Operator subadditivity of the D-logarithmic integral transform for positive operators in Hilbert spaces

Silvestru Sever Dragomir

Published: 2021-05-26

Vol. 35 No. 2 (2021)

Operator subadditivity of the D-logarithmic integral transform for positive operators in Hilbert spaces

Silvestru Sever Dragomir

Abstract

For a continuous and positive function w(λ), λ>0 and μ a positive measure on [0,∞) we consider the following D-logarithmic integral transform
DLog(w,μ)(T) :=∫₀^∞w(λ)ln(\frac{λ+T}{λ})dμ(λ),
where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.

We show among others that, if A, B>0 with BA+AB ≥ 0, then
DLog(w,μ)(A) +DLog(w,μ)(B) ≥ DLog(w,μ)(A+B).
In particular we have
\frac{1}{6}π²+dilog(A+B) ≥ dilog(A) + dilog(B),
where the dilogarithmic function dilog : [0,∞)→ℝ is defined by
dilog(t) :=∫₁^t\frac{ln s}{1-s}ds, t ≥ 0.
Some examples for integral transform DLog(·,·) related to the operator monotone functions are also provided.

Keywords:

operator monotone functions , operator inequalities , logarithmic operator inequalities , power inequalities

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Dragomir, S. S. (2021). Operator subadditivity of the D-logarithmic integral transform for positive operators in Hilbert spaces. Annales Mathematicae Silesianae, 35(2), 158–171. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13443

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Vol. 35 No. 2 (2021)
Published: 2021-09-10

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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