Hyperinvariant Subspaces Research Articles

Hyperinvariant subspaces for compact perturbations of operators whose spectrum has a Dini-smooth exposed arc

Hyperinvariant subspaces for compact perturbations of operators whose spectrum has a Dini-smooth exposed arc

Linear maps that preserve the commutant, double commutant or the lattice of invariant subspaces

For a linear map φ on either the algebra of all bounded operators on a Banach space of the algebra of all linear transformations on a vector space, we prove φ preserves any of the following functions of the operator if and only if it is of the form φ(T) = αT + f(T) I for a scalar α and a linear functional f:(i) the commutant, (ii) the double commutant, (iii) the algebra generated by the operator, (iv) the lattice of invariant subspaces, (v) the lattice of hyperinvariant subspace. (vi) the set of reducing subspaces.

The Unicellularity of Contractions of Class C 0

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class Co on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction T of class CO on Hilbert space is unicellular if and only if T has one-point spectrum and every invariant subspace for T is cyclic (Theorem 6). (3) A contraction T of class Co on Hilbert space is unicellular if and only if T has one-point spectrum and all invariant subspaces of T are hyperinvariant subspaces of T (Theorem 8). The aim of this paper is to discuss the unicellularity of contractions of class Co. We shall generalize the unicellularity of operators acting on finitedimensional spaces to that on infinite-dimensional spaces. Throughout this paper, we denote by H a complex, separable, and infinitedimensional Hilbert space. The related concepts and symbols can be found in [4]. Definition 1. Let H1 be a nontrivial invariant subspace of T. H1 is called a Banach reducible subspace of T if there exists another nontrivial invariant subspace H2 of T such that

The Integration Operator in Two Variables

In this paper we consider the integration operator in two variables on ${L_2}{[0,1]^2}$, determine its multiplicity and reducing subspaces, and make some observations about its invariant and hyperinvariant subspaces.

Operator Semigroups, Flows, and Their Invariant Sets

Let B( H) be the bounded operators on a Hubert space H. An operator semi-group Σ is an absolutely convex, unital subsemigroup of the ball of B( H). Such a Σ induces a flow on H 1, the unit ball of H, namely ( s, x) → sx. Let l(Σ) be the lattice of closed invariant sets of this flow. Both the weak and norm topologies on H 1 induce Hausdorff metrics on the hyperspace of closed convex subsets of H 1. We show that various subsets of I(Σ) are closed subspaces of these hyperspaces. There exist nontrivial operator semigroups which have the same invariant sets as the ball of B( H); i.e., the "transitive semigroup problem" has a negative solution. However, we can characterize the ball of B( H) in terms of certain transitivity properties. We obtain an analogue of Lomonosov′s Theorem for operator semigroups which are transitive on the unit sphere of H. Let A be a weakly closed unital subalgebra of B( H), and let A 1 be its unit ball. We show that the existence of "minimal invariant sets" for A yields hyperinvariant subspaces. We characterize various operator algebras for which A 1 is strongly precompact including von Neumann algebras, triangular algebras, CSL algebras, and the standard function algebras acting by multiplication on a L 2 space. This yields structural results for certain algebras generated by a subnormal operator.

The integration operator in two variables

In this paper we consider the integration operator in two variables on L 2 [ 0 , 1 ] 2 {L_2}{[0,1]^2} , determine its multiplicity and reducing subspaces, and make some observations about its invariant and hyperinvariant subspaces.

The unicellularity of contractions of class 𝐶₀

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class C 0 {C_0} on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction T T of class C 0 {C_0} on Hilbert space is unicellular if and only if T T has one-point spectrum and every invariant subspace for T T is cyclic (Theorem 6). (3) A contraction T T of class C 0 {C_0} on Hilbert space is unicellular if and only if T T has one-point spectrum and all invariant subspaces of T T are hyperinvariant subspaces of T T (Theorem 8).

Which linear transformations have isomorphic hyperinvariant subspace lattices?

Let A be a linear transformation on a finite-dimensional complex vector space with the associated algebra Alg A, commutant { A}′, and hyperinvariant subspace lattice Hyperlat A. We determine Alg A, { A}′ (up to algebra isomorphism), and Hyperlat A (up to lattice isomorphism) in terms of the parameters in the Jordan form of A.

The Jordan form of a bitriangular operator

A Jordan operator is a (infinite) direct sum of a norm bounded sequence of finite dimensional Jordan blocks. An operator T is quasisimilar to a Jordan operator if and only if both T and T ∗ are triangularizable, whence bitriangular. The canonical Jordan model of such an operator is determined just by the numerical data: dim [ ker (T − λI) n + 1 ker (T − λI) n ] . An analysis of hyperinvariant subspaces, similarity orbits, and other aspects of these operators are analyzed using this model.

Invariant subspaces for Bishop-type operators

Nontrivial hyperinvariant subspaces are exhibited for a class of weighted translation operators which includes many operators of the form M φ U α , on L 2[0, 1), where φ ϵ L α [0, 1) and U α , is translation, modulo one, by an irrational α. In particular, if φ is the restriction to [0, 1) of a function analytic in some open neighbourhood of [0, 1] in the complex plane, then M φ U α has a nontrivial hyperinvariant subspace for almost all α.

Each Hyperinvariant Subspace for Multiplication Operator is Spectral

Each Hyperinvariant Subspace for Multiplication Operator is Spectral

Each hyperinvariant subspace for a multiplication operator is spectral

We consider multiplication operators on general separable complex L p {L^p} -spaces, for 1 ≤ p > + ∞ 1 \leq p > + \infty , and obtain the result announced in the title. Moreover, a result of Douglas and Pearcy on normal operators is given an alternate proof.

Invariant subspaces for algebras of subnormal operators. II

We continue our study of hyperinvariant subspaces for rationally cyclic subnormal operators. We establish a connection between hyperinvariant subspaces and weak-star continuous point evaluations on the commutant.

Matrix Operators and Hyperinvariant Subspaces

In this paper we study the super-decomposability of some matrix operators as well as some other special properties. These matrix operators are derived from non-analytic functional calculi. As by-products, we obtain statements concerning the existence of (non-trivial) hyperinvariant subspaces.

The existence of invariant subspaces for operators with nonsymmetric growth of the resolvent

One proves the existence of invariant and hyperinvariant subspaces for certain new classes of continuous operators in a Banach space. These classes are defined by conditions on the spectrum— it has to be “thin” (while, in the interesting cases, a one-point set) — and by estimates of the resolvent (necessarily nonsymmetric). For example, one can take operators T such that σ(T)={0} and for some β ε (0,π), The hyperinvariant subspaces have the form Ker f(T), and f(T) is defined in some special operator calculus, constructed in the paper.

Invariant subspaces for algebras of subnormal operators

Every rationally cyclic subnormal operator has a hyperinvariant subspace.

Cohyponormal operators with the single valued extension property

It is proved that in order to find a nontrivial hyperinvariant subspace for a cohyponormal operator it suffices to make the further assumption that the operator have the single‐valued extension property.

Invariant subspaces of Toeplitz operators

This paper is devoted to the problem of the existence of invariant subspaces for Toeplitz operators. Let Γ be a Lipschitzian arc in the plane and let f be a non-constant continuous functions on the unit circumference. It is proved that if there exists an open circle such that and if the modulus of continuity ωf of the function f satisfies the condition then the Toeplitz operator Tf in the Hardy space H2 has a nontrivial hyperinvariant subspace. For the proof of this theorem one makes use of the Lyubich-Matsaev theorem.

Hyperinvariant subspaces and the topology on LatA

Hyperinvariant subspaces and the topology on Lat<i>A</i>

Sets of essentially unitary operators

Let U e {U_e} be the set of essentially unitary operators on a separable Hilbert space H H ; for 1 ⩽ p ⩽ ∞ 1 \leqslant p \leqslant \infty , let U p {U_p} be the set of operators T T such that I − T ∗ T I - T^{\ast }T lies in the Schatten p p -ideal and the spectrum of T T does not fill the unit disc; and let U e n U_e^n be the set of operators in U e {U_e} of Fredholm index n n . The author proves that each U e n U_e^n is closed and path connected, that U p {U_p} is dense in U e 0 {U_e}^0 and U p {U_p} is path connected for each p p , and that all these sets are invariant under Cayley transform. It is proved that the spectrum is continuous on U ∞ {U_\infty } but not on U e {U_e} , while the spectral radius is continuous on U e {U_e} . Sufficient conditions that an operator in U e {U_e} have a nontrivial hyperinvariant subspace are given, and it is proved that the general hyperinvariant subspace problem can be reduced to that problem for perturbations of the bilateral shift. The product of commuting operators in U p {U_p} is U p {U_p} , but this result is false in general. Quasisimilarity in U e {U_e} is also studied; quasisimilar operators in U e ∖ U ∞ {U_e}\backslash {U_\infty } are unitarily equivalent modulo the ideal of compacts, and this result also holds in U ∞ {U_\infty } if the spectrum is also preserved.