Cauchy-Schwarz for sums of products of matrices

The usual Cauchy-Schwarz inequality states that, for real sequences $a_i,b_i$ $$ \Big|\sum_{i=1}^n a_i b_i \Big| \leq \Big(\sum_{i=1}^n a_i^2\Big)^{1/2}\Big(\sum_{i=1}^n b_i^2\Big)^{1/2}. $$

My question is whether the same holds for matrices. More precisely, let $A_i,B_i$ be sequences of $m\times m$ matrices. Does it hold that $$ \Big\|\sum_{i=1}^n A_i B_i \Big\| \leq \Big\| \sum_{i=1}^n A_i^*A_i\Big\|^{1/2} \Big\| \sum_{i=1}^n B_i^*B_i\Big\|^{1/2}, $$ where $\|\cdot\|$ is the operator norm?

Solution 1:

The Cauchy–Bunyakovsky–Schwarz inequality does not generalise as proposed above.

For $m=2=n$ consider the concrete choices $$A_1=\begin{pmatrix}1 &0\\1&0\end{pmatrix},\; B_1=\begin{pmatrix}1 &0\\0&0\end{pmatrix},\; A_2=\begin{pmatrix}0 &1\\0&1\end{pmatrix},\; B_2=\begin{pmatrix}0 &0\\0&1\end{pmatrix}.$$