Lattice Of Invariant Subspaces Research Articles

An invariant subspace lattice of order type 𝜔+𝜔+1

An example is given of a bounded linear operator on a Hilbert space whose lattice of invariant subspaces is totally ordered and is of order-type ω + ω + 1 \omega + \omega + 1 .

The inaccessible invariant subspaces of certain 𝐶₀ operators

We extend the Douglas-Pearcy characterization of the inaccessible invariant subspaces of an operator on a finite-dimensional Hilbert space to the cases of algebraic operators and certain C 0 {C_0} operators on any Hilbert space. This characterization shows that the inaccessible invariant subspaces for such an operator form a lattice. In contrast to D. Herrero’s recent result on hyperinvariant subspaces, we show that quasisimilar operators in the classes under consideration have isomorphic lattices of inaccessible invariant subspaces.

Derivations of nest algebras

The results presented below are generalisations of some well known results about yon Neumann algebras. The first result, Theorem 2.1, yields a sufficient condition for automatic continuity of any derivation of an algebra of operators into a dual normal module. The theorem is followed by a corollary which shows that reflexive algebras with commutative subspace lattices have only continuous derivations into dual normal modules. The second result, Theorem 3.6, is quite independent of the former. In Section 3 we prove that, for any nest algebra ~¢ and any ultraweakly closed algebra ~ of operators, the first cohomology group Hl (d ,9~) always vanishes. We ought perhaps to mention, that nest algebras are reflexive and do have commutative lattices of invariant subspaces.

On Reflexive Compact Operators

Let A be a compact operator on a separable Hilbert space . The aim of this paper is to investigate the relationship between the weak closure of the algebra of polynomials in A (denoted by U(A)) and its invariant subspace lattice Lat A.

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.

A Lattice-Theoretic Description of the Lattice of Hyperinvariant Subspaces of a Linear Transformation

If A is a (linear) transformation acting on a (finitedimensional, non-zero, complex) Hilbert space H the family of (linear) subspaces of H which are invariant under A is denoted by Lat A. The family of subspaces of H which are invariant under every transformation commuting with A is denoted by Hyperlat A. Since A commutes with itself we have Hyperlat A ⊆ Lat A. Set-theoretic inclusion is an obvious partial order on both these families of subspaces. With this partial order each is a complete lattice; joins being (linear) spans and meets being set-theoretic intersections. Also, each has H as greatest element and the zero subspace (0) as least element. With this lattice structure being understood, Lat A (respectively Hyperlat A) is called the lattice of invariant (respectively, hyper invariant) subspaces of A.

The radical of a reflexive operator algebra

The radical of a reflexive operator algebra % whose lattice of invariant subspaces 2 is commutative is related to the space of lattice homomorphisms of 2 onto {0,1}. To each such homomorphism φ is associated a closed, two-sided ideal Wψ contained in H. The intersection of the ^ is contained in the radical; it is conjectured that equality always holds. The conjecture is proven for a variety of special cases: countable direct sums of nest algebras; finite direct sums of algebras which satisfy the conjecture; algebras whose lattice of invariant subspaces is finite; algebras whose lattice of invariant subspaces is isomorphic to the lattice of nonincreasing sequences with values in N U {°°}. 1* Introduction* This paper studies the radical of a certain class of non-self-ad joint operator algebras. Given an algebra 21 and a lattice S of orthogonal projections acting on a separable Hubert space φ, we use the standard notations, Sat 21 and SttgS, to denote, respectively, the lattice of all projections invariant under Sί and the algebra of all (bounded) operators which leave invariant each projection of 2. 2t and 2 are said to be reflexive if 2ί = Stlg Sat 2C and S = Sat 2CIg S, respectively. The algebras which we study are reflexive algebras which contain a maximal abelian self-adjoint algebra (m.a.s.a.). A commutative subspace lattice is a lattice of pairwise commuting, orthogonal projections on $ which contains 0 and 1 and which is closed in the strong operator topology. It follows automatically that a commutative subspace lattice is a complete lattice. If 2C is an operator algebra containing a m.a.s.a., then Sat SI is a commutative subspace lattice. Every commutative subspace lattice, S, is reflexive ([1], p. 468), and SίlgS is a reflexive algebra which contains a m.a.s.a. Henceforth, all lattices of projections in this paper will be commutative subspace lattices and all algebras will be reflexive algebras which contain a m.a.s.a. An incisive study of these lattices and algebras by Arveson is found in [1]. At least in certain special cases, the radical of a reflexive algebra, 2ί, containing a m.a.s.a. can be described in terms of the set of lattice homomorphisms from S = Sαt2t onto {0,1}. To each such homomorphism φ we shall associate a closed two-sided ideal %φ in St. The radical, % of Sf is equal to the intersection of these ideals. It appears reasonable to conjecture that this equality holds for all

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.

On strictly cyclic algebras, $\mathcal{P}$-algebras and reflexive operators

An operator algebra $\mathfrak {A} \subset \mathcal {L}(\mathcal {X})$ (the algebra of all operators in a Banach space $\mathcal {X}$ over the complex field C) is called a “strictly cyclic algebra” (s.c.a.) if there exists a vector ${x_0} \in \mathcal {X}$ such that $\mathfrak {A}({x_0}) = \{ A{x_0}:A \in \mathfrak {A}\} = \mathcal {X};{x_0}$ is called a “strictly cyclic vector” for $\mathfrak {A}$. If, moreover, ${x_0}$ separates elements of $\mathfrak {A}$ (i.e., if $A \in \mathfrak {A}$ and $A{x_0} = 0$, then $A = 0$), then $\mathfrak {A}$ is called a “separated s.c.a." $\mathfrak {A}$ is a $\mathcal {P}$-algebra if, given ${x_1}, \ldots ,{x_n} \in \mathcal {X}$, there exists ${x_0} \in \mathcal {X}$ such that $\left \| {A{x_j}} \right \| \leq \left \| {A{x_0}} \right \|$, for all $A \in \mathfrak {A}$ and for $j = 1, \ldots ,n$. Among other results, it is shown that if the commutant $\mathfrak {A}’$ of the algebra $\mathfrak {A}$ is an s.c.a., then $\mathfrak {A}$ is a $\mathcal {P}$-algebra and the strong and the uniform operator topology coincide on $\mathfrak {A}$; these results are specialized for the case when $\mathfrak {A}$ and $\mathfrak {A}’$ are separated s.c.a.’s. (Here, and in what follows, algebra means strongly closed subalgebra on $\mathcal {L}(\mathcal {X})$ containing the identity I on $\mathcal {X}$.) In the second part of the paper, it is shown that a large class of bilateral weighted shifts (which includes all the invertible ones) in a Hilbert space are reflexive. The result is used to show that “reflexivity” is neither a “restriction property” nor a “quotient property." Recall that an algebra $\mathfrak {A}$ is called reflexive if, whenever $T \in \mathcal {L}(\mathcal {X})$ and the lattice of invariant subspaces of T contains the corresponding lattice of $\mathfrak {A}$, then $T \in \mathfrak {A}$.

A pathological lattice of invariant subspaces

An explicit example of a Hilbert space operator whose lattice of invariant subspaces (under the metric topology “gap between subspaces”) contains an inaccessible point which is not isolated is constructed; the component of that inaccessible point is not arcwise connected and, moreover, no ball (of sufficiently small radius) about the point is connected. This example answers negatively three questions proposed by R. G. Douglas and Carl Pearcy.

Analytically invariant and bi-invariant subspaces

The purpose of this paper is to call attention to some interesting weakly closed algebras related to a bounded linear operator T T acting on a Banach space X \mathfrak {X} and their associated lattices of invariant subspaces, namely, the algebras generated by the polynomials and by the rational functions in T T , and the commutant and the double-commutant of T T . The relationship between those algebras and their lattices, as well as the ones corresponding to the operators induced by T T on an invariant subspace (restriction), or on the quotient space X / M \mathfrak {X}/\mathfrak {M} (where M \mathfrak {M} is an invariant subspace of a given type) are analyzed. Several results relative to the decomposition of invariant subspaces and the topological structure of the lattices (under the “gap-between-subspaces” metric topology) are also considered.

On the Unicellularity of Weighted Shifts

An operator acting on a Banach space is said to be unicellular if its lattice of invariant subspaces is totally ordered by inclusion. Each weighted shift on a sequence space has a natural totally ordered set of invariant subspaces, and is unicellular if these are its only invariant subspaces.

On Algebras of Operators with Totally Ordered Lattice of Invariant Subspaces

On Algebras of Operators with Totally Ordered Lattice of Invariant Subspaces

On algebras of operators with totally ordered lattice of invariant subspaces

For a Hilbert space H \mathcal {H} , let A \mathcal {A} be a weakly closed algebra of bounded operators on H \mathcal {H} which contains the identity. A \mathcal {A} is said to be transitive if no closed subspace of H \mathcal {H} is invariant under A \mathcal {A} . There are no known proper subalgebras of B ( H ) \mathcal {B}(\mathcal {H}) which are transitive. In this paper it is shown that the only transitive algebra which satisfies a certain condition β \beta is B ( H ) \mathcal {B}(\mathcal {H}) . Furthermore, a generalization of condition β \beta is given which characterizes those algebras with totally ordered lattice of invariant subspaces that are reflexive.

Weakly closed maximal triangular algebras are hyperreducible

In this note we observe that the assertion in the title follows easily from known results. Recall that an algebra 6R of bounded linear operators on a Hilbert space is called [1] triangular if (R 6* is a maximal abelian von Neumann algebra. A triangular algebra is maximal if it is not properly contained in another triangular algebra; it is hyperreducible if the projections onto its invariant subspaces generate a maximal abelian von Neumann algebra. Now the theorem stated in the title can be proven as follows. Let a1 be a weakly closed maximal triangular algebra. Since aR is maximal the lattice of invariant subspaces of 61 is totally ordered [1, Lemma 2.3.3]. It follows from Theorem 2 of [2] that 61 is refexive, i.e., 61 contains every operator which leaves invariant every invariant subspace of (R. Hence, by Lemma 5.1 of [3], 61 is hyperreducible. This result shows one reason that hyperreducible maximal triangular algebras are nice (see [1]). Note that the proof of Lemma 5.1 of [3] shows that every reflexive triangular algebra (not just those with totally ordered invariant subspace lattices), is hyperreducible. Thus it follows (by a proof similar to the above) from Theorem 3 of [2] that a weakly closed triangular algebra whose lattice of invariant subspaces is the direct product of a family of totally ordered lattices is hyperreducible. In view of these results it seems reasonable to conjecture that every weakly closed triangular algebra is hyperreducible. A proof of this conjecture would be a partial answer to question (i) of [2].

Remarks on Invariant Subspace Lattices

If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.

The Invariant Subspace Lattice of a Linear Transformation

The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.

The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation

The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation