Maximal, Minimal, and Primary Invariant Subspaces

Atzmon, Aharon

doi:10.1006/jfan.2001.3760

Cited by 4 publications

(3 citation statements)

References 45 publications

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“…Beurling had previously obtained analogous results in [5] in the (easier) case of some classes of locally convex spaces of holomorphic functions. A. Atzmon later showed in [1] that the nontrivial z-invariant subspaces M c generated by the functions ϕ c are the only z-invariant subspaces M of H N. Nikol skiȋ gave in [20] a method, based on the so-called "Keldysh lemma", to construct functions f ∈ H 2 σ (D) that have no zeros in D and generate a nontrivial z-invariant subspace for decreasing weights that tend to zero arbitrarily fast. In the case of the weights σ α : n → e −n α , α ∈ (1/2, 1), Nikol skiȋ's method becomes explicit, and he constructs a function ψ α of the type…”

Section: S Dubernet and J Esterlementioning

confidence: 96%

The spectrum of some compressions of unilateral shifts

Dubernet¹,

Esterlé²

2009

St. Petersburg Math. J.

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Abstract. Let E be a star-shaped Banach space of analytic functions on the open unit disk D. It is assumed that the unilateral shift S : z → zf and the backwardare bounded on E and that their spectrum is the closed unit disk.Let M be a closed z-invariant subspace of E such that dim(M/zM ) = 1, and let g ∈ M . The main result of the paper states that if g has an analytic extension to D∪D(ζ, r) for some r > 0, with g(ζ) = 0, and if S and T satisfy the "nonquasianalytic condition" n≥0 log S n + log T n 1 + n 2 < +∞, then ζ does not belong to the spectrum of the compression S M : f + M → zf + M of the unilateral shift to the quotient space E/M . This shows in particular that Spec(S M ) = {1} for some z-invariant subspaces M of weighted Hardy spaces that were constructed by N. K. Nikol skiȋ in the 1970s by using the Keldysh method.

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Section: S Dubernet and J Esterlementioning

confidence: 96%

The spectrum of some compressions of unilateral shifts

Dubernet¹,

Esterlé²

2009

St. Petersburg Math. J.

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“…For an operator T ∈ B(H) and an invariant subspace M for T , a T -invariant subspace N is called a maximal invariant subspace of M, if N M and there is no T -invariant subspace L such that N L M. Hedenmalm [6] obtained first the result that every maximal invariant subspace of the Bergman space is of codimension 1. For further generalizions of the Bergman space, we refer the interested readers to [1,16]. Later, Guo et al [5] extended the result to a much more general situation.…”

Section: Introductionmentioning

confidence: 95%

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

2019

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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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“…For instance, we might be better able to characterize an invariant subspace if we know something about its so-called weak spectrum (see [36,63] for a definition). As an example of this, we mention that Aharon Atzmon [14,15] has obtained a complete description of invariant subspaces in L 2 a (D) with one-point spectra, also in wide classes of radially weighted Bergman spaces on D. Another question which is tractable is to ask for a description of the maximal invariant subspaces in L 2 a (D); see Section 10 for details. There has been some progress on Problem 2.…”

Section: The Basic Projectmentioning

confidence: 99%

Recent Progress and Open Problems in the Bergman Space

Aleman

Hedenmalm

Richter

Quadrature Domains and Their Applications

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An initial remarkThe following text is a modified and updated version of the problem collection [39], which was written in 1993 but became publicly available only in 1995. It was a survey of various open problems; a general survey of the field was provided in [41, 42] in 1998, written in 1995 and 1996, respectively. Since then, a number of new developments have taken place, which in their turn have led to new questions. We feel it is time to update the problem collection.2 The basic project Here, dS denotes area measure in C, normalized by a constant factor:is said to be z-invariant, or simply invariant, provided the product zf belongs to J whenever f ∈ J. Here, we use the standard notation z for the coordinate function:A sequence A = {a j } j of points in D, is said to be an L By an effective factorization we mean one that is in some sense equivalent to that of the Hardy spaces, where Blaschke products, singular inner functions, and outer functions are involved.Of the above three problems, the second and third ones are more likely to find definite answer than the first one. In fact, from one point of view, Problem 1 is as difficult as the famous invariant subspace problem in Hilbert space. Indeed, it is shown [10] how to apply the the dilation theory of Apostol, Bercovici, Foiaş, and Pearcy [12] to obtain the following: If we could show that given two z-invariant subspaces I, J in L 2 a (D), with I ⊂ J, and dim(J I) = +∞, there exists another invariant subspace K, other than 1

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Maximal, Minimal, and Primary Invariant Subspaces

Cited by 4 publications

References 45 publications

The spectrum of some compressions of unilateral shifts

The spectrum of some compressions of unilateral shifts

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

Recent Progress and Open Problems in the Bergman Space

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