# Convexity and tuple of contractions

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 $$(B_1,B_2)\in S$$?

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 $$\|\lambda_1 B_1+\lambda_2 B_2\|=\left\lVert\begin{pmatrix}\lambda_1 I_n&\lambda_2 I_n\end{pmatrix}B\right\rVert\leq \|\begin{pmatrix}\lambda_1 I_n&\lambda_2 I_n\end{pmatrix}\|\|B\|\leq 1.$$