Description

• Title:
  Gradient inequalities for an integral transform of positive operators in Hilbert spaces
• Creator:
  Dragomir, Silvestru Sever
• Description:
  For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0,∞) we consider the following integral transformD(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. Then0 ≤ -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, then0 ≤ 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.
• Object availability:
  free access
• Rights: https://creativecommons.org/licenses/by/4.0
• Date:
  2023.07.26
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Annales Mathematicae Silesianae; Vol. 37 No. 2 (2023); 248-265;
  Annales Mathematicae Silesianae; Tom 37 Nr 2 (2023); 248-265;
  2391-4238;
  0860-2107
• Format:
  application/pdf
• Publisher:
  University of Silesia Press
• Relation: https://journals.us.edu.pl/index.php/AMSIL/article/view/15788/12187
• Subject:
  operator monotone functions;
  operator convex functions;
  operator inequalities;
  Löwner–Heinz inequality;
  logarithmic operator inequalities
• Identifier: https://journals.us.edu.pl/index.php/AMSIL/article/view/15788
• Data provider:
  Czasopisma Wydawnictwa Uniwersytetu Śląskiego
• Can I use it?:
  check license
• Type:
  other