Symbols of compact truncated Toeplitz operators

TL;DR

This paper characterizes the dual space of a specific model space intersected with a shifted Hardy space and establishes conditions under which bounded and compact truncated Toeplitz operators have symbols of particular forms.

Contribution

It provides a duality characterization for the space $K_I^1 igcap z H^1$ and links the existence of symbols for bounded and compact truncated Toeplitz operators.

Findings

Dual space of $K_I^1 igcap z H^1$ characterized.

Bounded truncated Toeplitz operators have symbols if and only if compact ones do.

Compact operators have symbols of the form $I$ times a continuous function.

Abstract

This paper characterises the dual of the model space $K_I^1$, where $I$ is an inner function, intersected with the shifted Hardy space, $z H^1$. With this duality result, it is then shown that every bounded truncated Toeplitz operator on the model space $K_I^2$ has a bounded symbol if and only if every compact truncated Toeplitz operator on $K_I^2$ has a symbol which is of the form $I$ multiplied by a continuous function.