Matrices with circular symmetry on their unitary orbits and $C$-numerical ranges

Abstract:

We give equivalent characterizations for those $n \times n$ complex matrices $A$ whose unitary orbits $\mathcal {U}(A)$ and $C$-numerical ranges ${W_C}(A)$ satisfy ${e^{i\theta }}\mathcal {U}(A) = \mathcal {U}(A)$ or ${e^{i\theta }}{W_C}(A) = {W_C}(A)$ for some real $\theta$ (or for all real $\theta$). In particular, we show that they are the block-cyclic or block-shift operators. Some of these results are extended to infinite-dimensional Hilbert spaces.