Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

For a continuous and positive function ω (λ); λ> 0 and μ a positive measure on [0; ∞) we consider the following 𝒟-logarithmic integral transform𝒟ℒog(w,μ)(T):=∫0∞w(λ)1n(λ+Tλ)dμ(λ),\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),} 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).\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right).

In particular we have 16π2+dilog(A+B)≥dilog(A)+dilog(B),{1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right), where the dilogarithmic function dilog : [0; ∞) → ℝ is defined by dilog(t):=∫1t1ns1-sds, t≥0.{\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0.

Some examples for integral transform 𝒟𝒧og (˙;˙) related to the operator monotone functions are also provided.