Norms of Truncated Toeplitz Operators and Numerical Radii of Restricted Shifts

Abstract

This paper gives a new approach to the calculation of the numerical radius of a restricted shift operator by linking it to the norm of a truncated Toeplitz operator (TTO), which can be calculated by various methods. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved.

Symbols of compact truncated Toeplitz operators

“…One question posed in [6] was whether there was a compact truncated Toeplitz operator with a symbol in IC(T) + IH ∞ that has no continuous symbol. This question was then answered affirmatively in [12] when a compact truncated Toeplitz operator with this property was then constructed.…”

Idempotent, model, and Toeplitz operators attaining their norms

“…The latter conclusion of Theorem 3.3 is not new, and it essentially follows from [8, Section 7, Corollary 3]. In fact, [8, Section 7] (also see [10]) deals with the problem of norm attaining symbols of truncated Toeplitz operators: given…”

Crouzeix's Conjecture and related problems

“…Note that extremal vectors can be used to build new bases of K Θ . Recent work by Gaaya and Gorkin-Partington has revealed precise formulas for some w(S Θ ), see [16,24]. This suggests that it might be possible to answer parts of this question for very specialized Θ.…”