The numerical ranges of the generalized quadratic operators

TL;DR

This paper characterizes the numerical range of a generalized quadratic operator on Hilbert spaces, establishing when it attains its norm and providing a new approach for the complete description of its numerical range.

Contribution

It introduces a new method to fully characterize the numerical range of generalized quadratic operators and links norm attainment of the operator to that of the underlying operator A.

Findings

T attains its norm iff A attains its norm.

A complete characterization of the numerical range of T is provided.

A new approach simplifies the analysis of the numerical range.

Abstract

We investigate the generalized quadratic operator defined by $$T =\left( \begin{array}{cc} a I_H & A \\ c A^* & bI_K \end{array} \right) ,$$ where $H$ and $K$ are Hilbert spaces, $A:K\to H$ is a bounded linear operator, $I_H$ and $I_K$ denote the identity operators on $H$ and $K$, respectively, and $a,b,c$ are complex numbers. It is shown that $T$ attains its norm if and only if $A$ attains its norm. Furthermore, a complete characterization of the numerical range of $T$ is provided by a new approach.