Norm of Schur multiplication for Schatten norm

Abstract:

Let ${\mathbb {M}_n}$ denote the algebra of all $n \times n$ complex matrices and $|| \cdot ||$ be the Schatten’s $p$-norm on ${\mathbb {M}_n}$. For each $A \in {\mathbb {M}_n}$, a linear operator ${S_A}$ on ${\mathbb {M}_n}$ is defined by ${S_A}\left ( X \right ): = A \circ X$ for all $X \in {\mathbb {M}_n}$, where $\circ$ denotes the Schur product and $||{S_A}|{|_{p,q}}$ is defined as the operator norm from $({\mathbb {M}_n},|| \cdot |{|_p})$ to $({\mathbb {M}_n},|| \cdot |{|_q})$ for $p,q \geq 1$. Given an $A \in {\mathbb {M}_n}$, suppose $0 < \lambda < 1$, and $p,{p_1},{p_2},q,{q_1}$, and ${q_2}$ are not smaller than 1 , and \[ \frac {1} {p} = \frac {\lambda } {{{p_1}}} + \frac {{1 - \lambda }} {{{p_2}}}\quad {\text {and}}\quad \frac {1} {q} = \frac {\lambda } {{{q_1}}} + \frac {{1 - \lambda }} {{{q_2}}}\] are satisfied. Then we will show that $||{S_A}|{|_{p,q}} \leq ||{S_A}||_{{p_1},{q_1}}^\lambda \cdot ||{S_A}||_{{p_2},{q_2}}^{1 - \lambda }$.