On simultaneous similarity of families of commuting operators

Characterization of simultaneously similarity for commuting $m$-tuples of operators is an open problem even in finite-dimensional spaces; known as ``A wild problem in linear algebra". In this paper we offer a criteria for simultaneously similarity of $m$-tuples of $k$-cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found in \cite{BS13} for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for $m$-tuples of commuting square matrices.