Fredholmness and compactness of truncated Toeplitz and Hankel operators

TL;DR

This paper establishes spectral mapping for Fredholm spectra of truncated Toeplitz operators, characterizes compactness of truncated Hankel operators via symbols, and explores Schatten class properties.

Contribution

It proves the spectral mapping theorem for Fredholm spectra of truncated Toeplitz operators and characterizes compact truncated Hankel operators with continuous symbols.

Findings

Spectral mapping theorem for $\sigma_e(A_)$ and $(\sigma_e(A_z))$

Characterization of compact truncated Hankel operators via continuous symbols

Description of truncated operators in Schatten classes $S^p$

Abstract

We prove the spectral mapping theorem $\sigma_e(A_\phi) = \phi(\sigma_e(A_z))$ for the Fredholm spectrum of a truncated Toeplitz operator $A_\phi$ with symbol $\phi$ in the Sarason algebra $C+H^\infty$ acting on a coinvariant subspace $K_\theta$ of the Hardy space $H^2$. Our second result says that a truncated Hankel operator on the subspace $K_\theta$ generated by a one-component inner function $\theta$ is compact if and only if it has a continuous symbol. We also suppose a description of truncated Toeplitz and Hankel operators in Schatten classes $S^p$.