A Sufficient Condition for Hyperinvariance

A linear transformation on a finite-dimensional complex linear space has the property that all of its invariant subspaces are hyperinvariant if and only if its lattice of invariant subspaces is distributive [1]. It is shown that an operator on a complex Hilbert space has this property if its lattice of invariant subspaces satisfies a certain distributivity condition. 1. Preliminaries. Throughout this paper 77 will denote an arbitrary complex Hilbert space. All operators are bounded and all subspaces are closed. By a subspace lattice on H is meant a family of subspaces of 77 which is closed under the formation of arbitrary intersections and arbitrary closed linear spans and which contains the zero subspace (0) and 77. The family of subspaces invariant under an operator T is denoted by Lat T. This is a subspace lattice as is the family of subspaces invariant under every operator commuting with T which we denote by Hyperlat T. The elements of Hyperlat T are called the hyperinvariant subspaces of T. Clearly Hyperlat T C Lat T. A subspace lattice § is called commutative if for every pair of subspaces M, N E V? the corresponding projections PM and PN commute. Let L be an abstract lattice. We say that L is (i) distributive if ay(bf\c) = (ayb)/\(ayc) (a, b, c E L) and its dual statement holds identically; (ii) ainfinitely meet distributive if L is a-complete and ay {/\bn:n> l) = /\{ayb„:n> 1} (a, bn E L) holds identically in L. That the dual equation defining distributivity are equivalent to each other is an elementary result of lattice theory. 2. A sufficient condition for hyperinvariance. The key to the sufficient condition is a result of Sarason and the following lattice-theoretic result. Proposition 2.1. 7/L is an abstract o-infinitely meet distributive lattice and 9: L —> L is a lattice automorphism with the properties (I) a < 9(a) \f 9 ~x(a) (a E L); (II) a, 9(a) comparable implies a = 9(a), then 9 is the identity automorphism. Proof. For every n > 1 let 9": L —» L be defined in the obvious way. Let a Received by the editors March 27, 1975. AMS (MOS) subject classifications (1970). Primary 47A15; Secondary 26A30. Copyright <> 1977, American Mathematical Society 26 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use A SUFFICIENT CONDITION FOR HYPERINVARIANCE 27 be an arbitrary fixed element of L. For n > 1 put an = a /\0(a) /\ 02(a) A • • • A 0"(a). Then a„ A0(an) = a„+1 (n > 1). Using (I) and the fact that L is distributive the statement 0(x) = (x/\0(x))v[0(x/\9(x))] holds identically in L. Using this identity it is easily shown that 8(a) = a, V 0(a„) (n > I). Thus 9(a) = A{a, V 0(an): n > 1} = a, V (A{0(«„): »>!})■ If c = A{«n: " > 1} then 0(c) = A{0(«„): « > 1} > A{«„+,: « > 1} > c and by (II), 6(c) = c. Hence 0(a) = ax V 9(c) = a, V c = a, < a and again by (II), 0 (a) = a. This completes the proof. Let T be an operator on H. Notice that if S is an invertible operator commuting with T then SM G Lat T (M G Lat T) and the mapping M -* SM is a lattice automorphism with the mapping M —> S~XM as its inverse. If the operator A commutes with T and it is a scalar with | ju| >||^||, the operator 5 = A ii is invertible, commutes with T and Lat A = Lat S. By a result of Sarason [4], Lat S = Lat S_1. It readily follows that Hyperlat T = Lat A if and only if for every invertible operator S commuting with T satisfying Lat S = Lat S_1 the mapping M -* SM (M G Lat T) is the identity automorphism. Proposition 2.2. If Lat T is distributive and S is an invertible operator commuting with T, then M C SM V S~XM (M G Lat A). Proof. Choose A with 0 < A < 1/||S||. The operator C = 1 + XS is invertible and commutes with T. Let M G Lat A. It is readily verified that CM n SM = C(M n SM) and CM n M = C(M n S_1M). Since CM C SM V M, by distributivity we have cm = (cm n sm) v(cm n M) = C(M n SM)\y c(M n S_1M) = C(M n[SM vs_1m]) and the result follows. Theorem 2.3. // Lat A is o-infinitely meet distributive Hyperlat A = Lat A. Proof. By our earlier remarks it suffices to show that if S is an invertible operator commuting with A and satisfying Lat S = Lat S ~' then the automorphism M -» SM of Lat A is the identity automorphism. Since Lat A is distributive, this automorphism satisfies condition (I) of Proposition 2.1 by Proposition 2.2. Since Lat S = Lat S~x, condition (II) is also satisfied. The result now follows from Proposition 2.1. Corollary 2.3.1. Hyperlat T = Lat T if Lat T is any one of the following: (i) commutative; (ii) isomorphic to the direct product of complete chains; (iii) totally ordered. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use