Multipliers and operator space structure of weak product spaces

Abstract

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space H 1 . For complete Nevanlinna-Pick spaces H, we characterize all multipliers of the weak product space H ⊙ H. In particular, we show that if H has the so-called column-row property, then the multipliers of H and of H ⊙ H coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on H ⊙ H, which enables the use of dilations of completely bounded maps.

Every complete Pick space satisfies the column-row property

“…As observed there, in the presence of the column-row property, this leads to the equality Mult(H ⊙ H) = Mult(H). Thus, combining [11,Corollary 1.3] with Theorem 1.2, we obtain the following result. for all ϕ ∈ Mult(H).…”

“…Instead, we will make use of the following lemma, which crucially uses a result of Jury and Martin [22]; see also [11,Lemma 3.3]. Lemma 3.9.…”

“…In recent years, the column-row property has emerged as an important technical property in the theory of complete Pick spaces. It plays a role in the context of corona theorems [23,34], weak products [7,11,22], interpolating sequences [5] and de Branges-Rovnyak spaces on the ball [17,19,21]. We will review some of these connections below.…”

“…In fact, the upper bound for c d obtained in [7] grows exponentially in d. Also in the case of H 2 d for d ≥ 2, there are sequences of multipliers that give unbounded column, but bounded row operators; see [7, Subsection 4.2]. In fact, the basic √ n-bound between column norm and row norm is sharp in this case; see [11,Lemma 4.8]. The column-row property with some finite constant was extended to radially weighted Besov spaces on the ball in [6].…”

“…Richter and Wick showed that if H is a first order Besov space on B d , then Mult(H ⊙ H) = Mult(H), with equivalence of norms [30]. For normalized complete Pick spaces H, a description of Mult(H ⊙ H) in terms of column multiplication operators of H was obtained by Clouâtre and the author in [11]. As observed there, in the presence of the column-row property, this leads to the equality Mult(H ⊙ H) = Mult(H).…”