The Spectral Radius of a Product of Two Hilbert-Space Operators

Question

I’m given a Hilbert space $ \mathcal{H} $ such that $ \dim(\mathcal{H}) > 1 $, and I’m supposed to construct two operators $ A $ and $ B $ on $ \mathcal{H} $ such that $ r(A B) \neq r(A) r(B) $. Is there any general approach to this? I can’t think of an example.

Answer 1

How about this for an example?

$$A = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)\quad,\quad B = \left(\begin{array}{cc} 0 & 0 \\ 0 & 1\end{array}\right).$$

Then $r(A) = 1 = r(B)$ but $AB$ is the zero matrix, so $r(AB) = 0$. My thought process was to take two matrices with nonzero eigenvalues which multiply to zero (which has eigenvalues of zero). Then in that case, we wouldn't have equality.