Operator inequalities of Malamud and Wielandt (Q5932204)

Let \(A\) be a positive invertible matrice or operator in Hilbert space and \(m(A) =\parallel A^{-1}\parallel ^{-1}\), \(M(A)=\parallel A \parallel\). It is shown that for \(\alpha\in\mathbb{R}\) the following conditions are equivalent: NEWLINENEWLINENEWLINE\(A+(\alpha -1) PAP \geq O\) for all projections, NEWLINENEWLINENEWLINE\(|(Ay,x)|^{2}\leq\alpha(Ax,x)(Ay,y)\) for every orthogonal pair \( x\) and \( y\) and NEWLINENEWLINENEWLINE\(\alpha \geq W_{n}(A)\), where Wielandt's constant \(W_{m}(A)= (M(A)- m(A))^{2}(M(A)+m(A))^{-2}\).NEWLINENEWLINENEWLINEMatrix represantations to the previous conditions are considered and applied to the Malamud's multivariable inequality [see \textit{S. M. Malamud}, Linear Algebra Appl. 257, 239-257; (1998; Zbl 0912.47008)].