Gradient inequalities for an integral transform of positive operators in Hilbert spaces

Silvestru Sever Dragomir

Published: 2023-07-26

Vol. 37 No. 2 (2023)

Gradient inequalities for an integral transform of positive operators in Hilbert spaces

Silvestru Sever Dragomir

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)^-1dμ(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 ≤ -mD'(w,μ)(δ) ≤ D(w,μ)(A)-D(w,μ)(B) ≤ -MD'(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/δ²[f(δ)-f'(δ)δ] ≤ f(A)A^-1-f(B)B^-1 ≤ M/α²[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.

Keywords:

operator monotone functions , operator convex functions , operator inequalities , Löwner–Heinz inequality , logarithmic operator inequalities

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Dragomir, S. S. (2023). Gradient inequalities for an integral transform of positive operators in Hilbert spaces. Annales Mathematicae Silesianae, 37(2), 248–265. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/15788

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Vol. 37 No. 2 (2023)
Published: 2023-09-21

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

Licence CC

This work is licensed under a Creative Commons Attribution 4.0 International License.

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