Second derivative Lipschitz type 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 ) ⁻¹ dµ (λ) ,

where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. We show among others that, if A ≥ m 1 > 0, B ≥ m 2 > 0, then

||D (w, µ) (B) − D (w, µ) (A) − D (D (w, µ)) (A) (B − A)||

≤|B − A|²×[D(w,µ)(m₂)−D(w,µ)(m₁)−(m₂- m₁)D’(w,µ)(m₁)]/(m₂−m₁)²    if m₁≠m₂,

≤ D’’(w, µ)(m)/2   if m₁=m₂=m,

where D (D (w, µ)) is the Fréchet derivative of D (w, µ) as a function of operator and D’’(w, µ) is the second derivative of D (w, µ) as a real function.

We also prove the norm integral inequalities for power r ∈ (0, 1] and A, B ≥ m > 0,

||∫₀¹((1−t)A+tB)^r−1dt−((A+B)/2)^r−1|| ≤ (1−r) (2−r) m^r−3||B−A||²/24

and

||((A^r−1+B^r−1 )/2) − ∫₀¹((1−t) A+tB)^r−1dt|| ≤ (1−r) (2−r) m^r−3||B − A||²/12.