About some interesting inequality

Question

Suppose A and B are some fixed $L\times L$ complex matrix and $\beta$ is a real variable, what is the upper bound of the absolute value of the trace of $(\mathrm{cos}\beta~ A+ i \mathrm{sin}\beta~ B)^k$?

Thank you very much!

Answer 1

Withot additional assumptions there are no bounds whatsoever — take $A$ with single large nonzero entry in upper right corner and $B$ with nonzero entry in lower left. Then they both are traceless, but your expression is arbitrarily large for $k = 2$. Throw in more nonzero off-diagonal entries for analogous construction for bigger $k$.

Answer 2

Naively, since \begin{align} |\operatorname{Tr} A|\leq \sum^L_{i=1}|\lambda_i| \end{align} where $|\lambda_i|$ are the singular values of $A$ and \begin{align} \sum^L_{i=1}|\lambda_i| \leq L \max_{1\le i\leq L}|\lambda_i|=L\|A\|_\infty, \end{align} then we have \begin{align} |\operatorname{Tr}(\cos\beta A+\sin\beta B)^k| \leq&\ (\operatorname{Tr}|\cos\beta A+\sin \beta B|)^k\\ \leq&\ L^k(\|\cos\beta A\|_\infty + \|\sin \beta B\|_\infty)^k\\ \leq&\ L^k(\cosh(\beta\|A\|_\infty)+\sinh(\beta\|B\|_\infty))^k \end{align}