Semigroups of composition operators and integral operators on mixed norm spaces

Abstract

We characterize the semigroups of composition operators that are strongly continuous on the mixed norm spaces H(p, q, α). First, we study the separable spaces H(p, q, α) with q < ∞, that behave as the Hardy and Bergman spaces. In the second part we deal with the spaces H(p, ∞, α), where polynomials are not dense. To undertake this study, we introduce the integral operators, characterize its boundedness and compactness, and use its properties to find the maximal closed linear subspace of H(p, ∞, α) in which the semigroups are strongly continuous. In particular, we obtain that this maximal space is either H(p, 0, α) or non-separable, being this result the deepest one in the paper.

Holomorphic semigroups and Sarason's characterization of vanishing mean oscillation

“…Conversly, suppose that (d) holds, i.e. the infinitesimal generator G satisfies (LVMO), which also implies (4). Then from [9, Corollary 2] the result follows.…”

Semigroups of Composition Operators in Analytic Morrey Spaces

“…There is a ζ ∈ T such that lim r→1 − h(rζ) = w, and the argument for the case of interior Denjoy-Wolff point repeats word-for-word to obtain a positive lower bound for the norm f ζ • φ t − f ζ B 3−λ 2 as in (3.2). Again replacing f ζ by 2 1−λ f ζ gives the assertion of the claim. Having proved the claim we continue to finish the proof of the theorem.…”

“…Additional information for strong continuity of (C t ) can be found in [17] for the Dirichlet space, in [9] for weighted Dirichlet spaces, in [20] for Q p spaces, in [1] and [8] for the disc algebra, and in [2] for mixed norm spaces.…”

On generators of C0-semigroups of composition operators

“…We refer to [13,Chapter 3] for H p spaces and to [9] for VMOA. Property (⋆⋆) also holds true for the case of mixed norm spaces H(p, q, α), see [2].…”

“…Moreover, for some spaces B, there is no nontrivial C 0 semigroups of composition operators on B. See [5] for spaces between H ∞ and the Bloch space, [2] for certain mixed norm spaces; in these papers one can find references to earlier results. For non-reflexive spaces, it would be desirable to find an analogue of Theorem 2, where the words "C 0 semigroup" are substituted by a weaker property, valid for all bounded composition semigroups.…”