Two mixed Hadamard type generalizations of Heinz inequality

Abstract:

We give two types of mixed Hadamard inequalities containing the terms $T,\left | T \right |$, and $\left | {{T^ * }} \right |$, where $T$ is a bounded linear operator on a complex Hilbert space. As an immediate consequence of these results, we can easily show some extensions of the Hadamard inequality and also the Heinze inequality: \[ \left ( * \right )\quad {\left | {\left ( {Tx,y} \right )} \right |^2} \leq \left ( {{{\left | T \right |}^{2\alpha }}x,x} \right )\left ( {{{\left | {{T^ * }} \right |}^{2\left ( {1 - \alpha } \right )}}y,y} \right )\] for any $T$, any $x,y$ in $H$, and any real number $\alpha$ with $0 \leq \alpha \leq 1$. And the following conditions are equivalent in case $0 < \alpha < 1$: (1) the equality in (*) holds; (2) ${\left | T \right |^{2\alpha }}x$ and ${T^ * }y$ are linearly dependent; (3) $Tx$ and ${\left | {{T^ * }} \right |^{2\left ( {1 - \alpha } \right )}}y$ are linearly dependent. Results in this paper would remain valid for unbounded operators under slight modifications.