A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces

Abstract

Abstract. Let (X, B, µ) be a σ-finite measure space and let H ⊂ L 2 (X, µ) be a separable reproducing kernel Hilbert space on X. We show that the multiplier algebra of H has property (A 1 (1)).

Pick interpolation in several variables

“…Property A 1 (1) is hereditary for weak- * closed subspaces ( [5], Proposition 2.04), and so we may combine the above result with Theorem 2.8, which proves Theorem 1.1. In order to prove Theorem 1.2, we employ a versatile factorization theorem of Punraru ( [11], Theorem 4.1). This result applies to any instance where H is a reproducing kernel Hilbert space on a measure space (X, B, µ) and H is a closed subspace of L 2 (X, µ).…”

“…A natural notion of equivalence between reproducing kernel Hilbert spaces is introduced and applied to cyclic subspaces of H 2 (Ω) and L 2 a (Ω) in Theorem 2.6. In order to prove the desired results, some dual algebra results of Bercovici-Westood [6] and Prunaru [11] are invoked which, when combined with the other results in Section 2, establish the desired results.…”