Norm attainment of a class of block operator matrices

Abstract

Given complex numbers $a,b,c$ and a non-negative continuous function $\phi$ defined on $[0,+\infty )$, consider the $2\times 2$ matrix $$M_t=\left(\begin{array}{cc}a& t\\ ct& b\phi (t)\end{array}\right),t\in [0,+\infty ).$$ We establish conditions for the strict monotonicity of the norm function $t\mapsto \Vert M_t\Vert$. As an application, we characterize the norm attainment of the corresponding block operator matrix $$T=\left(\begin{array}{cc}a{I}_H& A\\ c{A}^{\ast }& b\phi (|A|)\end{array}\right),$$ where $I_H$ is the identity operator on a Hilbert space $H$ and $A$ is a bounded linear operator from another Hilbert space to $H$.

arXiv commentIn the near future, a new section devoted to the numerical range of operators will be added; accordingly, this paper will be expanded