QUADRATIC LIPSCHITZIAN INEQUALITIES FOR THE \(\mathcal{AT}\)-INTEGRAL TRANSFORM OF SELFADJOINT OPERATORS IN HILBERT SPACES

Abstract

For \(\mu\) a positive measure on \(\left[ 0,1\right]\) and \(p:\left[ 0,1\right] \rightarrow \lbrack 0,\infty )\) a continuous function, we consider the following \(\mathcal{AT}\)-\textit{integral transform}% \begin{equation*} \mathcal{AT}\left( p,\mu \right) \left( T\right) :=\int_{0}^{1}\frac{p\left( \lambda \right) }{\lambda ^{2}}\left[ 1-\left( \lambda ^{2}T^{2}+1\right) ^{-1}\right] d\mu \left( \lambda \right), \end{equation*}% where the integral is assumed to exist for \(T\) a selfadjoint operator on a complex Hilbert space \(H\).   In this paper, we show among others that, if \(A, B\) are selfadjoint operators with \(A^{2}\geq m_{1}^{2}\), \(B^{2}\geq m_{2}^{2}\) and \(m_{1}\), \(m_{2}>0\), then \begin{align*} & \left\Vert \mathcal{AT}\left( p,\mu \right) \left( B\right) -\mathcal{AT}% \left( p,\mu \right) \left( A\right) \right\Vert \\ & \leq \left\Vert B^{2}-A^{2}\right\Vert \left\{  \begin{array}{l} \frac{\mathcal{AT}\left( p,\mu \right) \left( m_{2}\right) -\mathcal{AT}% \left( p,\mu \right) \left( m_{1}\right) }{m_{2}^{2}-m_{1}^{2}}\text{ if }% m_{2}^{2}\neq m_{1}^{2}, \\  \\  \frac{1}{2m}\mathcal{AT}^{\prime }\left( p,\mu \right) \left( m\right) \text{ if }m_{2}^{2}=m_{1}^{2}=m^{2},% \end{array}% \right. \end{align*}% where \(\mathcal{AT}^{\prime }\left( p,\mu \right) \left( t\right)\) is the derivative of \(\mathcal{AT}\left( p,\mu \right)\) as a function of \(t\). Some examples for integral transforms \(\mathcal{AT}\left( p,\mu \right)\) related to the \(\arctan\) function are also provided.