Abstract:

A Hardy kernel is an integral kernel k(x,y) in two variables x>0 and y>0 which is homogeneous in (x,y) of degree -1. Integral operators on the positive semi-axis with Hardy kernels are explicitly diagonalisable by the Mellin transform. It is however by no means clear how to diagonalise the infinite matrix {k(n,m)} which is obtained by restricting a Hardy kernel onto natural numbers n,m. In the talk, I will describe one specific explicit one-parametric family of Hardy kernels k when the spectral analysis of {k(n,m)} can be carried out. This matrix appears in the analysis of composition operators on the Hardy space of Dirichlet series; I will explain this connection at the end of the talk. This is joint work with Ole Brevig (Trondheim) and Kalle Perfekt (Reading).