Simple Operator Asynchronicity of an Integral Transform with Applications

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.

We show among others that, if B, A > 0, then [𝒟(w,μ)(A)−𝒟(w,μ)(B)](B−A)=∫0∞w(λ)(∫01[λ+(1−t)B+tA)−1(B−A)]2dt)dμ(λ). \matrix{{\left[{\mathcal{D}\left({w,\mu}\right)\left(A\right)-\mathcal{D}\left({w,\mu}\right)\left(B\right)}\right]\left({B-A}\right)}\cr{=\int_0^\infty{w\left(\lambda\right)\left({\int_0^1{{{\left[{\lambda+\left({1-t}\right)B+tA{)^{-1}}\left({B-A}\right)}\right]}^2}dt}}\right)d\mu\left(\lambda\right).}}\cr} We also provide some sufficient conditions for the operators A, B > 0 such that the inequality 𝒟(w,μ)(A)B+𝒟(w,μ)(B)A≥A𝒟(w,μ)(A)+B𝒟(w,μ)(B) \mathcal{D}\left({w,\mu}\right)\left(A\right)B+\mathcal{D}\left({w,\mu}\right)\left(B\right)A{\ge}A\mathcal{D}\left({w,\mu}\right)\left(A\right)+B\mathcal{D}\left({w,\mu}\right)\left(B\right) holds. Some examples for power and logarithmic functions are also provided.