Operator norm inequalities in semi-Hilbertian spaces
Let \(A \in B(H)\) be a positive operator and define the seminorm \(\|\xi\|_A:= \langle\xi,\xi\rangle^{1/2}:=\langle A\xi,\xi\rangle\) \((\xi\in H)\) and the operator norm \(\|T\|_A:=\sup\{\|T\xi\|_A: \|\xi\|_A=1\}\). An operator \(T\in B(H)\) is called \(A\)-positive if \(AT \in B(H)\) is positive. The author extends the Cordes inequality, see \textit{H.\,O.\thinspace Cordes} [``Spectral theory of linear differential operators and comparison algebras'' (London Mathematical Society Lecture Note Series 76; Cambridge University Press) (1987; Zbl 0727.35092)] by showing that, if \(A, V,W \in B(H)\) are positive and \(V\) and \(W\) are \(A\)-positive operators, then for every \(t \in [0, 1]\) it holds that \(\|W^tV^t\|_A \leq \|WV\|_A^t\). She then generalizes an arithmetic-geometric-mean inequality due to \textit{A.\,McIntosh} [``Heinz inequalities and perturbation of spectral families'', Macquarie Math.\ Reports 79--0006, Macquarie Univ.\ (1979)] to \(A\)-seminorms. She finally proves a Corach-Porta-Recht type inequality for the \(A\)-operator seminorm [see \textit{G.\,Corach, H.\,Porta} and \textit{L.\,Recht}, Linear Algebra Appl.\ 142, 153--158 (1990; Zbl 0733.47009)].
scientific article; zbMATH DE number 5821974
scientific article