Simple Operator Asynchronicity of an Integral Transform with Applications

Abstract

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 ) − 1 d μ ( 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 ) ( ∫ 0 1 [ λ + ( 1 − t ) B + t A ) − 1 ( B − A ) ] 2 d t ) d μ ( 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 ≥ A 𝒟 ( w , μ ) ( A ) + B 𝒟 ( 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.