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

• S. S. Dragomir

## 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 postive 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.

## Mathematics Subject Classification:

• Mathematics, College of Engineering & Science, Melbourne City, Australia.
• Pages: 195–209
• Date Published: 2023-07-19
• Vol. 25 No. 2 (2023)

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2023-07-19

## How to Cite

[1]
S. S. Dragomir, “Several inequalities for an integral transform of positive operators in Hilbert spaces with applications”, CUBO, pp. 195–209, Jul. 2023.

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