Several inequalities for an integral transform of positive operators in Hilbert spaces with applications

Abstract

For a continuous and positive function $w\left( \lambda \right) ,$ $\lambda >0$ and $\mu $ a positive measure on $(0,\infty )$ we consider the following integral transform % \begin{equation*} \mathcal{D}\left( w,\mu \right) \left( T\right) :=\int_{0}^{\infty }w\left( \lambda \right) \left( \lambda +T\right) ^{-1}d\mu \left( \lambda \right) , \end{equation*}% where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.\\ We show among others that, if $\beta \geq A\geq \alpha >0,$ $B>0$ with $% M\geq B-A\geq m>0$ for some constants $\alpha ,$ $\beta ,$ $m,$ $M$, then % \begin{align*} 0& \leq \frac{m^{2}}{M^{2}}\left[ \mathcal{D}\left( w,\mu \right) \left( \beta \right) -\mathcal{D}\left( w,\mu \right) \left( M+\beta \right) \right] \\ & \leq \frac{m^{2}}{M}\left[ \mathcal{D}\left( w,\mu \right) \left( \beta \right) -\mathcal{D}\left( w,\mu \right) \left( M+\beta \right) \right] \left( B-A\right) ^{-1} \\ & \leq \mathcal{D}\left( w,\mu \right) \left( A\right) -\mathcal{D}\left( w,\mu \right) \left( B\right) \\ & \leq \frac{M^{2}}{m}\left[ \mathcal{D}\left( w,\mu \right) \left( \alpha \right) -\mathcal{D}\left( w,\mu \right) \left( m+\alpha \right) \right] \left( B-A\right) ^{-1} \\ & \leq \frac{M^{2}}{m^{2}}\left[ \mathcal{D}\left( w,\mu \right) \left( \alpha \right) -\mathcal{D}\left( w,\mu \right) \left( m+\alpha \right) % \right] . \end{align*}% Some examples for operator monotone and operator convex functions as well as for integral transforms $\mathcal{D}\left( \cdot ,\cdot \right) $ 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