Tensorial and Hadamard product integral inequalities for convex functions of continuous fields of operators in Hilbert spaces

Year 2025, Volume: 54 Issue: 1, 115 - 124, 28.02.2025

Sever Dragomır

https://doi.org/10.15672/hujms.1362698

Abstract

Let $H$ be a Hilbert space and $\Omega $ a locally compact Hausdorff space endowed with a Radon measure $\mu $ with $\int_{\Omega }1d\mu \left( t\right) =1.$ In this paper we show among others that, if $f$ is continuous differentiable convex on the open interval $I$, $\left( A_{\tau }\right)_{\tau \in \Omega }$ is a continuous field of positive operators in $B\left( H\right) $ with spectra in $ I$ for each $\tau \in \Omega $ and $B$ an operator with spectrum in $I,$ then we have
\begin{align*}
&{\small\operatorname{\small\int}_{\Omega }\left( f^{\prime }\left( A_{\tau }\right) A_{\tau }\right) d\mu \left( \tau \right) \otimes 1-\operatorname{\small\int}_{\Omega }f^{\prime }\left( A_{\tau }\right) d\mu \left( \tau \right) \otimes B}\\
& \geq \int_{\Omega }f\left( A_{\tau }\right) d\mu \left( \tau \right)
\otimes 1-1\otimes f\left( B\right) \\
& \geq \left( \int_{\Omega }A_{\tau }d\mu \left( \tau \right) \otimes
1-\left( 1\otimes B\right) \right) \left( 1\otimes f^{\prime }\left(
B\right) \right)
\end{align*}
and the Hadamard product inequality
\begin{align*}
& {\small\operatorname{\small\int}_{\Omega }\left( f^{\prime }\left( A_{\tau }\right) A_{\tau }\right)
d\mu \left( \tau \right) \circ 1-\operatorname{\small\int}_{\Omega }f^{\prime }\left( A_{\tau
}\right) d\mu \left( \tau \right) \circ B} \\
& \geq \int_{\Omega }f\left( A_{\tau }\right) d\mu \left( \tau \right) \circ
1-1\circ f\left( B\right) \\
& \geq \int_{\Omega }A_{\tau }d\mu \left( \tau \right) \circ f^{\prime
}\left( B\right) -1\circ \left( f^{\prime }\left( B\right) B\right) .
\end{align*}

Keywords

tensorial product, Hadamard product, selfadjoint operators, convex functions

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