| dc.creator | Dragomir, S. S. | |
| dc.date | 2023-07-19 | |
| dc.date.accessioned | 2023-12-19T18:55:51Z | |
| dc.date.available | 2023-12-19T18:55:51Z | |
| dc.identifier | https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3420 | |
| dc.identifier | 10.56754/0719-0646.2502.195 | |
| dc.identifier.uri | https://revistaschilenas.uchile.cl/handle/2250/239185 | |
| dc.description | 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. | en-US |
| dc.format | application/pdf | |
| dc.language | eng | |
| dc.publisher | Universidad de La Frontera. Temuco, Chile. | en-US |
| dc.relation | https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3420/2298 | |
| dc.rights | Copyright (c) 2023 S. S. Dragomir | en-US |
| dc.rights | https://creativecommons.org/licenses/by-nc/4.0/ | en-US |
| dc.source | CUBO, A Mathematical Journal; Vol. 25 No. 2 (2023); 195–209 | en-US |
| dc.source | CUBO, A Mathematical Journal; Vol. 25 No. 2 (2023); 195–209 | es-ES |
| dc.source | 0719-0646 | |
| dc.source | 0716-7776 | |
| dc.subject | Operator monotone functions | en-US |
| dc.subject | Operator convex functions | en-US |
| dc.subject | Operator inequalities | en-US |
| dc.subject | Löwner- Heinz inequality | en-US |
| dc.subject | Logarithmic operator inequalities | en-US |
| dc.subject | 47A63 | en-US |
| dc.subject | 47A60 | en-US |
| dc.title | Several inequalities for an integral transform of positive operators in Hilbert spaces with applications | en-US |
| dc.type | info:eu-repo/semantics/article | |
| dc.type | info:eu-repo/semantics/publishedVersion | |
