The numerical ranges of the generalized quadratic operators
Kangjian Wu, Qingxiang Xu
TL;DR
The paper analyzes generalized quadratic operators $T=egin{pmatrix} a I_H & A \ c A^* & b I_K \\end{pmatrix}$ on $H\oplus K$, proving that $T$ attains its norm if and only if the off-diagonal entry $A$ attains its norm, thereby extending known results for quadratic operators. It then provides a complete description of the numerical range $W(T)$ by relating it to a family of $2\times2$ matrices $S_d$ and their unions $E_d$, showing $W(T)$ is either $W(S_d)$ or $E_d$ with $d=\|A\|$, and that $\overline{E_d}=W(S_d)$. The work gives detailed conditions and equivalent formulations for norm attainment, analyzes the structure of $W(T)$ via a two-point parametrization, and discusses applications including cases where $W(T)$ is a non-degenerate elliptical disk that is neither open nor closed, as well as representations of $T$ in the form $Q+cQ^*+kI$. These results deepen the understanding of norm attainment and numerical ranges for generalized quadratic operators and reveal fundamental differences from classical quadratic operator theory.
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.