SUT J. Math. 60 (1), 1-16, (June 2024) DOI: 10.55937/sut/1717241617
KEYWORDS: Tensorial product, Hadamard product, selfadjoint operators, Convex functions, Schwarz inequality, 47A63, 47A99

Let $H$ be a Hilbert space. In this paper we show among others that, if the functions $f,g:I\subset \mathrm{\mathbb{R}}\to [0,\infty )$ are continuous and $A,B$ are selfadjoint operators with spectra $\text{Sp}\left(A\right),\text{Sp}\left(B\right)\subset I$, then

$$\begin{array}{c}{f}^{2}\left(A\right)\otimes {g}^{2}\left(B\right)+{g}^{2}\left(A\right)\otimes {f}^{2}\left(B\right)\\ \ge \left[{f}^{2(1-\lambda )}\left(A\right){g}^{2\lambda}\left(A\right)\right]\otimes \left[{f}^{2\lambda}\left(B\right){g}^{2(1-\lambda )}\left(B\right)\right]\\ +\left[{f}^{2\lambda}\left(A\right){g}^{2(1-\lambda )}\left(A\right)\right]\otimes \left[{f}^{2(1-\lambda )}\left(B\right){g}^{2\lambda}\left(B\right)\right]\\ \ge 2\left[f\left(A\right)g\left(A\right)\right]\otimes \left[f\right(B)g(B\left)\right]\end{array}$$for all $\lambda \in [0,1]$. We also have the following inequalities for the Hadamard product

$$\begin{array}{c}{f}^{2}\left(A\right)\u25cb{g}^{2}\left(B\right)+{g}^{2}\left(A\right)\u25cb{f}^{2}\left(B\right)\\ \ge \left[{f}^{2(1-\lambda )}\left(A\right){g}^{2\lambda}\left(A\right)\right]\u25cb\left[{f}^{2\lambda}\left(B\right){g}^{2(1-\lambda )}\left(B\right)\right]\\ +\left[{f}^{2\lambda}\left(A\right){g}^{2(1-\lambda )}\left(A\right)\right]\u25cb\left[{f}^{2(1-\lambda )}\left(B\right){g}^{2\lambda}\left(B\right)\right]\\ \ge 2\left[f\left(A\right)g\left(A\right)\right]\u25cb\left[f\right(B)g(B\left)\right]\end{array}$$for all $\lambda \in [0,1]$.