On \(\mathbb{A}\)-numerical radius equalities and inequalities for certain operator matrices
Let \(H\) be a Hilbert space, \(A\) a positive bounded operator on \(H\), and let \(\mathbb A\) be the diagonal \(n{\times}n\) matrix with diagonal entries equal to \(A\). Then \(\mathbb A\) determines a semidefinite inner product on \(H^n\) by \(\langle x,y\rangle_\mathbb A=\langle \mathbb Ax,y\rangle\), \(x,y\in H^n\). Given a bounded operator \(T\) on \(H^n\), the \(\mathbb A\)-numerical radius is defined by \(w_\mathbb A(T)=\sup\{|\langle Tx,x\rangle_\mathbb A|:x\in H^n,\|x\|_\mathbb A\leq 1\}\). The authors prove some equalities and inequalities for the \(\mathbb A\)-numerical radius for circular and tri-diagonal matrices with entries being bounded operators on \(H\).
scientific article; zbMATH DE number 7376590
scientific article