Hyperinvariant Subspace Problem Research Articles

Quasiaffine transforms of Hilbert space operators

Ampliation quasisimilarity was applied as a tool in Foias and Pearcy (J Funct Anal 219:134–142, 2005) to reduce the hyperinvariant subspace problem to a particular class of operators. The seemingly weaker pluquasisimilarity relation was introduced in Bercovici et al. (Acta Sci Math Szeged 85:681–691, 2019) and studied also in Kérchy (Acta Sci Math Szeged 86:503–520, 2020). The problem whether these two relations are actually equivalent is addressed in the present paper. The following more general, related question is studied in details: under what conditions is the operator A a quasiaffine transform of B, whenever A can be injected into B and A can be also densely mapped into B.

On local spectral properties of extended Hamilton operators

This paper deals with local spectral properties of Extended Hamilton operators and their adjoint operators. The relationship between the local spectral properties (strongly decomposability, hyperinvariant subspace problem, etc.) of Extended Hamilton operators and the corresponding properties of their adjoint operators is obtained.

Generalizations of the relation of quasisimilarity for operators

In this note we first briefly review the progress on the hyperinvariant subspace problem for operators on Hilbert space made possible by the equivalence relation of ampliation quasisimilarity recently introduced in [7].Then we introduce another equivalence relation, which we call pluquasisimilarity, with bigger equivalence classes than ampliation quasisimilarity but verydifferent in appearance, which preserves the existence of hyperinvariant sub-spaces for operators, and thus may be useful in the future. We also comparethese with two other equivalence relations, injection-similarity and completeinjection-similarity, introduced long ago by Sz.-Nagy and Foias in [13].

Hyperinvariant Subspace Problem for Some Classes of Operators

An n-normal operator may be defined as an $$n \times n$$ operator matrix with entries that are mutually commuting normal operators and an operator $$T \in \mathcal {B(H)}$$ is quasi-nM-hyponormal (for $$n \in \mathbb {N}$$ ) if it is unitarily equivalent to an $$n \times n$$ upper triangular operator matrix $$(T_{ij})$$ acting on $$\mathcal {K}^{(n)}$$ , where $$\mathcal {K}$$ is a separable complex Hilbert space and the diagonal entries $$T_{jj}$$ $$(j = 1,2,\ldots , n)$$ are M-hyponormal operators in $$\mathcal {B(K)}$$ . This is an extended notion of n-normal operators. We prove a necessary and sufficient condition for an $$n \times n$$ triangular operator matrix to have Bishop’s property $$(\beta )$$ . This leads us to study the hyperinvariant subspace problem for an $$n \times n$$ triangular operator matrix.

Quasi-nilpotent Operators and Hyperinvariant Subspace Problem

In this note, we strengthen some well-known results on the hyperinvariant subspace problem for quasi-nilpotent operators. We show that if T is a quasi-nilpotent quasi-affinity on a Hilbert space and there is a sequence $$\{x_n\}$$ of unit vectors, such that the closure of $$\{x_n : n\in \mathbb {N}\}$$ is compact and zero is not a weak limit of any subsequence of $$\{\frac{T^nT^{*n}x_n}{\Vert T^nT^{*n}x_n\Vert }\}$$ , then T has a nontrivial hyperinvariant subspace.

Triangular n-isometric operators

In this paper, we introduce the class of triangular n-isometric operators and study various properties. We show that every triangular n-isometric operator is subscalar of order 2n; in particular, every isometric operator is subscalar of order two. Consequently, if the spectrum of a triangular n-isometric operator T has a nonempty interior in , then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for triangular n-isometric operators. Some spectral properties of this class of operators are also presented.

Spectral Properties of k-Quasi-M-hyponormal Operators

In this paper, we study the k-quasi-M-hyponormal operator and mainly prove that if T is a k-quasi-M-hyponormal operator, then \(\sigma _{ja}(T)\backslash \{0\}=\sigma _{a}(T)\backslash \{0\}\), and the spectrum is continuous on the class of all k-quasi-M-hyponormal operators; let \(d_{AB}\in B(B(H))\) denote either the generalized derivation \(\delta _{AB}= L_{A}-R_{B}\) or the elementary operator \(\Delta _{AB} =L_{A}R_{B}- I\), we show that if A and \(B^{*}\) are k-quasi-M-hyponormal operators, then \(d_{AB}\) is polaroid and generalized Weyl’s theorem holds for \(f(d_{AB})\), where f is an analytic function on \(\sigma (d_{AB})\) and f is not constant on each connected component of the open set U containing \(\sigma (d_{AB})\). In additon, we discuss the hyperinvariant subspace problem for k-quasi-M-hyponormal operators.

Subscalarity, Invariant, and Hyperinvariant Subspaces for Upper Triangular Operator Matrices

In this paper, we introduce the class of quasi-nM-hyponormal operators and study various properties. We show that every quasi-nM-hyponormal operator is subscalar of order 2n; in particular, every M-hyponormal operator is subscalar of order two. Consequently, if the spectrum of T has a nonempty interior in \(\mathbb {C}\), then T has a nontrivial invariant subspace. We also examine the hyperinvariant subspace problem for quasi-nM-hyponormal operators. Finally, we consider some of the spectral properties of this class and show that they share many spectral properties with normal operators on Hilbert spaces, mainly concerning Fredholm theory and local spectral theory. We are primarily interested in isolated points of the spectra of quasi-nM-hyponormal operators as well as the isolated points of the approximate point spectra. These properties lead to the concept of a polaroid-type operator, which when combined with the single-valued extension property, an important property in local spectral theory, produces a general framework for several versions of Weyl-type theorems.

Hyperinvariant subspaces for operators having a compact part

It is well known that if T = A ⊕ B , where A is compact, then T has a nontrivial hyperinvariant subspace. In this paper, we try to solve the hyperinvariant subspace problem for operators which have a compact part. Our main result is that if A is compact, then either ( A ⁎ 0 B ) or ( A 0 ⁎ B ) has a nontrivial hyperinvariant subspace.

Hyperinvariant subspaces for quasinilpotent operators on Hilbert spaces

In this paper, we employ the model theory due to C. Foias and C. Pearcy and the notion of Enflo's extremal vectors of quasinilpotent operators to study the hyperinvariant subspace problem for quasinilpotent operators. Our main result is that if a quasinilpotent quasiaffinity T has a sequence of “ c-eigenvectors” x n of T n T ∗ n such that the set cl { x n : n ∈ N } is compact, then T has a nontrivial hyperinvariant subspace.

Quasidiagonality and the hyperinvariant subspace problem

In a sequence of recent papers, [11], [13], [9] and [5], the authors (together with H. Bercovici and C. Foias) reduced the hyperinvariant subspace problem for operators on Hilbert space to the question whether every C00-(BCP)-contraction that is quasidiagonal and has spectrum the unit disc has a nontrivial hyperinvariant subspace (n.h.s.). An essential ingredient in this reduction was the introduction of two new equivalence relations, ampliation quasisimilarity and hyperquasisimilarity, defined below. This note discusses the question whether, by use of these relations, a further reduction of the hyperinvariant subspace problem to the much-studied class (N + K) (defined below) might be possible.

Some quasinilpotent generators of the hyperfinite II 1 factor

For each sequence { c n } n in l 1 ( N ) we define an operator A in the hyperfinite II 1-factor R . We prove that these operators are quasinilpotent and they generate the whole hyperfinite II 1-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of A are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of A ∗ A .

Hyperinvariant Subspace Problem for Quasinilpotent Operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let $$({\mathcal{CRQ}})$$ denote the class of quasinilpotent quasiaffinities Q in $${\mathcal{L}}(H)$$ such that Q * Q has an infinite dimensional reducing subspace M with Q * Q| M compact. It was known that if every quasinilpotent operator in $$({\mathcal{CRQ}})$$ has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in $$({\mathcal{CRQ}})$$ . The purpose of this paper is to provide sufficient conditions for elements in $$({\mathcal{CRQ}})$$ to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.

THE HYPERINVARIANT SUBSPACE PROBLEM FOR QUASI-n-HYPONORMAL OPERATORS

In this paper we examine the hyperinvariant subspace problem for quasi-n-hyponormal operators. The main result on this problem is as follows. If T = N + K is such that N is a quasi-n-hyponormal operator whose spectrum contains an exposed arc and K belongs to the Schatten p-ideal then T has a non-trivial hyperinvariant subspace.

Shift-type invariant subspaces of contractions

Using the Sz.-Nagy–Foias functional model it was shown in [L. Kérchy, Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes, Acta Sci. Math. (Szeged) 61 (1995) 443–476] that under certain conditions on a contraction T the natural embedding of a Hardy space of vector-valued functions into the corresponding L 2 space can be factored into the product of two transformations, intertwining T with a unilateral shift and with an absolutely continuous unitary operator, respectively. The norm estimates in the Factorization Theorem of this paper are sharpened to their best possible form by essential improvements in the proof. As a consequence we obtain that if the residual set of a contraction covers the whole unit circle then those invariant subspaces, where the restriction is similar to the unilateral shift with a similarity constant arbitrarily close to 1, span the whole space. Furthermore, the hyperinvariant subspace problem for asymptotically non-vanishing contractions is reduced to these special circumstances.

Hyperinvariant subspaces for some subnormal operators

In this article we employ a technique originated by Enflo in 1998 and later modified by the authors to study the hyperinvariant subspace problem for subnormal operators. We show that every “normalized” subnormal operatorSSsuch that either{(S∗nSn)1/n}\{(S^{\ast n}S^{n})^{1/n}\}does not converge in the SOT to the identity operator or{(SnS∗n)1/n}\{(S^{n}S^{\ast n})^{1/n}\}does not converge in the SOT to zero has a nontrivial hyperinvariant subspace.

On the hyperinvariant subspace problem III

In two recent papers (Foias and Pearcy, J. Funct. Anal., in press, Hamid et al., Indiana Univ. Math. J., to appear), the authors reduced the hyperinvariant subspace problem for operators on Hilbert space to the question whether every C 00 - ( BCP ) -operator that is quasidiagonal and has spectrum the unit disc has a nontrivial hyperinvariant subspace (n.h.s.). In this note, we continue this study by showing, with the help of a new equivalence relation, that every operator whose spectrum is uncountable, as well as every nonalgebraic operator with finite spectrum, has a hyperlattice (i.e., lattice of hyperinvariant subspaces) that is isomorphic to the hyperlattice of a C 00 , quasidiagonal, (BCP)-operator whose spectrum is the closed unit disc.

On the hyperinvariant subspace problem

In this paper, we introduce a new equivalence relation, ampliation quasisimilarity, on L( H) , more general than quasisimilarity, that preserves the existence of nontrivial hyperinvariant subspaces. We show that if T does not have nontrivial hyperinvariant subspaces for elementary reasons, then T is ampliation quasisimilar to a (BCP)-operator in the class C 00. This reduces the hyperinvariant subspace problem for operators in L( H) to a very special subcase of itself.

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.

Sets of Essentially Unitary Operators

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