Hyperinvariant Subspaces Research Articles

The Lomonosov type theorems and the invariant subspace problem for non-archimedean Banach spaces

In this paper, we study the existence of invariant (and even hyperinvariant) subspaces of bounded operators on a non-archimedean Banach space E=(E,‖.‖) over a valued field K equipped with a non-trivial non-archimedean valuation |.|. Specifically, we consider compact operators and operators that commute with a compact operator. First we show that if E has a base, then any compact operator T such that limn⁡‖Tn‖1n>0 has a finite-dimensional hyperinvariant subspace. Next we show that if K is locally compact, then every compact operator T on E has a hyperinvariant subspace. Afterward, assuming that K is spherically complete or E is of countable type, we provide a necessary condition for a bounded operator on E to have a hyperinvariant subspace. We demonstrate that the classical Lomonosov Invariant Subspace theorem does not hold in the case where K is non-spherically complete. Finally, we prove Lomonosov type theorem for spectral quasinilpotent operators, when K is locally compact.

Invariant subspaces of contractions with constant characteristic function

We describe all the invariant subspaces of a completely nonunitary contraction T c T_c with nonzero scalar constant characteristic function c c , where | c | > 1 |c|>1 . Moreover, we show that T c T_c and restriction to its invariant subspaces are essentially normal. Using this description of T c T_c -invariant subspaces, we then identify the hyperinvariant subspaces.

Hypercyclictty and Countable Hypercyclicity for Adjoint of Operators

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .

Isomorphisms between lattices of hyperinvariant subspaces

Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.

Hyperinvariant subspaces for operators intertwined with weighted shift operators

Hyperinvariant subspaces for operators intertwined with weighted shift operators

Power set of some quasinilpotent weighted shifts on ℓp

For a quasinilpotent operator T on a Banach space X, Douglas and Yang defined kx=limsupz→0ln⁡‖(z−T)−1x‖ln⁡‖(z−T)−1‖ for each nonzero vector x∈X, and call Λ(T)={kx:x≠0} the power set of T. They proved that the power set have a close link with T's lattice of hyperinvariant subspaces. In this paper, we compute the power set of some weighted shifts on ℓp for 1≤p<∞. The following results are obtained: (1) If T is an injective quasinilpotent forward unilateral weighted shift on ℓp(N), then Λ(T)={1} when ke0=1, where {en}n=0∞ be the canonical basis for ℓp(N); (2) There is a class of backward unilateral weighted shifts on ℓp(N) whose power set is [0,1]; (3) There exists a bilateral weighted shift on ℓp(Z) with power set [12,1].

Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy

Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

We prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5–7], Fang and J. Xia [3] and the authors [8,9] on an open question posed by Pearcy in the seventies.

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.

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.

Finite rank perturbations of normal operators: Spectral subspaces and Borel series

We characterize the spectral subspaces associated to closed sets of rank-one perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space by means of functional equations involving Borel series. As a particular instance, if T=DΛ+u⊗v is a rank-one perturbation of a diagonalizable normal operator DΛ with respect to a basis E=(en)n≥1 and the vectors u and v have Fourier coefficients (αn)n≥1 and (βn)n≥1 with respect to E, respectively, it is shown that T has non-trivial closed invariant subspaces provided that either (αn)n≥1∈ℓ1 or (βn)n≥1∈ℓ1. Likewise, analogous results hold for finite rank perturbations of DΛ. Moreover, such operators T have non-trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity extending previous theorems of Foiaş, Jung, Ko and Pearcy [8] and of Fang and J. Xia [6] on an open question of at least forty years.

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.

Local commutants and ultrainvariant subspaces

For an operator A on a complex Banach space X and a closed subspace M⊆X, the local commutant of A at M is the set C(A;M) of all operators T on X such that TAx=ATx for every x∈M. It is clear that C(A;M) is a closed linear space of operators, however it is not an algebra, in general. For a given A, we show that C(A;M) is an algebra if and only if the largest subspace MA of X such that C(A;M)=C(A;MA) is invariant for every operator in C(A;M). We say that a closed subspace M⊆X is ultrainvariant for A∈B(X) if it is invariant for every operator in C(A;M). For several types of operators we prove that they have non-trivial ultrainvariant subspaces. For a normal operator on a Hilbert space, every hyperinvariant subspace is ultrainvariant. On the other hand, the family of all ultrainvariant subspaces of a nilpotent operator can be a proper subset of the lattice of all hyperinvariant subspaces.

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.

Spectral Decompositions Arising from Atzmon’s Hyperinvariant Subspace Theorem

By means of a weaker functional model, we prove the existence of non-trivial closed hyperinvariant subspaces for linear bounded operators generalizing, in particular, a classical theorem of Atzmon and revealing the spectral nature of the hyperinvariant subspaces involved. As an application, we show non-trivial spectral subspaces for Bishop operators on $$L^p[0,1)$$ , $$1\le p<\infty $$ , as long as they satisfy Atzmon’s Theorem, providing, in turns, a local spectral decomposition.

Hermitian geometry on the resolvent set (II)

For an element A in a unital C*-algebra ℬ, the operator-valued 1-form ωa(z) = (z − A)−1dz is analytic on the resolvent set ρ(A), which plays an important role in the functional calculus of A. This paper defines a class of Hermitian metrics on ρ(A) through the coupling of the operator-valued (1, 1)-form ΩA = −ω*A Λ ωa with tracial and vector states. Its main goal is to study the connection between A and the properties of the metric concerning curvature, arc length, completeness and singularity. A particular example is when A is quasi-nilpotent, in which case the metric lives on the punctured complex plane ℂ {0}. The notion of the power set is defined to gauge the “blow-up” rate of the metric at 0, and examples are given to indicate a likely link with A’s hyper-invariant subspaces.

Hyperinvariant subspaces for some 2 × 2 operator matrices, II

Hyperinvariant subspaces for some 2 × 2 operator matrices, II

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.