Lipschitz Type Inequalities for an Integral Transform of Positive Operators With Applications

We introduce the following integral transform: \[ D^{(\mu)}(T) := -\int_{0}^{\infty} (\lambda+T)^{-1} d\mu(\lambda), \quad t > 0, \] where \(\mu\) is a positive measure on \((0,\infty)\) 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 \( A \geq m_1 > 0 \) and \( B \geq m_2 > 0 \), then: \[ \| D^{(\mu)}(B) - D^{(\mu)}(A) \| \leq \| B - A \|_{[m_1,m_2]} D^{(\mu)}(\cdot), \] where \( D^{(\mu)}(\cdot) \) is a function of \( t \), and \( [m_1,m_2]D^{(\mu)}(\cdot) \) is its divided difference. If \( f: [0,\infty) \to \mathbb{R} \) is an operator monotone function with \( f(0) = 0 \), then: \[ \| f(A)A^{-1} - f(B)B^{-1} \| \leq \| B - A \|_{[m_1,m_2]} f(\cdot)(\cdot)^{-1}. \] Similar inequalities for operator convex functions and some particular examples of interest are also given.