Invariant Subspaces Of Operators Research Articles

Invariant subspaces of operators via Berezin symbols and Duhamel product

Invariant subspaces of operators via Berezin symbols and Duhamel product

Transformation Semigroups of the Space of Functions that are Square Integrable with Respect to a Translation-Invariant Measure on a Banach Space

We examine measures on a Banach space E that are invariant under shifts by arbitrary vectors of the space and are additive extensions of a set function defined on the family of bars with converging products of edge lengths that do not satisfy the σ-finiteness condition and, perhaps, the countable additivity condition. We introduce the Hilbert space ℋ of complex-valued functions of the space E of functions that are square integrable with respect to a shift-invariant measure. We analyze properties of semigroups of shift operators in the space ℋ and the corresponding generators and resolvents. We obtain a criterion of the strong continuity of such semigroups. We introduce and examine mathematical expectations of operators of shifts along random vectors by a one-parameter family of Gaussian measures that form a semigroup with respect to the convolution. We prove that the family of mathematical expectations is a one-parameter semigroup of linear self-adjoint contraction mappings of the space ℋ, find invariant subspaces of operators of this semigroup, and obtain conditions of its strong continuity.

Convex analysis and non-trivial invariant subspaces

We establish sufficient conditions for the existence of invariant subspaces for operators on real Banach spaces, and we investigate the behaviour of operators without such subspaces. All proofs are elementary.

Localizing algebras and diagonal operators

The notion of a localizing algebra was introduced by Lomonosov, Radjavi, and Troitsky as a side condition to build invariant subspaces for operators on Banach spaces. The goal of this paper is to show that the unital algebra generated by a single diagonal operator on a separable Hilbert space is localizing if and only if there is a non-zero compact operator in the weak closure of the unit ball of the algebra .

Quasiaffinity and invariant subspaces

Special classes of intertwining transformations between Hilbert spaces are introduced and investigated, whose purposes are to provide partial answers to some classical questions on the existence of nontrivial invariant subspaces for operators acting on separable Hilbert spaces. The main result ensures that if an operator is \({{\mathcal D}}\)-intertwined to a normal operator, then it has a nontrivial invariant subspace.

On invariant subspaces of operators in the class [formula omitted

In this paper, we consider invariant subspaces of operators in the class θ, which is the set of operators T such that T∗T and T+T∗ commute. It is shown that every operator in the class θ such that the outer boundary of its spectrum is the outer boundary of a Carathéodory domain has a nontrivial invariant subspace. We also give a family of operators in the class θ which are reductive, i.e., their invariant subspaces are reducing. In addition, we give a condition on spectra of operators in the class θ which gives some information about invariant subspaces.

Invariant subspaces for operators whose spectra are Carathéodory regions

In this paper it is shown that if an operator T satisfies ‖ p ( T ) ‖ ⩽ ‖ p ‖ σ ( T ) for every polynomial p and the polynomially convex hull of σ ( T ) is a Carathéodory region whose accessible boundary points lie in rectifiable Jordan arcs on its boundary, then T has a nontrivial invariant subspace. As a corollary, it is also shown that if T is a hyponormal operator and the outer boundary of σ ( T ) has at most finitely many prime ends corresponding to singular points on ∂ D and has a tangent at almost every point on each Jordan arc, then T has a nontrivial invariant subspace.

Invariant subspaces for operators in a general II1-factor

Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μT (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace \({\mathcal{K}}={\mathcal{K}}_{\mathrm{T}}(B)\) affiliated with ℳ, such that the Brown measure of \(\mathrm {T}|_{{\mathcal{K}}}\) is concentrated on B and the Brown measure of \(\mathrm{P}_{{\mathcal{K}}^{\bot}}\mathrm{T}|_{{\mathcal{K}}^{\bot}}\) is concentrated on ℂ∖B. Moreover, \({\mathcal{K}}\) is T-hyperinvariant and the trace of \(\mathrm{P}_{\mathcal{K}}\) is equal to μT(B). In particular, if T∈ℳ has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T∈ℳ the limit \(A:=\lim _{n\rightarrow\infty}[(\mathrm{T}^{n})^{*}\mathrm{T}^{n}]^{\frac{1}{2n}}\) exists in the strong operator topology, and the projection onto \({\mathcal{K}}_{\mathrm{T}}(\overline{B(0,r)})\) is equal to 1[0,r](A), for every r>0.

Cyclic vectors of diagonal operators on the space of functions analytic on a disk

The purpose of this paper is to study cyclic vectors and invariant subspaces of operators on the space of functions analytic on an open disk in the complex plane having as eigenvectors the monomials z n .

Brown measures of sets of commuting operators in a type II 1 factor

Using the spectral subspaces obtained in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II 1-factor, preprint, 2005], Brown's results (cf. [L.G. Brown, Lidskii's theorem in the type II case, in: H. Araki, E. Effros (Eds.), Geometric Methods in Operator Algebras, Kyoto, 1983, in: Pitman Res. Notes Math. Ser., vol. 123, Longman Sci. Tech., 1986, pp. 1–35]) on the Brown measure of an operator in a type II 1 factor ( M , τ ) are generalized to finite sets of commuting operators in M . It is shown that whenever T 1 , … , T n ∈ M are mutually commuting operators, there exists one and only one compactly supported Borel probability measure μ T 1 , … , T n on B ( C n ) such that for all α 1 , … , α n ∈ C , τ ( log | α 1 T 1 + ⋯ + α n T n − 1 | ) = ∫ C n log | α 1 z 1 + ⋯ + α n z n − 1 | d μ T 1 , … , T n ( z 1 , … , z n ) . Moreover, for every polynomial q in n commuting variables, μ q ( T 1 , … , T n ) is the push-forward measure of μ T 1 , … , T n via the map q : C n → C . In addition it is shown that, as in [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II 1-factor, preprint, 2005], for every Borel set B ⊆ C n there is a maximal closed T 1 - , … , T n -invariant subspace K affiliated with M , such that μ T 1 | K , … , T n | K is concentrated on B. Moreover, τ ( P K ) = μ T 1 , … , T n ( B ) . This generalizes the main result from [U. Haagerup, H. Schultz, Invariant subspaces of operators in a general II 1-factor, preprint, 2005] to n-tuples of commuting operators in M .

Cyclic vectors of diagonal operators on the space of entire functions

The purpose of this paper is to study cyclic vectors and invariant subspaces of operators on the space of entire functions having as eigenvectors the monomials z n .

On some uses of factorizations in linear system theory

We present a short, and far from exhaustive, survey of some of the uses of factorization theory in the study of linear systems. The whole development is based on model theory, be it in the context of vectorial polynomial, rational or Hardy spaces. Invariant subspaces of operators are related to factorizations of matrix functions. The connection to system theory is via realizations based on various coprime factorizations. Among topics touched upon are the study of factorizations of rational functions and in particular the important cases of inner and all-pass functions. We survey the subject of coprime factorizations and their connection, via spectral factorizations, to various Riccati equations. In this connection we describe some geometric aspects of Wiener-Hopf factorizations. We proceed to describe connections with geometric control theory and give a Hilbert space characterization of stabilizability and detectability subspaces. Finally we describe a complete parametrization of all minimal spectral factors in the case of a coercive spectral density function.

SOME RESULTS RELATED TO THE SEPARABLE QUOTIENT PROBLEM

The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimensional separable quotient if and only if X has an infinite dimensional separable quasicomplemented subspace, also if and only if there exists a Banach space Y and a bounded linear operator T∈B(Y,X) such that the range of T is nonclosed and dense in X. Besides, the other relevant questions for such spaces e.g. the question on operator ideals that on H.I.(hereditarily indecomposable) spaces, that on invariant subspaces of operators, etc. are also discussed.

On Some Uses of Factorizations in Linear System Theory

We present a short, and far from exhaustive, survey of some of the uses of factorization theory in the study of linear systems. The whole development is based on model theory, be it in the context of vectorial polynomial, rational or Hardy spaces. Invariant subspaces of operators are related to factorizations of matrix functions. The connection to system theory is via realizations based on various coprime factorizations. Among topics touched upon are the study of factorizations of rational functions and in particular the important cases of inner and all-pass functions. We survey the subject of coprime factorizations and their connection, via spectral factorizations, to various Riccati equations. In this connection we describe some geometric aspects of Wiener-Hopf factorizations. We proceed to describe connections with geometric control theory and give a Hilbert space characterization of stabilizability and detectability subspaces. Finally we describe a complete parametrization of all minimal spectral factors in the case of a coercive spectral density function.

Invariant subspaces for operators with Bishop's property (β) and thick spectrum

Invariant subspaces for operators with Bishop's property (β) and thick spectrum

Invariant subspaces of operators which act in a space with indefinite metric

Invariant subspaces of operators which act in a space with indefinite metric

Algebras Intertwining Normal and Decomposable Operators

The celebrated result of Lomonosov [6] on the existence of invariant subspaces for operators commuting with a compact operator have been generalized in different directions (for example see [2], [7], [8], [9]). The main result of [9] (see also [7]) is: If is a norm closed algebra of (bounded) operators on an infinite dimensional (complex) Banach space , if K is a nonzero compact operator on , and if then has a non-trivial (closed) invariant subspace. In [7], it is mentioned that the above result holds if instead of compactness for K we assume that K is a non-invertible injective operator with a non-zero eigenvalue belonging to the class of decomposable, hyponormal, or subspectral operators.