Operator subadditivity of the D-logarithmic integral transform for positive operators in Hilbert spaces
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) :=∫1t\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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Mathematics, College of Engineering & Science, Victoria University, Australia Australia
https://orcid.org/0000-0003-2902-6805
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