Gradient Inequalities for an Integral Transform of 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 integral transform 𝒟(w,μ)(T):=∫0∞w(λ)(λ+T)-1dμ(λ),\mathcal{D}\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right){{\left( {\lambda + T} \right)}^{ - 1}}d\mu \left( \lambda \right),} where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.

Keywords

• operator monotone functions
• operator convex functions
• operator inequalities
• löwner–heinz inequality
• logarithmic operator inequalities
• 47a63
• 47a60