Lattice Of Invariant Subspaces Research Articles

Counterexamples in the theory of nonselfadjoint operator algebras

In this note we announce the answers to several questions which involve nonselfadjoint operator algebras. Detailed proofs will appear elsewhere. We use the following notation. M is a separable Hilbert space, B(#) is the algebra of bounded linear operators on )/, and Bi(H) is the ideal of trace class operators on M. For T € 0(#), {T}' is the commutant of T and {T} is the double commutant of T. B(M) is the dual of Bi(#) (see [2]) so that B{M) has a weak * topology. A(T) denotes the smallest weak * closed algebra containing T and 7, while IV (T) is the smallest weak operator closed algebra containing T and I. LatT is the lattice of (closed) invariant subspaces of T, and AlgLatT = {B £ B{X): LatT C La t£} . It is elementary that A{T) C IV {T) C {T} C {T}', that IV (T) C AlgLatT, and that all of these sets except A(T) are weakly closed algebras. Further, T is said to be reflexive if IV (T) = Alg Lat T. We will consider the following questions. QUESTION 1. Does 1V{T) = {T}' n AlgLatT, VT e B{U)t QUESTION 2. Does 1V{T) = {T} n AlgLatT, VT 1? (Here T denotes the direct sum of n copies of T.) QUESTION 4. If Ti and T2 are reflexive operators, must Ti0T2 be reflexive? QUESTION 5. Does A{T) = W(T), VT e B(#)? QUESTION 6. Does IV(T) have a separating vector, VT c S(>/)? Before stating the last question, we need some additional notation. Since TV(T) is weak * closed in B(#), 1V(T) is a dual space, with predual Tj/(T)* = B I ( ^ ) / W ( T ) _ L . Here W(T)j. denotes the preannihilator of U/(T). For each n, let jPn C Bi(#) denote the set of operators of rank < n. QUESTION 7. Is Fi/U/(T)i. dense in W(T)*, VT C B(#)? Some remarks regarding these questions are in order. There are some relations among the questions. For n = 1,2, or 6, an affirmative answer to Question n implies an affirmative answer to Question n + 1. Question 1 was raised independently by D. Sarason and P. Rosenthal (see [6, p. 195] and [7]). Rosenthal also asked Question 2 in [7]. In [4], J. Deddens listed several open questions, including Questions 3 and 4, concerning reflexive operators. Question 5 has been raised by many people. The question appears in [2]. In [8], D. Westwood gave an example of an operator T so that A(T) = IV (T) but so that the weak and weak * topologies are different on A(T).

Is a self-adjoint operator determined by its invariant subspace lattice?

If A and B are self-adjoint operators, this paper shows that A and B have order isomorphic invariant subspace lattices if and only if there are Borel subsets E and F of σ( A) and σ( B), respectively, whose complements have spectral measure zero, and there is a bijective function φ: E → F such that (i) Δ is a Borel subset of E if and only if φ(Δ) is a Borel subset of F; (ii) a Borel subset Δ of E has A-spectral measure zero if and only if φ(Δ) has B-spectral measure zero; (iii) B is unitarily equivalent to φ( A). If A is any self-adjoint operator, there is an associated function κ A : N ∪ {∞} → ( N ∪ {0, ∞}) × {0,1} defined in this paper. If F denotes the collection of all functions from N ∪ {∞} into ( N ∪ {0,∞}) × {0,1}, then F is a parameter space for the isomorphism classes of the invariant subspace lattices of self-adjoint operators. That is, two self-adjoint operators A and B have isomorphic invariant subspace lattices if and only if κ A = κ B . The paper ends with some comments on the corresponding problem for normal operators.

A Characterization of the Invariant Subspaces of Direct Sums of Strictly Cyclic Algebras

Two characterizations of the invariant subspace lattice of ${A^{(n)}}$ for a strictly cyclic operator algebra $A$ on a separable Hilbert space are proven.

Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ∗-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg( L 1 ⊗ L 2) = Alg L 1 ⊗ Alg L 2, where each L i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each L i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if L is a finite-width commutative subspace lattice and K is the set of compact operators then the quasitriangular algebra Alg L + K is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.

Unicellular operators

An operator is unicellular if its lattice of invariant subspaces is totally ordered by inclusion. The list of nests which are known to be the set of invariant subspaces of a unicellular operator is surprisingly short. We construct unicellular operators on l p , 1 ⩽ p &gt; ∞ {l^p},1 \leqslant p &gt; \infty , and on c 0 {c_0} with lattices isomorphic to α + X + β ∗ \alpha + X + {\beta ^{\ast }} where α \alpha and β \beta are countable (finite or zero) ordinals, and X X is in this short list. Certain other nests are attained as well.

Picturing the lattice of invariant subspaces of a nilpotent complex matrix

Let Q be a nilpotent transformation acting on a finite-dimensional complex vector space. A method is given by which a diagrammatic representation of Lat Q, the lattice of invariant subspaces of Q, can be obtained. Basically the method consists of adding to the Hasse diagram of Hyperlat Q the finite lattice of hyperinvariant subspaces of Q in a specified way. Some applications are given.

Invariant subspaces of direct sums of finite convolution operators

In general, a direct sum of finite convolution operators has a very complicated invariant subspace structure. In some cases, however, the invariant subspaces split and can be exactly described. This phenomenon is studied with the aid of a theorem that relates lattices of invariant subspaces and weakly closed algebras of operators.

Lattices of invariant subspaces of certain operators

Lattices of invariant subspaces of certain operators

A note on rank-one operators in reflexive algebras

It is shown that if the invariant subspace lattice of a reflexive algebra A \mathcal {A} , acting on a separable Hilbert space, is both commutative and completely distributive, then the algebra generated by the rank-one operators of A \mathcal {A} is dense in A \mathcal {A} is any of the strong, weak, ultrastrong or ultraweak topologies. Some related density results are also obtained.

Invariant subspace lattices of Lambert's weighted shifts

AbstractLet B(H) be the Banach algebra of all (bounded linear) operators on an infinite-dimensional separable complex Hilbert space H and let be a bounded sequence of positive real numbers. For a given injective operator A in B(H) and a non-zero vector f in H, we put We define a weighted shift Tw with the weight sequence on the Hilbert space 12 of all square-summable complex sequences by . The main object of this paper is to characterize the invariant subspace lattice of Tw under various nice conditions on the operator A and the sequence .

On a Characterization of Invariant Subspace Lattices of Weighted Shifts

The paper concerns itself with the characterization of invariant subspace lattices of weighted shift operators on the Hilbert space ${l^2}$ with suitable conditions on their weights. This characterization is also extended to the case of Banach spaces ${l^p}$, $1 < p < \infty$.

On projective equivalence of invariant subspace lattices

For i = 1,2, let A i be a linear transformation on a complex vector space and let σ be a lattice isomorphism from the invariant subspace lattice of A 1 onto the invariant subspace lattice of A 2. We determine the conditions under which σ is implemented by a linear or conjugate linear transformation (or a sum of these two kinds).

Remarks on invariant subspaces for finite dimensional operators

Every invariant subspace of the commutant { A′} of an operator A is the range of some operator in { A′}. If two operators have the same lattice of invariant subspaces, then each is similar to a polynomial in the other.

On a characterization of invariant subspace lattices of weighted shifts

The paper concerns itself with the characterization of invariant subspace lattices of weighted shift operators on the Hilbert space l 2 {l^2} with suitable conditions on their weights. This characterization is also extended to the case of Banach spaces l p {l^p} , 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty .

On Reflexivity of Algebras

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat )(n) are the same. (If A is an operator, A(n) denotes the direct sum of n copies of A; if is a collection of operators,. Also, alg lat denotes the algebra of all operators leaving all invariant subspaces of invariant.) In the first section we show that . In Section 2 we prove that every weakly closed algebra containing a maximal abelian self adjoint algebra (m.a.s.a.) is , and that . It is also shown that certain algebras containing a m.a.s.a. are necessarily reflexive.

Lattices of invariant subspaces of group representations

Lattices of invariant subspaces of group representations

Compact Perturbations of Reflexive Algebras

In this paper we study lattice properties of operator algebras which are invariant under compact perturbations. It is easy to see that if and are two operator algebras with contained in , then the reverse inclusion holds for their lattices of invariant subspaces. We will show that in certain cases, the assumption thats is contained in , where is the ideal of compact operators, implies that the lattice of is “approximately” contained in the lattice of . In particular, supposed and are reflexive and have commutative subspace lattices containing “enough” finite dimensional elements. We show (Corollary 2.8) that if is unitarily equivalent to a subalgebra of , then there is a unitary operator which carries all “sufficiently large” subspaces in lat into lat .

Invariant subspace lattices for a class of operators

Invariant subspace lattices for a class of operators

Compact perturbations of reflexive algebras

We study compact perturbations of reflexive algebras with commutative invariant subspace lattices. This leads us to study lattice isomorphisms that are uniform limits of spatially induced mappings and satisfy a compactness condition. We prove that two lattices that are nested or complemented are isomorphic in this way iff their corresponding perturbed algebras are unitary equivalent. While for complemented lattices this relation implies unitary equivalence, we prove that any two continuous nests are equivalent in this way.

Invariant subspace lattices and compact operators

Invariant subspace lattices and compact operators