Lipschitz Type Inequalities for an Integral Transform of Positive Operators With Applications - Sever Dragomir, Silvestru (Sarajevo Journal of Mathematics, 2025)
We introduce the following integral transform: \[ D^(μ)(T) := -_0^∞ (λ+T)^-1 dμ(λ), t > 0, \] where \(μ\) is a positive measure on \((0,∞)\) and the integral is assumed to exist for \(T\) as a positive operator on a complex Hilbert space \(H\). In this paper, we show, among other results, that if \( A ≥ m_1 > 0 \) and \( B ≥ m_2 > 0 \), then: \[ \| D^(μ)(B) - D^(μ)(A) \| ≤ \| B - A \|_[m_1,m_2] D^(μ)(·), \] where \( D^(μ)(·) \) is a function of \( t \), and \( [m_1,m_2]D^(μ)(·) \) is its divided difference. If \( f: [0,∞) → R \) is an operator monotone function with \( f(0) = 0 \), then: \[ \| f(A)A^-1 - f(B)B^-1 \| ≤ \| B - A \|_[m_1,m_2] f(·)(·)^-1. \] Similar inequalities for operator convex functions and some particular examples of interest are also given.