Every operator has almost-invariant subspaces

In this paper, we show that for T ∈ B(H), if M is almost-invariant for T , then every maximal almost-invariant subspace of M is of codimension 1 in M, where H is a separable, infinite-dimensional Hilbert space. We also describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most 1 acting on H. Our result is that for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension 1 in it.

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“…The following lemma, proved by Popov and Tcaciuc in [14], is quite important to get the main result of this section.…”

Section: Maximal Almost-invariant Subspacesmentioning

confidence: 93%

“…So we only need to consider a half-space, that is, a subspace of H which is infinite-dimensional and infinite-codimensional. For further information about almost-invariant subspaces, we refer the interested readers to [2,[12][13][14].…”

Section: Introductionmentioning

confidence: 99%

A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces

2019

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“…In their paper [10] the authors were able, in the special case of complex Hilbert spaces, to prove a quantitative version of this result, albeit at some cost. Namely, for any bounded operator T there exists k ∈ N depending only on T , such that for any > 0 there exists an operator K of rank at most k, ||K|| < , such that T − K has an invariant half-space (see [10,Theorem 3.3 and Proposition 3.4]). It would be of interest to know whether this result can be generalized to pairs of idempotents as well.…”

Section: Theoremmentioning

confidence: 98%

“…Recently, the notion of almost invariant half-subspaces has attracted a considerable amount of attention, see [3,7,10,14]. Recall that a subspace of a Banach space is called a half-space if it has infinite dimension and codimension.…”

Section: Theoremmentioning

confidence: 99%

Invariant and Almost-Invariant Subspaces for Pairs of Idempotents

Bernik

Radjavi

2015

Integr. Equ. Oper. Theory

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It is known that every bounded operator on an infinite dimensional separable Hilbert space H has an invariant subspace if and only if each pair of idempotents on H has a common invariant subspace. We show that the same equivalence holds for operators and pairs of idempotents that are essentially selfadjoint. We also show that each pair of idempotents on H has a common almost-invariant half-space.Mathematics Subject Classification. 47A15.

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“…It is worth observing that there is one inherent weakness in the definition of orthogonally reductive operators: it is entirely possible that there might exist an operator with no non-trivial closed, invariant subspace, in which case the operator is reductive for trivial reasons. On the other hand, it was shown by Popov and Tcaciuc [16] that given any operator T acting on an infinite-dimensional, complex, separable Hilbert space H, there exists an orthogonal projection P of infinite rank and co-rank such that rank P T P ⊥ ≤ 1. (Their result actually holds for operators acting on reflexive Banach spaces and beyond, but we do not require that here.)…”

Section: The Infinite-dimensional Setting -Property (Cr)mentioning

confidence: 99%

Hilbert space operators with compatible off-diagonal corners

Livshits

Macdonald

Marcoux

et al. 2018

Journal of Functional Analysis

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Given a complex, separable Hilbert space H, we characterize those operators for which P T (I − P ) = (I − P )T P for all orthogonal projections P on H. When H is finitedimensional, we also obtain a complete characterization of those operators for which rank (I − P )T P = rank P T (I − P ) for all orthogonal projections P . When H is infinite-dimensional, we show that any operator with the latter property is normal, and its spectrum is contained in either a line or a circle in the complex plane.2010 Mathematics Subject Classification. 15A60, 47A20, 47A30, 47B15. 1 Research supported in part by NSERC (Canada). Aug. 7, 2018. 1 1.2. Definition. Let T ∈ B(H). We say that T has the common norm property (property (CN)) if for any projection P ∈ B(H) we have that P T P ⊥ = P ⊥ T P .We denote by G norm the set of operators with property (CN). We say that T has the common rank property (property (CR)) if for any projection P ∈ B(H) we have that rank P T P ⊥ = rank P ⊥ T P.We denote by G rank the set of operators with property (CR). As we shall see, our results depend upon whether or not H is finite-dimensional. When the Hilbert space is finite-dimensional and of dimension at least four, then we shall show that the set of operators satisfying property (CN) coincides with the set of operators satisfying property (CR), and that this consists of those operators which are scalar translates of scalar multiples of hermitian (or of unitary) operators. (See Theorem 3.15 below.)In the infinite-dimensional setting, we obtain a complete characterization of those operators satisfying property (CN). Again, any scalar translate of a scalar multiple of a hermitian operator will suffice. This time, however, the unitary operators involved must have essential spectrum contained in only half of a circle. (See Theorem 4.13 below.)The problem of characterizing those operators acting on an infinite-dimensional Hilbert space which enjoy property (CR) is much more delicate. We are able to demonstrate that any operator T satisfying property (CR) must once again be a scalar translate of a scalar multiple of a hermitian (or of a unitary) operator. In particular, such operators are normal. However, an obstruction occurs in that it is not the case that every unitary operator has property (CR). Indeed, as is well-known (see Section 5 for an example) -not every unitary operator is reductive.1.3. We shall need some standard notation and definitions in what follows.If T = [ A B C D ] is a block-matrix in M n (C), and A is invertible, then the matrix D − CA −1 B is said to be the Schur complement of A in T and is denoted by T |A. In such a case T is invertible if and only if T |A is, and when this happens, the SE block-corner T −1 SE of T −1 is (T |A) −1 . Furthermore:Similarly, if B is invertible then C − DB −1 A is the Schur complement T |B of B in T , and T is invertible if and only if T |B is, in which caseCorresponding statements and concepts apply to C and D as well.As always, T = {z ∈ C : |z| = 1}. A subset of C is circlinear if it is contained in a ...

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Every operator has almost-invariant subspaces

Cited by 24 publications

References 12 publications

A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces

A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces

Invariant and Almost-Invariant Subspaces for Pairs of Idempotents

Hilbert space operators with compatible off-diagonal corners

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