Gradient Inequalities for an Integral Transform of Positive Operators in Hilbert Spaces

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.

Assume that A ≥ α > 0, δ ≥ B > 0 and 0 < m ≤ B − A ≤ M for some constants α, δ, m, M. Then 0≤-m𝒟′(w,μ)(δ)≤𝒟(w,μ)(A)-𝒟(w,μ)(B)≤-M𝒟′(w,μ)(α), 0 \le - m\mathcal{D}'\left( {w,\mu } \right)\left( \delta \right) \le \mathcal{D}\left( {w,\mu } \right)\left( A \right) - \mathcal{D}\left( {w,\mu } \right)\left( B \right) \le - M\mathcal{D}'\left( {w,\mu } \right)\left( \alpha \right), where D^′(w, µ) (t) is the derivative of D(w, µ) (t) as a function of t > 0.

If f : [0, ∞) → ℝ is operator monotone on [0, ∞) with f (0) = 0, then 0≤mδ2[ f(δ)-f′(δ)δ≤f(A)A-1-f(B)B-1 ]≤Mα2[ f(α)-f′(α)α ]. \matrix{ {0 \le {m \over {{\delta ^2}}}\left[ {f\left( \delta \right) - f'\left( \delta \right)\delta \le f\left( A \right){A^{ - 1}} - f{{\left( B \right)}^{B - 1}}} \right]} \cr { \le {M \over {{\alpha ^2}}}\left[ {f\left( \alpha \right) - f'\left( \alpha \right)\alpha } \right].} \cr }

Some examples for operator convex functions as well as for integral transforms D (·, ·) related to the exponential and logarithmic functions are also provided.