On the Bochner theorem on orthogonal operators

Zinoviy Grinshpun

doi:10.1090/S0002-9939-02-06707-2

On the Bochner theorem on orthogonal operators

Zinoviy Grinshpun

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We prove the following theorem. Any isometric operator U, that acts from the Hubert space H₁(Ω) with nonnegative weight p(x) to the Hilbert space H₂(Ω) with nonnegative weight q(x), allows for the integral representation Uf = 1/q(ζ) ∂ⁿ/∂ζ₁...∂ζ_n ∫_Ω L(ζ, t)f(t)p(t)dt, U^-1f = 1/p(ζ) ∂ⁿ/∂ζ₁...∂ζ_n ∫_Ω K(ζ, t)f(t)p(t)dt, where the kernels L(ζ,t) and K(ζ,t) satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

Original language	English
Pages (from-to)	1591-1600
Number of pages	10
Journal	Proceedings of the American Mathematical Society
Volume	131
Issue number	5
DOIs	https://doi.org/10.1090/S0002-9939-02-06707-2
State	Published - May 2003

Keywords

Bochner Theorem
Hilbert space
Isometric operator
Orthogonal polynomials
Weight function

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10.1090/S0002-9939-02-06707-2

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abstract = "We prove the following theorem. Any isometric operator U, that acts from the Hubert space H1(Ω) with nonnegative weight p(x) to the Hilbert space H2(Ω) with nonnegative weight q(x), allows for the integral representation Uf = 1/q(ζ) ∂n/∂ζ1...∂ζn ∫Ω L(ζ, t)f(t)p(t)dt, U-1f = 1/p(ζ) ∂n/∂ζ1...∂ζn ∫Ω K(ζ, t)f(t)p(t)dt, where the kernels L(ζ,t) and K(ζ,t) satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.",

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N2 - We prove the following theorem. Any isometric operator U, that acts from the Hubert space H1(Ω) with nonnegative weight p(x) to the Hilbert space H2(Ω) with nonnegative weight q(x), allows for the integral representation Uf = 1/q(ζ) ∂n/∂ζ1...∂ζn ∫Ω L(ζ, t)f(t)p(t)dt, U-1f = 1/p(ζ) ∂n/∂ζ1...∂ζn ∫Ω K(ζ, t)f(t)p(t)dt, where the kernels L(ζ,t) and K(ζ,t) satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

AB - We prove the following theorem. Any isometric operator U, that acts from the Hubert space H1(Ω) with nonnegative weight p(x) to the Hilbert space H2(Ω) with nonnegative weight q(x), allows for the integral representation Uf = 1/q(ζ) ∂n/∂ζ1...∂ζn ∫Ω L(ζ, t)f(t)p(t)dt, U-1f = 1/p(ζ) ∂n/∂ζ1...∂ζn ∫Ω K(ζ, t)f(t)p(t)dt, where the kernels L(ζ,t) and K(ζ,t) satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

KW - Bochner Theorem

KW - Hilbert space

KW - Isometric operator

KW - Orthogonal polynomials

KW - Weight function

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

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On the Bochner theorem on orthogonal operators

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