Constructive approximation of convergent sequences by eigenvalue sequences of radial Toeplitz--Fock operators
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| Format: | Preprint |
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2025
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| _version_ | 1866916666231750656 |
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| author | García, Kevin Esmeral Maximenko, Egor A. |
| author_facet | García, Kevin Esmeral Maximenko, Egor A. |
| contents | It is well known that for every measurable function $a$, essentially bounded on the positive halfline, the corresponding radial Toeplitz operator $T_a$, acting in the Segal--Bargmann--Fock space, is diagonal with respect to the canonical orthonormal basis consisting of normalized monomials. We denote by $γ_a$ the corresponding eigenvalues sequence. Given an arbitrary convergent sequence, we uniformly approximate it by sequences of the form $γ_a$ with any desired precision. We give a simple recipe for constructing $a$ in terms of Laguerre polynomials. Previously, we proved this approximation result with nonconstructive tools (Esmeral and Maximenko, ``Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences'', Complex Anal. Oper. Theory 10, 2016). In the present paper, we also include some properties of the sequences $γ_a$ and some properties of bounded sequences, uniformly continuous with respect to the sqrt-distance on natural numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_23276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructive approximation of convergent sequences by eigenvalue sequences of radial Toeplitz--Fock operators García, Kevin Esmeral Maximenko, Egor A. Functional Analysis Operator Algebras 47B35, 30H20, 41A10, 30E05 It is well known that for every measurable function $a$, essentially bounded on the positive halfline, the corresponding radial Toeplitz operator $T_a$, acting in the Segal--Bargmann--Fock space, is diagonal with respect to the canonical orthonormal basis consisting of normalized monomials. We denote by $γ_a$ the corresponding eigenvalues sequence. Given an arbitrary convergent sequence, we uniformly approximate it by sequences of the form $γ_a$ with any desired precision. We give a simple recipe for constructing $a$ in terms of Laguerre polynomials. Previously, we proved this approximation result with nonconstructive tools (Esmeral and Maximenko, ``Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences'', Complex Anal. Oper. Theory 10, 2016). In the present paper, we also include some properties of the sequences $γ_a$ and some properties of bounded sequences, uniformly continuous with respect to the sqrt-distance on natural numbers. |
| title | Constructive approximation of convergent sequences by eigenvalue sequences of radial Toeplitz--Fock operators |
| topic | Functional Analysis Operator Algebras 47B35, 30H20, 41A10, 30E05 |
| url | https://arxiv.org/abs/2503.23276 |