Commutators of the Unilateral Shift and Adjoint for Reproducing Kernel Hilbert Spaces on the Disk

TL;DR

This paper extends the trace formula for commutators of the unilateral shift and its adjoint from Hardy spaces to a broader class of rotation invariant Hilbert spaces on the disk, including weighted shifts.

Contribution

It generalizes the Berger-Shaw trace formula to a larger class of Hilbert spaces and weighted shifts, with explicit examples and computations.

Findings

Trace formula extended to new Hilbert spaces

Computed inner products for various examples

Broadened the class of weighted shifts analyzed

Abstract

We generalise the result of Berger and Shaw the trace formula for Hardy Hilbert space to a larger class of rotation invariant Hilbert function spaces on the unit disk. We also demonstrate many meaningful examples of these Hilbert spaces by computing the inner products. We also extend to a wider class than the unilateral shift, that is, weighted shifts under certain restrictions.