Multiplication and composition operators on the derivative Hardy space S²(\mathbbD)

2018/08/29 by Gu, Caixing, Luo, Shuaibing
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1808.10041

Abstract

In this paper we propose a different (and equivalent) norm on S² (\mathbbD) which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of S²(\mathbbD) 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 S²(\mathbbD) when φ is a finite Blaschke product of order 2.

Related

• Composition-differentiation operators on S²(\mathbbD)
• Complex symmetric weighted composition operators on the space H²₂(\mathbbD)
• Spectra of Composition Operators with Symbols in S(2)
• Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators
• Some examples of composition operators and their approximation numbers on the Hardy space of the bi-disk