Spectral and numerical radius preservers on pairs of unitarily similar operators

Let $\mathcal{H}$ be a complex Hilbert space. We characterize the bijective linear maps $\mathrm{\Phi }:\mathcal{B}(\mathcal{H})\to \mathcal{B}(\mathcal{H})$ that preserve the spectral radius, respectively the numerical radius, on pairs of unitarily similar operators. We also characterize bijective linear maps that transform unitary similarity into similarity. Our results cover both finite and infinite-dimensional Hilbert spaces, with no separability assumption in the latter case.