Summary: The Wielandt inequality asserts that if a positive operator \(A\) on a Hilbert space \(H\) satisfies \(0< m\leq A\leq M\) for some \(0< m< M\), then \[ |(Ax, y)|^2\leq \Biggl({M-m\over M+m}\Biggr)^2\;(Ax,x)(Ay,y) \] for every orthogonal pair \(x\) and \(y\). In this paper, we show Wielandt type extensions of the Heinz-Kato-Kuruta inequality, which is based on some generalizations of the Wielandt inequality by Fujii-Katayama-Nakamoto and Bauer-Householder. The obtained inequalities are simultaneous extensions of the Heinz-Kato-Furuta and the Wielandt inequalities. Related to our extensions, we discuss some applications of the Furuta inequality and the grand Furuta inequality.
For the entire collection see [Zbl 0971.00017].