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 풟 (w , μ) (T) : = ∫ 0 ∞ w (λ) (λ + T) - 1 d μ (λ) , 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 ≤ - m 풟 ′ (w , μ) (δ) ≤ 풟 (w , μ) (A) - 풟 (w , μ) (B) ≤ - M 풟 ′ (w , μ) (α) , 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 ≤ m δ 2 [ f (δ) - f ′ (δ) δ ≤ f (A) A - 1 - f (B) B - 1 ] ≤ M α 2 [ f (α) - f ′ (α) α ]. Some examples for operator convex functions as well as for integral transforms D (·, ·) related to the exponential and logarithmic functions are also provided. [ABSTRACT FROM AUTHOR]