Fredholmness and compactness of truncated Toeplitz and Hankel operators

Abstract

We prove the spectral mapping theorem ${\sigma }_e(A_{\varphi })=\varphi ({\sigma }_e(A_z))$ for the Fredholm spectrum of a truncated Toeplitz operator $A_{\varphi }$ with symbol $\varphi$ 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 $S^p$.

10 pages, 1 conjecture

References in corpus (1)

• Duality theorems for coinvariant subspaces of $H^1$

Cited by in corpus (3)

• A Characterization of One-component Inner Functions
• Schatten properties of Toeplitz operators on the Paley-Wiener space
• Recent results on truncated Toeplitz operators