Convexity and tuple of contractions

Question

Let us define $S\colon=\{(A_1,A_2):\sup_{|\lambda_1|^2+|\lambda_2|^2=1}\|\lambda_1A_1+\lambda_2A_2\|\leq 1\}$ where $A_1,A_2$ are $n\times n$ matrices $M_n$ endowed with the usual operator norm ($(M_n,\|.\|_{op})$). Let $B_1,B_2\in M_n$ with $B_1^*B_1+B_2^*B_2\leq I_n$ where $I_n$ denotes the identity operator. Is it true that $(B_1,B_2)\in S$?

Answer 1

Yes. Let $B=\begin{pmatrix}B_1\\B_2\end{pmatrix}$, viewed as a linear map from $\mathbb{C}^n$ to $\mathbb{C}^{2n}$. Then the condition $B_1^\ast B_1+B_2^\ast B_2\leq I_n$ reads $B^\ast B\leq 1$, which implies $\|B\|\leq 1$. Similarly, $|\lambda_1|^2+|\lambda_2|^2\leq 1$ translates to $\|\begin{pmatrix}\lambda_1 I_n&\lambda_2 I_n\end{pmatrix}\|\leq 1$. Thus $$ \|\lambda_1 B_1+\lambda_2 B_2\|=\left\lVert\begin{pmatrix}\lambda_1 I_n&\lambda_2 I_n\end{pmatrix}B\right\rVert\leq \|\begin{pmatrix}\lambda_1 I_n&\lambda_2 I_n\end{pmatrix}\|\|B\|\leq 1. $$