Gradient 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)-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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Mathematics, College of Engineering & Science, Victoria University Australia
https://orcid.org/0000-0003-2902-6805
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