On the Bochner theorem on orthogonal operators

Zinoviy Grinshpun

Research output: Contribution to journalArticlepeer-review


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...∂ζnΩ L(ζ, t)f(t)p(t)dt, U-1f = 1/p(ζ) ∂n/∂ζ1...∂ζnΩ 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 languageEnglish
Pages (from-to)1591-1600
Number of pages10
JournalProceedings of the American Mathematical Society
Issue number5
StatePublished - May 2003


  • Bochner Theorem
  • Hilbert space
  • Isometric operator
  • Orthogonal polynomials
  • Weight function


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