Let $(X,\mathcal B,\mu)$ be a $\sigma$-finite measure space and let $H\subset L^2(X,\mu)$ be a separable reproducing kernel Hilbert space on $X$. We show that the multiplier algebra of $H$ has property $(A_1(1))$.

Keywords:

reproducing kernel Hilbert space, Berezin transform, dual algebra reproducing kernel Hilbert space, Berezin transform, dual algebra

MSC Classifications:

46E22, 47B32, 47L45 show english descriptions Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22]
Dual algebras; weakly closed singly generated operator algebras 46E22 - Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
47B32 - Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22]
47L45 - Dual algebras; weakly closed singly generated operator algebras

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