Spectral asymptotics for Toeplitz operators and an application to banded matrices

Pushnitski, Alexander

doi:10.1007/978-3-319-75996-8_21

Cited by 5 publications

(7 citation statements)

References 18 publications

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“…The above result was obtained in [10] in slightly more general form. Defining the counting function n(., T ϕ ) for the sequence of singular numbers of T ϕ as n(s, T ϕ ) := #{n : s n (T ϕ ) > s}, s > 0, we can rewrite (0.5) in an equivalent manner lim…”

Section: Introductionmentioning

confidence: 54%

“…The corollaries presented in this Section mainly follow Pushnitski [10]. Once again, remind the notation for functions ϕ 0 , ϕ 1 , ϕ, (0.6), (0.7).…”

Section: Applications and Concluding Remarksmentioning

confidence: 99%

“…Its singular values {s n (T ϕ )} n form a decreasing sequence and converge to zero. An important result to come was obtained in Pushnitski [10].…”

Section: Introductionmentioning

confidence: 98%

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On spectral properties of compact Toeplitz operators on Bergman space with logarithmically decaying symbol and applications to banded matrices

Koïta¹,

Kupin²,

Naboko³

et al. 2020

Preprint

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Let L 2 (D) be the space of measurable square-summable functions on the unit disk. Let L 2 a (D) be the Bergman space, i.e., the (closed) subspace of analytic functions in L 2 (D). P + stays for the orthogonal projection going from L 2 (D) to L 2 a (D). For a function ϕ ∈ L ∞ (D), the Toeplitz operatorThe main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is ϕ(z) = ϕ 1 (e iθ ) (1 + log(1/(1 − r))) −γ , γ > 0, where z = re iθ and ϕ 1 is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

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Section: Introductionmentioning

confidence: 54%

“…The corollaries presented in this Section mainly follow Pushnitski [10]. Once again, remind the notation for functions ϕ 0 , ϕ 1 , ϕ, (0.6), (0.7).…”

Section: Applications and Concluding Remarksmentioning

confidence: 99%

See 1 more Smart Citation

On spectral properties of compact Toeplitz operators on Bergman space with logarithmically decaying symbol and applications to banded matrices

Koïta¹,

Kupin²,

Naboko³

et al. 2020

Preprint

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“…To prove this result, we will introduce the following functionals (see [5,25]). Let T be a compact operator between two Hilbert spaces.…”

Section: Asymptoticsmentioning

confidence: 99%

“…Let T be a compact operator between two complex Hilbert spaces. The decreasing sequence of singular values of T will be denoted by (s n (T )) The goal of this Annex is to extend the results obtained for ψ(t) = t p , [5,25], to the class C p . We give here the proof for completeness.…”

Section: Appendix: Asymptotic Orthogonalitymentioning

confidence: 99%

On singular values of Hankel operators on Bergman spaces

Bourass¹,

El-Fallah²,

Marrhich³

et al. 2021

Preprint

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In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces A 2 ωϕ , where ω ϕ = e −ϕ and ϕ is a subharmonic function. We consider compact Hankel operators H φ , with anti-analytic symbols φ, and give estimates of the trace of h(|H φ |) for any convex function h. This allows us to give asymptotic estimates of the singular values (s n (H φ )) n in terms of decreasing rearrangement of |φ ′ |/ √ ∆ϕ. For the radial weights, we first prove that the critical decay of (s n (H φ )) n is achieved by (s n (H z )) n . Namely, we establish that ifand only if φ ′ belongs to the Hardy space H p , where p = 2 (1+β) 2+β . Finally, we compute the asymptotics of s n (H φ ) whenever φ ′ ∈ H p .

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