Multipliers of Hilbert spaces of analytic functions on the complex half-plane

Abstract

Abstract. It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted L 2 functions defined on (0,∞) and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences.Mathematics subject classification (2010): Primary 30H50, 46J15, 47B99; Secondary 46E22, 46J20.

Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane

“…The classical Hardy and weighted Bergman spaces defined on the open unit disk of the complex plane and the open right complex half-plane may be viewed as discrete and continuous counterparts (this is discussed for example in [13] and [14]). The continuous case is sometimes more appropriate when we consider applications of these spaces (see [10], [11] and [15]).…”

Laplace–Carleson embeddings and weighted infinite-time admissibility

“…If L(L 2 w (0, ∞)) is a Banach algebra with respect to the pointwise multiplication (for example L(L 2 1+t 2 (0, ∞)), then, by Theorem 3 from [12], we know that L(L 2 w (0, ∞)) must also be a reproducing kernel Hilbert space (with kernel k z , say) and sup…”

“…More examples of L(L 2 w (0, ∞)) can be easily produced from criteria given for example in [12], [14] or [15].…”

“…The Zen spaces generalise Hardy spaces on C + (these correspond toν = 1 2π δ 0 ) and weighted Bergman spaces (corresponding to dν = r α dr, α > −1). If p = 2, then Zen spaces are Hilbert spaces (see [16]), and in fact a reproducing kernel Hilbert spaces (see [12]). In [9] it was proved that the Laplace transform defines an isometric map L :…”

“…Although in most common physical applications the norms used are usually the unweighted L 1 or L 2 norms, it may sometimes be useful to consider weighted L p norms. The main purpose of this paper is to generalise admissibility criteria, obtained in [10] for weighted L 2 -and unweighted L p -admissibility (given in terms of Carleson measures and Carleson-Laplace embeddings described in [9]) to weighted L p (0, ∞) case, applying and generalising recent results from [13], concerning the spaces defined and studied in [12]. A powerful boundedness criterion for the Laplace-Carleson embeddings for weighted L p spaces, containing the earlier version from [9] as a special case, is also proved here.…”