Numerical radius and distance from unitary operators

TL;DR

This paper establishes bounds on how close an invertible operator with bounded numerical radius is to a unitary operator, generalizing previous results and discussing the operator -radius.

Contribution

It extends Stampfli's result by providing a bound involving the numerical radius and inverse radius, including the case of the -radius.

Findings

Distance from unitary operators is bounded by a constant times e^{1/4}.

The exponent 1/4 in the bound is proven to be optimal.

The results are generalized to the operator -radius for  between 1 and 2.

Abstract

Denote by w(A) the numerical radius of a bounded linear operator A acting on Hilbert space. Suppose that A is invertible and that the numerical radius of A and of its inverse are no greater than 1+e for some non-negative e. It is shown that the distance of A from unitary operators is less or equal than a constant times $e^{1/4}$. This generalizes a result due to J.G. Stampfli, which is obtained for e = 0. An example is given showing that the exponent 1/4 is optimal. The more general case of the operator $\rho$-radius is discussed for $\rho$ between 1 and 2.