2 problems

Let $\mathcal{A}$ be a unital C-algebra. For $x\in\mathcal{A}$, define the C-algebraic cosine by … where $e^x=\sum_{m=0}^{\infty}x^m/m!$. Write $\mathcal{A}_{\mathrm{sa}}$ for the…

Let $M_n(\mathbb C)$ denote the complex $n\times n$ matrices, let $\|\cdot\|_1$ be the first Schatten norm, and let $R\circ C$ denote the entrywise Schur product of…