Abstract
In this paper, we introduce two variables norm functionals of τ-measurable operators and establish their joint log-convexity. Applications of this log-convexity will include interpolated Young, Heinz and Trace inequalities related to τ-measurable operators. Additionally, interpolated versions and their monotonicity will be presented as well.
Highlights
Let Mn be the space of n × n complex matrices and Mn+ be the class of Mn consisting of positive semi-definite matrices
By an application of this log-convexity theorem, Bourin and Lee improved many remarkable matrix inequalities, most of which were related to Araki type inequalities
In the following theorem we prove the joint log-convexity of the function f (t, s) = xtzys E(M)(r), which is one of our main results
Summary
Let Mn be the space of n × n complex matrices and Mn+ be the class of Mn consisting of positive semi-definite matrices. The following monotonicity results that the interpolated inequalities obey were proved: Let A, B ∈ Mn+, X ∈ Mn and p ≥ q > 0. The main results in this work are the log-convexity theorems of a two variables functional of measurable operators (see Lemma 3.10 and Theorem 3.11), which are generalizations of the results of Bourin and Lee [1].
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