Two Mixed Hadamard Type Generalizations of Heinz Inequality

Abstract

We give two types of mixed Hadamard inequalities containing the terms T, \T\, and |T*|, where T is a bounded linear operator on a complex Hubert space.As an immediate consequence of these results, we can easily show some extensions of the Hadamard inequality and also the Heinze inequality: M \(Tx,y)\2 < (\T\2ax,x)(.\T*\2^-"îy,y)for any T, any x,y in H, and any real number a with 0 < a < 1.And the following conditions are equivalent in case 0 < a < 1:(1) the equality in (*) holds;(2) |T|2ai and T'y are linearly dependent;(3) Tx and \T*\2(l~a}y are linearly dependent.Results in this paper would remain valid for unbounded operators under slight modifications.

Locations

• Proceedings of the American Mathematical Society - View - PDF