Nontrivial Hyperinvariant Subspace Research Articles

Square roots of complex symmetric operators

In this paper we study square roots of complex symmetric operators. In particular, we prove that if T∈L(H) is a square root of a complex symmetric operator, then T∗ has the single-valued extension property if and only if so does T. Moreover, in this case, T has the Bishop's property (β) if and only if T is decomposable. Finally, we show that if T has a nontrivial hyperinvariant subspace, then T∗ has a nontrivial invariant subspace.

Hyperinvariant subspaces for sets of polynomially compact operators

We prove the existence of a non-trivial hyperinvariant subspace for several sets of polynomially compact operators. The main results of the paper are: (i) a non-trivial norm closed algebra \(\mathcal {A}\subseteq \mathcal {B}(\mathscr {X})\) which consists of polynomially compact quasinilpotent operators has a non-trivial hyperinvariant subspace; (ii) if there exists a non-zero compact operator in the norm closure of the algebra generated by an operator band \(\mathcal {S}\), then \(\mathcal {S}\) has a non-trivial hyperinvariant subspace.

Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

Abstract The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

On local spectral properties of operator matrices

In this paper, we focus on a 2 times 2 operator matrix T_{epsilon _{k}} as follows: \t\t\tTϵk=(ACϵkDB),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} T_{\\epsilon _{k}}= \\begin{pmatrix} A & C \\\\ \\epsilon _{k} D & B\\end{pmatrix}, \\end{aligned}$$ \\end{document} where epsilon _{k} is a positive sequence such that lim_{krightarrow infty }epsilon _{k}=0. We first explore how T_{epsilon _{k}} has several local spectral properties such as the single-valued extension property, the property (beta ), and decomposable. We next study the relationship between some spectra of T_{epsilon _{k}} and spectra of its diagonal entries, and find some hypotheses by which T_{epsilon _{k}} satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix T_{epsilon _{k}} has a nontrivial hyperinvariant subspace.

Compact perturbations of scalar type spectral operators

We consider compact perturbations S=DΛ+K of normal diagonal operators DΛ by certain compact operators. Sufficient conditions for K to ensure the existence of non-trivial hyperinvariant subspaces for S have recently been obtained by Foia\c{s} et al. in C.\ Foia\c{s}, I.B.\ Jung, E.\ Ko, C. Pearcy, \textit{J.\ Funct. Anal.} \textbf{253}(2007), 628--646, C.\ Foia\c{s}, I.B.\ Jung, E.\ Ko, C.~Pearcy, \textit{Indiana Univ.\ Math.\ J.} \textbf{57}(2008), 2745--2760, {C.\ Foia\c{s}, I.B.\ Jung, E.\ Ko, C.Pearcy}, \textit{J.\ Math.\ Anal.\ Appl.} \textbf{375}(2011), 602--609 (followed by Fang--Xia \textit{J.\ Funct. Anal} \textbf{263}(2012), 135-1377, and Klaja \textit{J.\ Operator Theory} \textbf{73}(2015), 127--142, by using certain spectral integrals along straight lines through the spectrum of S. In this note, the authors use circular cuts and get positive results under less restrictive local conditions for K.

Invertibility of functions of operators and existence of hyperinvariant subspaces

Let $T$ be an absolutely continuous polynomially bounded operator, and let $\theta$ be a singular inner function. It is shown that if $\theta(T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial hyperinvariant subspaces.

Perturbation of normal quaternionic operators

The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two S S -resolvent operators and the S S -spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the S S -spectrum of a quaternionic linear operator T T is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S S -spectrum and of slice hyperholomorphicity of the S S -resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.

Some sufficient conditions for the existence of hyperinvariant subspaces for operators intertwined with unitaries

For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. These hyperinvariant subspaces are the closures of the range of $\varphi (T)$, where $\varphi $ is a singul

Quasinilpotent operators and non-Euclidean metrics

The power setΛ(V) of quasinilpotent operator V on a Hilbert space H is defined in [4] to study the singularity of the non-Euclidean metrics ‖(V−z)−1x‖2dz⊗dz¯ at σ(V)={0}, and it is shown that if Λ(V) contains more than one point then V has a nontrivial hyperinvariant subspace. This paper first proves that the Volterra integral operator on the classical Hardy–Hilbert space has singleton power set, thus answering a question raised in [4]. Then, it studies the length of circles under the metrics and its connection with power set. In particular, it determines the maximal length of the unit circle with respect to the change of x in the metrics. Moreover, it shows that an extremal value integral equation is able to detect representing functions for all the invariant subspaces of the Volterra operator.

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.

On Analytic Roots of Hyponormal Operators

In this paper, we provide a condition under which analytic roots of hyponormal operators (defined below) have scalar extensions. As an application, we show that such analytic roots of hyponormal operators have nontrivial hyperinvariant subspaces. We also give some structures for analytic roots of hyponormal operators. Finally, we verify that the product of some analytic root of a hyponormal operator and an algebraic operator which are commuting is subscalar.

WEYL'S THEOREM, TENSOR PRODUCT, FUGLEDE-PUTNAM THEOREM AND CONTINUITY SPECTRUM FOR k-QUASI CLASS An*OPERATO

An operator <TEX>$T{\in}L(H)$</TEX>, is said to belong to k-quasi class <TEX>$A_n^*$</TEX> operator if <TEX>$$T^{*k}({\mid}T^{n+1}{\mid}^{\frac{2}{n+1}}-{\mid}T^*{\mid}^2)T^k{\geq}O$$</TEX> for some positive integer n and some positive integer k. First, we will see some properties of this class of operators and prove Weyl's theorem for algebraically k-quasi class <TEX>$A_n^*$</TEX>. Second, we consider the tensor product for k-quasi class <TEX>$A_n^*$</TEX>, giving a necessary and sufficient condition for <TEX>$T{\otimes}S$</TEX> to be a k-quasi class <TEX>$A_n^*$</TEX>, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class <TEX>$A_n^*$</TEX> operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and <TEX>$(B^*)^{-1}$</TEX> are k-quasi class <TEX>$A_n^*$</TEX> operators such that AX = XB, then <TEX>$A^*X=XB^*$</TEX>. Finally, we will prove the spectrum continuity of this class of operators.

A Class of Invertible Bilateral Weighted Shifts

Abstract. In this note we study a class of invertible weighted bilateral shifts on Hilbertspace introduced by Haskell Rosenthal recently. We show that every Rosenthal shift isunitarily equivalent to its inverse, not quasisimilar to its adjoint, and has a nontrivialhyperinvariant subspace. We write, as usual, Z for the set of integers and N(N 0 ) for the set of positive(nonnegative) integers. We also write l 2 (Z) for the separable, infinite dimensional,complex Hilbert space l 2 (Z) := ff n g n2 Z :  n 2 C ; ∑ n2 Z j n j 2 0, then the operator B w 2 L( l 2 (Z)) definedby(1) B w e n = w n e n +1 ; n 2 Z ; is a (invertible, forward) weighted bilateral shift. A trivial calculation gives thedefining equations(2)

Invariant and hyperinvariant subspaces for amenable operators

There has been a long-standing conjecture in Banach algebras that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to the statement that every non-scalar amenable operator has a non-trivial hyperinvariant subspace. It is also equiv- alent to the statement that every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two de- compositions for amenable operators, supporting the conjecture.

Invariant subspaces for certain finite-rank perturbations of diagonal operators

Suppose that {ek} is an orthonormal basis for a separable, infinite-dimensional Hilbert space H. Let D be a diagonal operator with respect to the orthonormal basis {ek}. That is, D=∑k=1∞λkek⊗ek, where {λk} is a bounded sequence of complex numbers. LetT=D+u1⊗v1+⋯+un⊗vn. Improving a result of Foias et al. (2007) [3], we show that if the vectors u1,…,un and v1,…,vn satisfy an ℓ1-condition with respect to the orthonormal basis {ek}, and if T is not a scalar multiple of the identity operator, then T has a non-trivial hyperinvariant subspace.

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.

Spectral decomposability of rank-one perturbations of normal operators

This paper is a continuation of the study by Foias, Jung, Ko, and Pearcy (2007) [4] and Foias, Jung, Ko, and Pearcy (2008) [5] of rank-one perturbations of diagonalizable normal operators. In Foias, Jung, Ko, and Pearcy (2007) [4] we showed that there is a large class of such operators each of which has a nontrivial hyperinvariant subspace, and in Foias, Jung, Ko, and Pearcy (2008) [5] we proved that the commutant of each of these rank-one perturbations is abelian. In this paper we show that the operators considered in Foias, Jung, Ko, and Pearcy (2007) [4] have more structure – namely, that they are decomposable operators in the sense of Colojoară and Foias (1968) [1].

A Reflexivity Result Concerning Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λj with radius rj for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reflexive Banach space whose spectrum contains ∂Ω and does not contain the points λ1, λ2, . . . , λn, and the operators T and rj(T − λjI)−1 are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f(T) : f is a rational function with poles off \({\overline\Omega}\)} is reflexive.

A note on transitive localizing algebras

A simple proof is provided for a theorem of Troitsky that every nonzero quasinilpotent operator on a Banach space whose commutant is a localizing algebra 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.