SOME FUNCTIONS PRESERVING THE ORDER OF POSITIVE OPERATORS

Josip Pečarić Faculty of Textile Technology, University of Zagreb Jadranka Mićić Technical College Zagreb, University of Zagreb Yuki Seo Senior Highschool, Osaka Kyoiku University

ABSTRACT We show some functions preserving the order of positive operators based on the Mond-Pečarić method as follows: Let A and B be positive operators on a Hilbert space H satisfying MI ≥ B ≥ mI > 0. Let f(t) be a continuous convex function on [m, M]. If g(t) is a continuous increasing convex function on [m, M] ∪ Sp(A), then for a given α > 0

A ≥ B ≥ 0 implies αg(A) + βI ≥ f(B)

where β = max_{m≤t≤M}{f(m) + [(f(M) − f(m))/(M − m)](t − m) − αg(t)}. We extend Kantorovich type operator inequalities via Ky Fan-Furuta constant as applications. Among others, we show the following inequality: If A ≥ B > 0 and MI ≥ B ≥ mI > 0, then

\frac{M^{p−1}}{m^{q−1}} A^{q} ≥ \frac{(q − 1)^{q−1}}{q^{q}} \frac{(M^{p} − m^{p})^{q}}{(M − m)(mM^{p} − Mm^{p})^{q−1}} A^{q} ≥ B^{p}

holds for all p > 1 and q > 1.