Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\lambda\in\mathbb C$ with $|\lambda|=1$ such that $\cosh(t)(a^2+b^2)+\sinh(t)(\lambda ab+\overline\lambda ba)$ is a contraction, then it's possible to choose $\lambda=\pm1$ and still have a contraction (maybe for a different value of $t$).
There's obviously something in the background to do with hyperbolic metrics etc. etc. but I can't see how to leverage any general theory. So I've asked the question in a simple, clean way, but in a way which is maybe lacking context. The hope is that because a and b are self-adjoint, it should be enough to choose $\lambda$ real (i.e. self-adjoint) but I've no idea if this is true...