Composition and multiplication operators on the derivative Hardy space

Abstract

In this paper we propose a different (and equivalent) norm on S 2 (D) which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of S 2 (D) in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with 3-isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of C ϕ for a class of composition operators. We completely characterize multiplication operators which are m-isometries. As an application of the 3-isometry, we describe the reducing subspaces of M ϕ on S 2 (D) when ϕ is a finite Blaschke product of order 2.

Complex symmetry of multiplication and composition operators on A12(D)

“…Equivalent norms on the set of all holomorphic functions on D whose derivative is also in H 2 (D) have been defined by several authors [8,10]. Some examples include:…”

Weighted composition, Volterra and Integral operators on Hardy Zygmund-type spaces

“…These spaces may be called Hardy Zygmund-type spaces. In particular, S p 1 = S p , the space of analytic function with derivative in Hardy space, which is investigated along with weighted composition operators and Volterra operators in [4,8,11,12].…”

“…A characterization for boundedenss, (weak) compactness and complete continuity of weighted composition operators can be found in [4]. Composition and multiplication operators with some different norms on S 2 were studied in [8]. Also Volterra type operators on S p spaces studied by authors of [10].…”

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

“…Still, for each m ∈ N there are examples of norm expansive strict m-isometries, see e.g. [25]. For norm expansive m isometries with finite rank defect operator ∆ there are some restrictions.…”