NEW ZEALAND JOURNAL OF MATHEMATICSVolume 31 (2002), 153-158

IN VARIAN T SUBSPACE LATTICES CONCERNING SUBDECOMPOSABLE OPERATORS

M i n g x u e L iu

(Received October 2000)

Abstract. In 2000, we showed the Mohebi-Radjabalipour Conjecture under an additional condition, and obtained an invariant subspace theorem concerning subdecomposable operators. In this paper, we obtain a stronger result that the invariant subspace lattice for a class of these operators is rich. The result accurately characterize the invariant subspace lattice for the class of operators.

In [10], Mohebi and Radjabalipour raised the following conjecture.

The Mohebi-Radjabalipour Conjecture (see [10], p. 236). Assume the operators T G B ( X ) and B G B(Z) on Banach spaces X and Z, and the nonempty open set G in the complex plane C, satisfy the following conditions:(1) qT = Bq for some injective q G B(X, Z ) with a closed range qX\

(2) There exist sequences {G(n)} of open sets and (M (n )} of invariant subspaces of B such that G(n) C G(n + 1), G = UnG(n), a[B\M{n)) C C\G(n) and cr(B/M(n)) c G(n), n = 1,2,--- ;

(3) cr(T) is dominating in G.Then T has a nontrivial invariant subspace.

It is easy to see that the Mohebi-Radjabalipour Conjecture, if true, will contain the main results of [1-4, 6-8, 10] (and others) as special cases.

In [8], using the S. Brown Technique, we proved the Mohebi-Radjabalipour Conjecture under an additional condition, and obtained the invariant subspace theorem concerning subdecomposable operators as follows:

Theorem A. Assume the operators T G B { X ) and B G B(Z) on Banach spaces X and Z, and the nonempty open set G in C, satisfy conditions (1), (2) and (3) in the Mohebi-Radjabalipour Conjecture and the following additional condition:

(4) {q X + M(n)} is a sequence of closed sets in Z.Then T has a nontrivial invariant subspace.

In this paper, we obtain a stronger result, which accurately characterize the invariant subspace lattice for a class of operators in Theorem A. This result is as follows.

2000 A M S Mathematics Subject Classification: 47A15, 47B40.K ey words and phrases: Banach space, bounded linear operator, invariant subspace lattice, essential spectrum.The research was supported by the Foundation of Education Committee of Fujian Province of P.R. China.

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Theorem 1. Assume the operators T G B (X ) and B G B(Z) on Banach spaces X and Z, and the nonempty open set G in C , satisfy the conditions (1), (2), (3) and (4) in Theorem A, then T has a non-trivial invariant subspace. In particular, if the essential spectrum cre(T) of T is dominating in G, then the invariant subspace lattice Lat(T) for T is rich.

To prove Theorem 1 we first recall some basic notation and facts, and give some lemmas.

We denote by H °°(G ) the Banach algebra of all bounded analytic functions on G equipped with the norm ||/|| = sup{|/(A)|; A G G}. It is well known that H °°(G ) is a w*-closed subspace of L°°{G ) relative to the duality (L\(G), L°°(G)) and that a sequence {/& } in H °°(G ) converges to zero relative to w*-topology if and only if it is norm-bounded and converges to zero uniformly on each compact subset of G. In particular, we can identify H°°(G) with the dual space of the Banach space Q = L\(G)/(H °°(G ))L. Since Q is separable, it follows from the characterization of it?*-convergent sequences in H °°(G ) that for each A 6 G, the point evaluation E\ : H°°(G) —► C, / —> /(A ), is a u>*-continuous linear functional.

A subset a of the complex plane C will be called dominating in the open set G if D/ll = sup{|/(A)|; A G f f f lG } holds for all / G H °°(G ),

Let £ be a Banach space, then Lat(-E') denote the lattice of all closed linear subspace of E. Let X be a Banach space and let T G B (X ), then the invariant subspace lattice Lat(T) for T is called to be rich if there exists an infinite dimensional Banach space E such that Lat(T) contains a sublattice order isomorphic to Lat(E').

It is easy to see that if cr(T) ^ &e{T) and d im (X ) > 2, Then T has an invariant subspace.

Throughout the rest of this paper, we shall assume that X, Z,T, B ,q ,G ,G (n) and M (n ) are as in Theorem 1, and that cre(T ) is dominating in G.

Lemma 2 ([8], Lemma 3). Define B : qX —> qX by Bz = Bz, and define q : X —> qX by qx = qx. Then B G B (qX), q G B (X , qX) and(1) For any polynomial p, for any vectors z G qX and z* G Z*, we have

(z,p(B*)z*) = Cz,p(B*)z*),

where z* denotes the restriction of z* onto qX , that is, z* — z*\qX.

(2) For any x G X and z* G Z*, we have (x,q*z*) = (x,q*z*) and 110*2*11 = ||g*2*||, where z* = z*\qX.

By [8, p.20], we have M (n)± \qX C M (n -f l^ lg X , a (^B*\(M(n)± \qX)'j C

G(n) C G for n = 1,2, • • •. Set

M (G ) = U n i M i n ^ q X ) .

Then for any x G X , z* G M (G), there exists a natural number n > uq such that 2 * G M (n)-1- \qX, where no is defined by [8, Note 2]. Therefore we can define a functional x ® z* : H °°(G ) —> C by

x ® z * ( f ) = (x,q* f(Bn)z*),

INVARIANT SUBSPACE LATTICES 155

where B* denotes B*\(M(n)± \qX), and f(B ^) is defined by the Riesz-Dunford functional calculus with analytic functions. By [8], x z* is a well-defined w*- continuous linear functional which is independent of the particular choice of n.

Lemma 3 ([8], Lemma 8). Let r,s be natural numbers. Consider non-negative real numbers c\, C2, • • • , cr with Ci+C2+- ■ -+cr = 1 and complex numbers Ai, A2, • • •, Ar E aie(T*) f l G, where aie(T*) denotes the left essential spectrum of T*. If ai, a2, • • •, as e l , b\, b2, • • •, b* E M (G ) and e > 0 are arbitrary, then there are vectors x E X , z* E M (G) such that ||x|| < 3, ||(f*z*|| < 2 and(1) ||(ci£Al + c2EX2 H--------f crEXr) - x ® z*\\ < e ,

(2) ||x

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Lemma 6. If the right essential spectrum crre(T) of T is dominating in G, then for each matrix L = (Ljk)j,k> 1 G M (oo,Q ), there are sequences {x m} and { z*n} such that(1) e X m, ^ n e [M(G)]m;

(2) for each natural number j , the limits

x (j) = lim x m(j) G X , x*(j) = lim q*z^(j) G X*m —►oo m —► oo

exist, where xm(j) and z^ (j) denote the jth components of xm and z^ respectively;

(3) for all natural number j , k, we have

Ljk = lim x m(j) z^(k),rn—►oo

where the limit is taken in Q.

Proof. Using Lemma 5, the proof of Lemma 6 is similar to that of Proposition 2.6 in [7] and is therefore omitted. □

Proof of Theorem 1. We consider the two cases separately:

Case 1. If 1 = (SjkEn)j,k>i € M (oo,Q ), where 5jk denotes the Kronecker delta. Therefore for any natural numbers m, k there exists a natural number n = n(m, k) such thatz^(k) = 2̂ (fc)|qX with z^{k) G M (n(m , k ))± . Consequently for each polynomial p G C[z\ in one complex variable, by Lemma 2 and the condition (1) in Theorem 1 we obtain

8jkP(X) = SjkEx{P) = lim x m(j) ® z^{k){p)771— ► OO

= lim (xm{j),q*p{B^{ k))z^{k)}m —»-oo v ’ '

= lim (xm(j),q*p{B*)z?n(k))m — > 00

= lim (;xrn(j),q*p(B*)z*m(k))7 n — ► 00

= lim (xm(j),p(T*)q*z*m(k))m —► 00

= lim (p(T)xrn(j),q*z?n(k))771—► OO

= (p{T )x(j), x*(k)) (1)

INVARIANT SUBSPACE LATTICES

for all natural numbers j , k. Set

U = span{p(T)x(j)-,p € C[z\,j = 1, 2, • • • },

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= s p a n {p (T ) (A -T )x ( j ) ;p e C [z ]J = 1,2, •••}.

Then U,V E Lat(T), and V C U, (A — T)E/ C V. For every natural number k, define the functional (j)k : UJV — > C by

c/)k(x + V) = (x ,x* (k )).

It is easy to see that 4>k is a well-defined bounded linear functional on U/V. Also, by (1) we obtain

k(x(j) + V) = (x(j),x*(k )) = Sjk.

Therefore U/V is a infinite dimensional Banach space. If 7r : U —> U/V is the canonical quotient map, then the map r : Lat{U/V) Lat(T), S —> 7r—1 (-S'), is a lattice embedding, where Lat (U/V) denotes the lattice of all closed linear subspaces of the Banach space (U/V). Consequently Lat(T) is rich, and this concludes the proof of Theorem 1.

Acknowledgment. The author thanks the referee for his valuable suggestion.

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Mingxue LiuDepartment of Mathematics Changsha University Changsha 410003 HunanPEO PLE’S REPUBLIC of CHINA liumingxue888@163.com

mailto:liumingxue888@163.com