A note on the generalized maximal numerical range of operators

Abstract

This study explores the $A$-maximal numerical range of operators, represented as $W_{\max}^A(\cdot)$, where $A$ is a positive bounded linear operator on a complex Hilbert space $\mathcal{H}$. The research provides new insights into the properties and characterizations of $A$-normaloid operators, including an extension of a recent result by Spitkovsky in [A note on the maximal numerical range, Oper. Matrices {\bf13} (2019), 601--605]. Specifically, it is demonstrated that an $A$-bounded linear operator $T$ on $\mathcal{H}$ is $A$-normaloid if and only if $W_{\max}^A(T)\cap \partial W_A(T)\neq\emptyset$, where $\partial W_A(T)$ denotes the boundary of the $A$-numerical range of $T$. Furthermore, novel $A$-numerical radius inequalities are introduced that generalize and enhance prior well-known results.