Wielandt type extensions of the Heinz—Kato—Furuta inequality

Abstract

The Wielandt inequality asserts that if a positive operator A on a Hilbert space H satisfies 0 < m ≤ A ≤ M for some 0 < m < M, then $$|(Ax,y){|^{2}} \leqslant {\left( {\frac{{M - m}}{{M + m}}} \right)^{2}}(Ax,x)(Ay,y)$$ for every orthogonal pair x and y. In this paper, we show Wielandt type extensions of the Heinz-Kato-Furuta 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.