A note on the generalized maximal numerical range of operators

Abstract

This study explores the A-maximal numerical range of operators, represented as WAmax(?), where A is a positive bounded linear operator on a complex Hilbert space 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 13 (2019), 601-605]. Specifically, it is demonstrated that an A-bounded linear operator T on H is A-normaloid if and only if WA,max (T) ? ?WA(T)?0, where ?AW(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.