) denote the space of compact operators on H, and let n : B(H) —> = C denote the Calkin map of 13(H) onto the Calkin algebra. Let E = {B e B(H) : ||7r(5)|| < 1 }. For A C B(H), let£A = £ n A.

T h eorem 1.1 (L om on osov ). Let A be a commutative subalgebra of 13(H). Then there exists nonzero x € H such that &a x = {Ax \ A € £a } is not dense in H.

P ro o f. It suffices to find x / 0 such that j$q£a x is n°t dense. Suppose j^ S a x is dense for all x ^ 0. Then choose xq € H with ||#o|| = 2. Let D = {.x € H :11̂ — £o|| < !}• For each x € D choose Ax e such that

\\Axx - x0|| < 1/4.

Now Ax = Tx + K x, where |]T3;|| < 1/50 and K x is compact. So, choose a weak neighborhood Vx of x such that \\Kxy — K xx\\ < 1 /4 for all y e Vx. Set Ux = VxnD.Then for y £ Ux,

II A xy - Axx\\ < ||Txy - Txx\\ + \\Kxy - K xx\\ < (1/50)2 + 1/4 < 1 / 2.

1991 A M S Mathematics Subject Classification: Primary 47B15, Secondary 47B20, 47A15.

12 SCOTT W. BROWN

Hence ||Axy — x0\\ < 1 for all y G Ux. So

AXUX C D. (1)

Next note that {Ux}x&d is a weak open cover of the weakly compact space D. Choose a finite subcover Ui, U2 , . . . , ,Un of { U x } x € e>. Let Ai be the operator associated with Ui by (1). Next choose weakly continuous functions fi, f 2 , ■ ,, fn on D such that 0 < /* < 1, supp(/i) C Ui, and XlILi fi = 1- Set

n

9(x ) = ^ 2 fi(x)Ai(x) for all x G D.i=i

By Schauder’s fixed point theorem, there exists point y G D such that g(y) = y. Set

n

B = Y ,h (y )A i-i= 1

So = y. Note that B is not the identity operator since it is in j^E ^x , and so its eigenspace with eigenvalue one is a nontrivial closed invariant subspace for the entire algebra A. For any element, x, in this eigenspace, j^ S ^ x is not dense, and a contradiction has been found. I

C oro lla ry 1.2. Let A be a commutative subalgebra of B{TL). Then there exist x ,y E H neither of which is zero such that \(Ax,y)\ < ||7r(A)|| for all A £ A.

P ro o f. Using the theorem choose x such that -^Sj^x is not dense. By the Hahn- Banach theorem, there exists y G H such that \(Ax,y)\ < 1 for all A G £ 4 . From this the result follows. |

Given x, y G Ti let x <8> y denote that linear functional on A given by (A , x<S>y) = (Ax,y) for all A € A. Now 1.2 can be restated:

C oro lla ry 1.3. Let A be a commutative subalgebra of B(TL). Then there exist nonzero x ,y G H , and there exists tp G C* such that

(A ,x ® y ) = (A,n*(<p))

for all A G A.

P ro o f. Let x ,y G 7i be as provided by 1.2. Consider the map on ir{A) defined by tt(A) —> (Ax , y) for all A G A. This map is well defined and continuous by 1 .2. By the Hahn-Banach theorem, there exists ip G C* such that

(Ax,y) = {?r(A),<p)

for all A € A. The result follows. I

The reasoning of 1.3 is repeated below for a specific case. An operator T G B(H) is called essentially normal if 7r(T) is normal.

LOMONOSOV’S THEOREM AND ESSENTIALLY NORMAL OPERATORS 13

T h eorem 1.4. Let T E B(H) be essentially normal. Then there exist nonzero x ,y E H and Borel measure fi on cr(T) such that

P ro o f. Let A = {p(T) | p is a polynomial}. Let C (cr(ir(T ))) denote the Banach space of continuous functions on <r(7r(T)) using the sup norm. Let V denote the subspace of C (cr(7r(T))) consisting of polynomials restricted to cr(ir(T )). Now 7r(T) is normal, so the map given by p —* p(7r(T)) is an isometry from V into 7t(A). Apply 1.2 to A to find x ,y E H such that \(Ax,y)\ < ||7t(̂ 4)|| for all A E A. Then the map defined by p(jr(T)) —> (p(T)x, y) for all polynomials p is continuous on n{A). And so the map on V given by p —> (p(T)x,y) is continuous. By the Hahn-Banach theorem and the Riesz Representation theorem there exists a Borel measure fx on a{ir(T)) such that

With minor modification with respect to notation, this theorem can be made to work for rational functions in T (not just polynomials in T).

U ltraw eakly C on tin u ou s R epresentation sThe general case will be considered in this section. Later essentially normal

operators will be examined. The basic notions of dual algebras make it easier to state some of the results, and these notions are outlined below.

The space of bounded operators B(7i) on Hilbert space H is isometrically iso­morphic to the space of trace class operators on H endowed with the trace norm, || ||i. The reader is referred to [6] for more information than that provided directly below. The dual pairing is given by (A , K) = trace A K for A E B(H) and K E C\(H). Hence B(7i) has a weak* (i.e. a(B(7i),Ci(7i))) topology on it. Often this topology is referred to as the ultraweak or cr-weak topology, and all three names will be used interchangeably below. Another topology on B(H), called the weak topology, is that induced by only the finite rank trace class operators. Any subalgebra of B(7i) that contains the identity operator and is weak* closed is called a dual algebra. If A is a dual algebra, then define A ± = {K E Ci{7i) | (A, K) = 0}. Define A * = )• Now (.4*)* ~ A. So A is a dual Banach space and has a weak* topology on it. O f course there may be other preduals to a given dual algebra, but the one just defined will be the only one considered throughout this paper.

A typical rank one trace class operator has the form x <g> y for some x ,y E H, where (x <g> y)z = (z,y)x for all 2 E TL. This induces a linear functional on B(H) as provided directly above, and (by restriction) induces a linear functional on any given dual algebra A. The linear functional so defined coincides with that defined just prior to Corollary 1.3; so, the notation is consistent.

pd/i = (p(T)x,y) for all polynomials p.

/ pdfi = (p(T)x,y) for all polynomials p. Jcr(T)

Here, the fact that <j(tt(T)) c cf(T) has been used. I

14 SCOTT W. BROWN

If A and B are dual algebras then a map $ : A —> B will be called a dual algebra homomorphism if it is a unital algebra homomorphism, is continuous when domain and range are endowed with their respective weak* topologies, and if ||3>|| = 1. If there exists such a $ which is also an isometry with range equal to B, then A and B will be said to be dual algebra isomorphic, written A ~ B.

Given a dual algebra A , a pair (p, K,) is called a representation of A if /C is a Hilbert space and p : A ^ B(K) is a, unital algebra homomorphism of norm 1. If in addition p is a dual algebra homomorphism (i.e. if p : (A, weak*) —> (B(K), weak*) is continuous) then (p, 1C) will be called an ultraweakly continuous representation. There is an obvious construction that shows that if (p, 1C) is any representation of A , and if p*(si <S> S2) is weak* continuous on A for some s i ,s 2 € /C, then (p, K) can be modified to yield an ultraweakly continuous representation. In Theorem 2.2, the ultraweakly continuous representation (p o 7r,<S) is constructed using this technique.

L em m a 2.1. Let X be a Banach space that is a dual space. Let linear T : (.X,weak*) —► (B(H),weak) be continuous with ||T|| finite. Then T : (X : weak*) —> (B(Ti), weak*) is continuous.

P ro o f. Assume wolog ||T|| < 1. View the predual X* of X in canonical fashion as a closed subspace of X * . Let e > 0 be arbitrary. Let K € C\(H). Then there exists a finite rank operator F such that ||F — K ||i < e. Note that T*F e X* by hypothesis. Furthermore ||T*F — T *K || < e. Hence T*K is within e of X*. Since e > 0 is arbitrary, the result follows. I

T h eorem 2.2. Let A C B(H) be a dual algebra. Let <p be any linear functional on C, the Calkin algebra, such that 7r*(<p) is weak* continuous on A. Then there exists a Hilbert space S, and r,s € *5, and a map p : n(A) —>• B(S) such that

(1) p : ^{A) —> B(S) is a unital algebra homomorphism, and ||p|| < 1.(2) The map p o n : A —» B(S) is a dual algebra homomorphism.(3) For all A e A,

(A, 7T*(<p)) = ((po7r(A ))r,s).

P ro o f. By the Gelfand-Naimark-Segal theorem (see [2] p. 257, or [7] p. 322), there exist a Hilbert space /C, and vectors r, s € /C, and ^-homomorphism (3 of C into B(K) such that

(T,<p) = (0(T)r,s)

for all T € C.Now set M. — norm closure in /C of {(3 o Tr(A))r. Set J\f = {n e M \ ((/3 o

7r(A))n,s) — 0 for all A E ^4}. Let P be the orthogonal projection of /C onto S where S = M .Q N . Define p by p(n(A)) = P(J3 o 7x(A))P. In natural fashion, p can now be viewed as a map into B(S).

Now ||/3|| < 1, so ||p|| < 1. The space M is invariant under /3o-k(A), and therefore statement (1) of the theorem holds. Setting r = Pr and s = Ps yields the final statement of the theorem.

LOMONOSOV’S THEOREM AND ESSENTIALLY NORMAL OPERATORS 15

To prove (2), it only has to be shown that p o 7r is ultraweakly continuous. Let w , v G <5, then w <8> v is an ultraweakly continuous linear functional on p o n ( A ) . Let W = {w(8)v G Ci(S) | w , v G S and ( p o 7r) *(w®v) G A * } . Now ||(/9° 7r)*|| < l ? so W is norm closed in 71 = {w <g) v G Ci(<S) | w, v G <S}. For fixed Ai,^42 £ -4, the map given by A —> ,A2 4̂̂ 4i —> ((/3 o 7r(A2A A i) ) f , s) = ((p o 7r(A2 4̂A i))r , s) = (p o 7r(^4), (p o n(Ai))r ® (p o 7r(A2))*s) is weak* continuous [note (p o 7r(A2))* is the Hilbert space adjoint of (p o 7r( 4̂2))]. So, Q = {(p o 7r(^4i))r ® (p o 7r(A 2))* s | A i,A 2 G .4} is a subset of W . By definition of .M and A/", the set Q is dense in 7Z. Hence W = 7Z. From this it follows that any finite linear combination of elements from 7Z is mapped via (p o n ) * to an ultraweakly continuous linear functional onA. The hypotheses of 2.1 are therefore satisfied and the result is established. |

T h eorem 2.3. Let A C B(H) be a commutative dual algebra. Then there exist nonzero x, y G H and p, S, and r,s G S as in the statement of 2.2 such that conclusions ( 1 ) and (2) hold as well as

(3') For all A 6 A,(Ax, y) = ( ( p o 7r(A ))r,s).

P ro o f. Just use 2.2 and 1.3. I

Theorem 2.3 can be modified slightly.

T h eorem 2.4. If A is a commutative dual algebra with no nontrivial invariant subspace, then the conclusion on 2.3 can be made to hold with r = s.

S ketch o f P ro o f. Choose x, y G H as in the conclusion of 1.2. Then the norm closure, S a x , of Ea % cannot be a subspace, or an invariant subspace has been found. Choose a supporting hyperplane (given by inner product with y G H) which does not pass through the origin; and, let x G S a x be a support point for this hyperplane. Assume wolog (x,y) = 1. Note that S a x C S a x and that y provides a supporting hyperplane for S a x with support point x. Using the reasoning of 1.3 and the Hahn-Banach theorem, choose <p G C* of norm one such that (w(I),<p) — 1 whereI is the identity of B(7i), and 7r*(<p) and x <8> y agree as linear functionals on A. By Gelfand-Naimark-Segal there exists a representation {(3, K) of C such that is represented by some r <g) r for some r G 1C. Continue as in the proof of 2.2. |

Finally, it is noted that very often for an arbitrary dual algebra A there ex­ist nonzero ip G C* such that 7r*(tp) is weak* continuous on A. The power of Lomonosov’s theorem is to force the weak* continuous linear functional to be of the form x ® y . The following theorem is really one concerning weak* closed subspaces of X** where X is any Banach space. In this case, X** — B(7i) and X = K,(H). However, the result is stated and proved in terms relevant to the situation at hand.

T h eorem 2.5. Suppose A C B(H) is a dual algebra. Let V be the set of compact operators in A. If A is not the weak* closure of T>, then there exists ip G C* such that 7T*(ip) is nonzero and weak* continuous on A.

16 SCOTT W. BROWN

P ro o f. The map 7r* is an isometry, and the dual of B(H) can be written as C\ (H) © A typical element of [6(H))* will be written as (K, 7r*(</?)) where K G

C\{H) and <p € C*. For this direct sum decomposition there exists M > 0 such that IMI ^ -^'ll(-^'>7r*(y>))ll f°r K E Ci('H) and ip € C*. (Actually it can be proved that this direct sum has the very strong decomposition || (K, 7r but this fact will not be needed.) Let A 1' denote the annihilator of A in Note that: (0 ) If {K,n*(ip)) € A 1- can be found such that K does not annihilate A then the desired result has been reached.

So assume that ( if , 7r* ((/?)) £ A 1- implies that K € A ± and 7r*(< )̂ € A ± . Let r : C* —■> A* be the map defined by setting T(cp) — n*(ip) \a , the restriction of 7r*((p) to A. Now if \\r{<p)\\ < 1, then tt*(ip) + 7r*(ip) + K e A L for some 'ip E C* and K £ Ci with \\n*(xp) + K\\ < 1. Hence ||tt*(̂ )|| < M . By the assumption, 7r*((p) + 7r*(t/;) € A ± . Hence T(—ip) = T((/?) with ||t/>|| < M . This shows that T is open and that T(C*) is norm closed.

Since V is continuous when domain and range are endowed with their weak* topologies, r(C*) is weak* closed (see [7], p. 96). It is easy to verify that V — (the annihilator in A of T(C*)). Hence V 1- = T(C*). But if V is not weak* dense in A, then there exists a nonzero weak* continuous linear functional on A which annihilates V. It has just been show that such a linear functional must be in the range of T and is therefore of the form 7r*(</?) I for some ip e C*. I

1 A

3. T w o Sim ple A p p lica tion s

Essentially normal operators with thin spectra will be considered. This has obvious bearing on the invariant subspace problem for hyponormal operators with thin spectra. Following this, some comments about compact pertubations of the shift will be made.

Let K be a subset of the complex plane. Then C(K) will be used to denote the Banach space of continuous functions on K using the sup norm. Let R(K) denote the set of all rational functions (restricted to K) with poles off K. Let R(K) denote the norm closure in C(K) of R(K). The set K will be called thin if R(K) = C(K).

For a positive Borel measure /i of compact support on the complex plane, it is not hard to show that L°°(n) can be identified with a dual algebra (consisting of multiplication operators on L2{H)).

An operator T € (13(H)) is called hyponormal if ||T*x|| < ||Tx|| for all x £ H. The invariant subspace problem for hyponormal operators has been solved (p. 95 Theorem 2.1' of [5], or [1]) in the case that the spectrum is not thin (i.e. R(a(T)) ^ C (a (T ))) . In settling the invariant subspace problem for hyponormal operators, it may be assumed that tt(T) is normal. This follows from the Berger-Shaw theorem (see [5], p. 130). Hence, 2.4 and the notions of 1.3 have bearing on the problem.

LOMONOSOV’S THEOREM AND ESSENTIALLY NORMAL OPERATORS 17

T h eorem 3.1. Let T be an essentially normal operator on a Hilbert space H with thin spectrum and no nontrivial invariant subspaces. Let A be the weak* closure of (r (T ) | r G R(a(T))}. Then there exists x ,y e W and positive Borel measure p on cr(T) such that

(1) There exists a one-to-one dual algebra homomorphism $ : A —> L°°(p) such that $ (r (T )) = r for all r G i?(cr(T)).

(2) For all r G R(a(T)),

(■r ( T ) x , y ) = f rdp.Ja{T)

Sketch o f P ro o f. Apply 2.4 to select x ,y G H, and Hilbert space <S, and s G S such that (Ax, y) = (/9 0 7r(>l)s, s) for all A G A. Note that this relation shows that p o 7r is one to one, for p o tt(A) — 0 implies that (Ax) 0 y = 0 G A* which yields an invariant subspace for T.

In the process of defining p, subspaces M and M were found (see the proof of 2.2). Now both M. and M are invariant under the normal operator (3 o 7r(T). But2 G R(a(T)), so it follows from the spectral theorem that M and Af are invariant under (/3 o 7r(T))*. Therefore S is invariant under (3 o 7x(T) and its Hilbert space adjoint. And N = P(/3on(T))P = po7r(T) is normal. Also from the construction, s is a cyclic vector for N. So iV is unitarily equivalent to the operator of multiplication by 2 on L2(p) for some positive Borel measure p with support on cr(7r(T)) C a(T ), and under this unitary equivalence s is taken to the constant function of value one. The weak* closure of { r(N ) | r G R(a(T ) ) } in B(S) can then be identified with L°°(p ) (by viewing each of the elements in L°°(p ) as a multiplication operator).

Let 3 be the weak* closure in B(S) of p(A). Now {r (p o 7r(T)) | r G i?(cr(T ))} is weak* dense in B , since {r(T) \ r G R (K ) } is weak* dense in A and p o tx is a dual algebra homomorphism. So B ~ L°°(p).

Statement (1) now follows with $ appropriately defined (note $ is one-to-one since p o n is).

Statement (2) follows using

(r(N)s,s) = / rdp.Ja(T)

I

Finally, the results 1.4 and 3.1 only provide for the existence of certain measures. Several questions remain unanswered:

(1) Can the measures satisfying the results 1.4 and 3.1 be selected such that their support lies within a given disc (which meets o (T ))?

(2) Can p be found with supp(p) = cr(T)?(3) Is p — 0 a measure satisfying 1.4 or 3.1 (this is equivalent to the invariant

subspace problem for the particular operators involved)?Let H be a Hilbert space with basis {e ^ }^ x. Let S be the unilateral shift given

by Sei = ei+i- We close with another direct application. The proof can easily be constructed using the techniques outlined above since 7r(S) is normal.

18 SCOTT W. BROWN

T h eorem 3.2. If K is any compact operator on H, and if S + K does not have a nontrivial invariant subspace, then there exists a positive Borel measure /x on the unit circle and x, y € H such that

J p d fi = (p(S + K)x, y) for all polynomials p.

Again an interesting question arises as to which measures might satisfy the state­ment of 3.2. Here the question is not so intractable and might be worthy of further pursuit.

R eferen ces

1. S.W. Brown, Hyponormal operators with thick spectra have invariant sub­spaces, Ann. of Math. 125 (1987), 93-103.

2. J. Conway, A Course in Functional Analysis, Springer-Verlag, 1985.3. V. Lomonosov, Invariant subspaces for operators commuting with compact op­

erators (Russian), Funkcional. Anal. Prilozen 7 : 3 (1973), 55-56.4. V. Lomonosov, An extension of Burnside’s Theorem to infinite-dimensional

spaces, (to appear).5. M. Martin and M. Putinar, Lectures on Hyponormal Operators, Birkhaiiser-Verlag,

1989.6. J.R. Ringrose, Compact Non-Self-Adjoint Operators, Van Nostrand Reinhold

Co., 1971.7. W . Rudin, Functional Analysis, McGraw-Hill, 1973.

Scott W . Brown Indiana University Rawles Hall Bloomington Indiana 47405