Decay estimates connected with Toeplitzness of composition operators

TL;DR

This paper proves that the arithmetic means of entries along each diagonal of certain composition operators decay at a rate of 1/(log N loglog N), extending previous results on their Toeplitzness properties.

Contribution

It establishes a uniform decay rate for all diagonal entries of the matrix associated with composition operators, generalizing earlier partial results.

Findings

Diagonal entries decay like 1/(log N loglog N)

Decay rate is uniform across all diagonals

Supports conjectures relating to Toeplitzness of composition operators

Abstract

In a higher dimensional version of an earlier conjecture of Nazarov and Shapiro, the truth of which would imply that any composition operator on the second Hardy space is weakly asymptotically Toeplitz, Shayya proved that the arithmetic means of the main diagonal entries of the matrix of coefficients connected with the conjecture, decay like 1/log N. But the proof does not extend to other diagonal entries. We prove that the arithmetic means of the entries along each diagonal decay like 1/(log N loglog N) uniformly.