An approximate approach to the structured distance to normality of Toeplitz operators

Abstract

A classical theorem from Brown and Halmos asserts that a Toeplitz operator T ( f ) T(f) is normal if and only if the range of its generator f : T → C f:\mathbb {T}\rightarrow \mathbb {C} is included in a straight line. In this paper, discretizing f ( T ) f(\mathbb {T}) and using the Principal Component Analysis method to project it onto a ‘best’ line segment in L 2 L^2 -norm, we propose a numerical method to find the nearest normal Toeplitz operator from T ( f ) T(f) in the norm | | | T ( f ) | | | ≔ ‖ f ‖ L 2 ( T ) {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert T(f)\right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }\coloneq \Vert f \Vert _{L^2(\mathbb {T})} which is weaker than the operator norm. Besides, we introduce an index for the distance from normality of Toeplitz operators which is invariant under the transformations f ↦ a f + b f\mapsto a f+b for all a ∈ R a\in \mathbb {R} and b ∈ C b \in \mathbb {C} with a ≠ 0 a\neq 0 .