Invariant Gaussian measures for operators on Banach spaces and linear dynamics

Abstract

We give conditions for an operator T on a complex separable Banach space X with sufficiently many eigenvectors associated to eigenvalues of modulus 1 to admit a non-degenerate invariant Gaussian measure with respect to which it is weak-mixing. The existence of such a measure depends on the geometry of the Banach space and on the possibility of parametrizing the 𝕋-eigenvector fields of T in a regular way. We also investigate the connection with frequent hypercyclicity.