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

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

References

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