Reverse Carleson embeddings for model spaces

Blandignères, Alain; Fricain, Emmanuel; Gaunard, Frédéric; Hartmann, Andreas; Ross, William T.

doi:10.1112/jlms/jdt018

Cited by 14 publications

(16 citation statements)

References 32 publications

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“…However, the space GK 2 I is in general neither contained in H 2 , nor closed even it it were contained in H 2 . For this we would need that |G| 2 dm is a Carleson and a reverse Carleson measure for K 2 I (the first condition guarantees that GK 2 I is a subspace of H 2 and the second one that it is closed; we refer to [13], [49], [6]).…”

Section: Kernels Of Toeplitz Operators and Extremal Functionsmentioning

confidence: 99%

Kernels of Toeplitz operators

Hartmann¹,

Mitkovski²

2016

Recent Progress on Operator Theory And Approximation in Spaces of Analytic Functions

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Toeplitz operators are met in different fields of mathematics such as stochastic processes, signal theory, completeness problems, operator theory, etc. In applications, spectral and mapping properties are of particular interest. In this survey we will focus on kernels of Toeplitz operators. This raises two questions. First, how can one decide whether such a kernel is non trivial? We will discuss in some details the results starting with Makarov and Poltoratski in 2005 and their succeeding authors concerning this topic. In connection with these results we will also mention some intimately related applications to completeness problems, spectral gap problems and Pólya sequences. Second, if the kernel is nontrivial, what can be said about the structure of the kernel, and what kind of information on the Toeplitz operator can be deduced from its kernel? In this connection we will review a certain number of results starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.2010 Mathematics Subject Classification. 30J05, 30H10, 46E22.

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Section: Kernels Of Toeplitz Operators and Extremal Functionsmentioning

confidence: 99%

Kernels of Toeplitz operators

Hartmann¹,

Mitkovski²

2016

Recent Progress on Operator Theory And Approximation in Spaces of Analytic Functions

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“…Then H (b) = (bH 2 ) ⊥ is the classical model space and is certainly a closed subspace of H 2 . There is a concept developed in [5] of a dominating set. Here a Borel set E ⊂ T with m(E) < 1 is called a dominating set for (bH 2 ) ⊥ if T |f | 2 dm E |f | 2 dm, f ∈ (bH 2 ) ⊥ .…”

Section: Final Remarkmentioning

confidence: 99%

“…In [5] it is shown that dominating sets exist for every model space (bH 2 ) ⊥ and can be used to give sufficient conditions for reverse Carleson embeddings for these spaces.…”

Section: Final Remarkmentioning

confidence: 99%

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Direct and reverse Carleson measures for $\mathscr{H}(b)$ spaces

Blandignères¹,

Fricain²,

Gaunard³

et al. 2015

Indiana Univ. Math. J.

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In this paper we discuss direct and reverse Carleson measures for the de Branges-Rovnyak spaces H (b), mainly when b is a non-extreme point of the unit ball of H ∞ .Here b belongs to H ∞ 1 , the unit ball in H ∞ , and H ∞ is the Banach algebra of bounded analytic functions on D normed with the supremum norm · ∞ . The space H (b) is often known as the de Branges-Rovnyak space and we will review the basics of this space in a moment. For now, note that when b ∞ < 1, then H (b) is just the classical Hardy space H 2 [16,18] with an equivalent norm while if b is an inner function, meaning |b| = 1 almost everywhere on T = ∂D, then H (b) is the classical model space (bH 2 ) ⊥ = H 2 ⊖ bH 2 . For any b, the space H (b) is contractively contained in H 2 . As is often the case, the properties of H (b) spaces, including direct and reverse Carleson measures, depend on whether or not b is an extreme point ofwhere m is normalized Lebesgue measure on the unit circle T.Let M + (D − ) denote the positive finite Borel measures on the closed unit disk D − . By a reverse Carleson measure for H (b) we mean a 2010 Mathematics Subject Classification. 30J05, 30H10, 46E22.

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“…In fact, the problem of rank one perturbations has connections to many interesting topics in analysis, such as model theory including deBranges-Rovnyak and Sz.-Nagy-Foiaş model spaces [7,17,19], Nehari interpolation [24], Carleson embeddings [5], singular integral operators [18], and truncated Toeplitz operators [4].…”

Section: Introductionmentioning

confidence: 99%

Rank-one perturbations and Anderson-type Hamiltonians

Liaw¹

2019

Banach J. Math. Anal.

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To the memory of R.G. Douglas. You were not only a vast source of knowledge. Words cannot fully express my appreciation for your steady advice, your unfaltering support and the many hours of mathematical discussions we shared.Abstract. Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts, began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators Hω = H + Vω on a separable Hilbert space H, where the perturbation is given by Vω = n ωn( · , ϕn)ϕn with a sequence {ϕn} ⊂ H and independent identically distributed random variables ωn.We show that the the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank one perturbation. This result connects one of the least trackable perturbation problem (with almost surely non-compact perturbations) with one where the perturbation is 'only' of rank one perturbations. The latter presents a basic application of model theory.We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has non-zero Lebesgue measure.2010 Mathematics Subject Classification. Primary 47A55; Secondary 82B44, 81Q10.

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Reverse Carleson embeddings for model spaces

Cited by 14 publications

References 32 publications

Kernels of Toeplitz operators

Kernels of Toeplitz operators

Direct and reverse Carleson measures for $\mathscr{H}(b)$ spaces

Rank-one perturbations and Anderson-type Hamiltonians

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