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- W3200149743 abstract "Let G be a locally compact abelian group with a Haar measure, and Y be a measure space. Suppose that H is a reproducing kernel Hilbert space of functions on $$Gtimes Y$$ , such that H is naturally embedded into $$L^2(Gtimes Y)$$ and is invariant under the translations associated with the elements of G. Under some additional technical assumptions, we study the W*-algebra $${mathcal {V}}$$ of translation-invariant bounded linear operators acting on H. First, we decompose $${mathcal {V}}$$ into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces $${widehat{H}}_xi $$ , $$xi in {widehat{G}}$$ , generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of $${mathcal {V}}$$ . Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to $${mathcal {V}}$$ , i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors." @default.
- W3200149743 created "2021-09-27" @default.
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- W3200149743 date "2022-08-12" @default.
- W3200149743 modified "2023-10-11" @default.
- W3200149743 title "Translation-invariant Operators in Reproducing Kernel Hilbert Spaces" @default.
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- W3200149743 doi "https://doi.org/10.1007/s00020-022-02705-4" @default.
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