瀬尾 祐貴

Let φ and ψ be unital positive linear maps satisfying some conditions with respect to positive scalars α and β. It is shown that if a real valued function f is operator concave on an interval J, then β (f (ψ(A)) - ψ(f (A))) ≤ f (φ(A)) - φ(f (A)) ≤ α (f (ψ(A)) - ψ(f (A))) for every selfadjoint operator A with spectrum σ(A) ⊂ J. Moreover, an external version of estimates above is presented.

We present a Diaz-Metcalf type operator inequality as a reverse Cauchy-Schwarz inequality and then apply it to get some operator versions of Pólya-Szegö's, Greub-Rheinboldt's, antorovich's, Shisha-Mond's, Schweitzer's, Cassels' and Klamkin-McLenaghan's inequalities via a unified approach. We also give some operator Grüss type inequalities and an operator Ozeki- Izumino-Mori-Seo type inequality. Several applications are included as well.