Nontrivial Invariant Subspace Research Articles

ON INVARIANT SUBSPACES FOR POWER-BOUNDED OPERATORS OF CLASS $C_{1\cdot}$

We prove that if $T$ is a power-bounded operator of class $C_{*\cdot}$ on a Hilbert space which commutes with a nonzero quasinilpotent operator, then $T$ has a nontrivial invariant subspace. Connections with the questions of convergence of $T^n$ to $0$ in the strong operator topology and of cyclicity of power-bounded operators of class $C_{1\cdot}$ are discussed.

Quasireducible operators

We introduce the concept of quasireducible operators. Basic properties and illustrative examples are considered in some detail in order to situate the class of quasireducible operators in its due place. In particular, it is shown that every quasinormal operator is quasireducible. The following result links this class with the invariant subspace problem: essentially normal quasireducible operators have a nontrivial invariant subspace, which implies that quasireducible hyponormal operators have a nontrivial invariant subspace. The paper ends with some open questions on the characterization of the class of all quasireducible operators.

On quasi-affine transforms of Read’s operator

We show that C. J. Read’s example of an operator T T on ℓ 1 \ell _1 which does not have any non-trivial invariant subspaces is not the adjoint of an operator on a predual of ℓ 1 \ell _1 . Furthermore, we present a bounded diagonal operator D D such that even though D − 1 D^{-1} is unbounded, the operator D − 1 T D D^{-1}TD is a bounded operator on ℓ 1 \ell _1 with invariant subspaces, and is adjoint to an operator on c 0 c_0 .

Invariant subspaces and localizable spectrum

LetT ∈L(X) be a continuous linear operator on a complex Banach spaceX. We show thatT possesses non-trivial closed invariant subspaces if its localizable spectrum σloc(T) is thick in the sense of the Scott Brown theory. Since for quotients of decomposable operators the spectrum and the localizable spectrum coincide, it follows that each quasiaffine transformation of a Banach-space operator with Bishop's property (β) and thick spectrum has a non-trivial invariant subspace. In particular it follows that invariant-subspace results previously known for restrictions and quotients of decomposable operators are preserved under quasisimilarity.

Common properties of operatorsRS andSR andp-hyponormal operators

LetR andS be bounded linear operators on a Bananch space. We discuss the spectral and subdecomposable properties and properties concerning invariant subspaces common toRS andSR. We prove that, by these properties,p-hyponormal and log-hyponormal operators and their generalized Aluthge transformations are all subdecomposable operators;T andT(r, 1−r)(0<r<1) have same spectral structure and equal spectral parts ifT denotesp-hyponormal or dominant operator; for everyT eL(H), 0<r<1,T has nontrivial (hyper-)invariant subspace ifT(r, 1−r) does.

A generalization of Lomonosov’s inequality and its applications to invariant subspaces

In this article we generalize Victor Lomonosov’s famous inequality so as to be applicable to a wider class of functions. Then using it we prove that the adjoint of an algebra with a compactness property which is weaker than the PS property, employed by Victor Lomonosov, has nontrivial invariant subspaces.

Problem of invariant subspaces in C*-algebras

Let A be a separable simple C * -algebra. For each a (≠0) in A, there exists a separable faithful and irreducible * representation ( π, H π ) on A such that π(a) has a non-trivial invariant subspace in H π .

Invariant Subspaces of Operator Lie Algebras and the Theory of K -Algebras

It is proved that if a Lie algebra of compact operators contains a nonzero ideal consisting of quasinilpotent operators then this Lie algebra has a nontrivial invariant subspace. Some applications of this result to lattices of invariant subspaces for families of compact operators and to structures of ideals of Banach Lie algebras with compact adjoint action are given.

Asymptotically holomorphic functions and translation invariant subspaces of weighted Hilbert spaces of sequences

Let ω be a weight on Z and assume that the spectrum of the usual shift operator on the weighted space ℓ ω 2( Z) equals the unit circle. We show that if ∑ n<0 logω(n) n 2 =+∞ , and if the sequence ( ω(− n)) n⩾0 satisfies suitable growth and regularity conditions, then all nontrivial translation invariant subspaces of ℓ ω 2( Z) are generated by their intersection with ℓ ω 2( Z +)= u=(u n) n∈ Z ∈ℓ ω 2( Z)∣u n=0 (n<0) . When ω( n)=1 for n⩾0 and ω( n)=e | n|/log(| n|+1) for n<0, this shows that every nontrivial translation invariant subspace of ℓ ω 2( Z) is generated by the translates of the Fourier sequence of some singular inner function. The proofs are based on a priori estimates on the growth of the solutions of some convolution equations, obtained by using the theory of asymptotically holomorphic functions in the disc.

Rank-one perturbations of diagonal operators

We study rank-one perturbations of diagonal Hilbert space operators mainly from the standpoint of invariant subspace problem. In addition to proving some general properties of these operators, we identify the normal operators and contractions in this class. We show that two well known results about the eigenvalues of rank-one perturbations and one-codimension compressions of self-adjoint compact operators are equivalent. Sufficient conditions are given for existence of nontrivial invariant subspaces for this class of operators.

On hypo-Jordan operators

In this paper, we show that if T=S+N, where S is similar to a hyponormal operator, S and N commute and N is a nilpotent operator of order m (i.e., N^m=0), then T is a subscalar operator of order 2m. As a corollary, we get that such a T has a nontrivial invariant subspace if its spectrum \sigma(T\hskip1) has the property that there exists some non-empty open set U such that \sigma(T\hskip1)\capU is dominating for U.

An application of the smooth variational principle to the existence of nontrivial invariant subspaces

We show that the consideration of Gâteaux smooth functions on Banach spaces which admit an equivalent Gâteaux smooth norm allows us to show that certain linear operators have nontrivial closed invariant subspaces. It is in particular the case of all operators on a real Banach space which admit a moment sequence.

Invariant subspaces for bounded operators with large localizable spectrum

SupposeHHis a complex Hilbert space andT∈L(H)T\in L(H)is a bounded operator. For each closed setF⊂CF\subset \mathbf {C}letHT(F)H_{T}(F)denote the corresponding spectral manifold. Letσloc(T)\sigma _{loc}(T)denote the set of all pointsλ∈σ(T)\lambda \in \sigma (T)with the property thatHT(V¯)≠0H_{T}(\overline {V})\neq 0for any open neighborhoodVVofλ.\lambda .In this paper we show that ifσloc(T)\sigma _{loc}(T)is dominating in some bounded open set, thenTThas a nontrivial invariant subspace. As a corollary, every Hilbert space operator which is a quasiaffine transform of a subdecomposable operator with large spectrum has a nontrivial invariant subspace.

Proper contractions and invariant subspaces

Let T be a contraction and A the strong limit of {T∗nTn}n ≥ 1. We prove the following theorem: if a hyponormal contraction T does not have a nontrivial invariant subspace, then T is either a proper contraction of class 𝒞00 or a nonstrict proper contraction of class 𝒞10 for which A is a completely nonprojective nonstrict proper contraction. Moreover, its self‐commutator [T*, T] is a strict contraction.

Some remarks on the invariant subspace problem for hyponormal operators

We make some remarks concerning the invariant subspace problem for hyponormal operators. In particular, we bring together various hypotheses that must hold for a hyponormal operator without nontrivial invariant subspaces, and we discuss the existence of such operators.

The Existence of Translation Invariant Subspaces of Symmetric Self-Adjoint Sequence Spaces on [formula omitted

We prove that if X is a reflexive translation invariant Banach space of complex sequences on Z that contains all finitely supported sequences, in which the coordinate functionals are continuous, and for every sequence {c(n)} in the space the sequences {c(n)} and {c(−n)} are also in the space, then X has a nontrivial translation invariant subspace. This provides in particular a positive solution to the translation invariant subspace problem for weighted ℓp spaces on Z with even weights, for 1<p<∞. The proof is based on an intermediate result that asserts that if A is an operator on a reflexive real Banach space of dimension greater than one, and there exist non-zero vectors, u in the space and v in the dual space, such that {〈Anu, v〉}∞n=0 is a moment sequence of a finite positive Borel measure on a bounded interval on the real line, then A has a nontrivial invariant subspace.

Aluthge transforms of operators

Associated with every operatorT on Hilbert space is its Aluthge transform $$\tilde T$$ (defined below). In this note we study various connections betweenT and $$\tilde T$$ , including relations between various spectra, numerical ranges, and lattices of invariant subspaces. In particular, we show that if $$\tilde T$$ has a nontrivial invariant subspace, then so doesT, and we give various applications of our results.

Decomposability and structure of nonnegative bands in ℳn(ℝ)

A standard subspace of ℝ n is a space spanned by a subset of the standard basis { e 1 , e 2 ,…, e n }. A multiplicative semigroup. ℒ in ℳ n (ℝ) is said to be decomposable if its members have a common nontrivial standard invariant subspace. Necessary and sufficient conditions for decomposability of nonnegative semigroups are given. In particular, decomposability of nonnegative bands (semigroups of idempotents) and their structure is discussed. It is proved that a nonnegative band with each member having rank greater than 1 is decomposable. Also, a geometric characterization of maximal, rank-one nonnegative bands is given.

Quasinilpotent Operators and the Invariant Subspace Problem

In this paper we construct quasinilpotent operators without nontrivial invariant subspaces. As well as being interesting in its own right, we believe that this is a sensible (perhaps necessary) first step for addressing the difficult problem of constructing topologically simple Banach algebras.

K-Quasihyponormal operators are subscalar

In this paper we shall prove that if an operatorT∈L(⊕ 1 2 H) is an operator matrix of the form $$T = \left( {\begin{array}{*{20}c} {T_1 } & {T_2 } \\ 0 & {T_3 } \\ \end{array} } \right)$$ whereT 1 is hyponormal andT 3 k =0, thenT is subscalar of order 2(k+1). Hence non-trivial invariant subspaces are known to exist if the spectrum ofT has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that anyk-quasihyponormal operators are subscalar.