On pairs of commuting operators

We investigate various problems associated with the structure theory of pairs of commuting (polynomially bounded) operators on Hilbert space. In particular, we develop a technique that enables us to expand the joint spectra of certain pairs of commuting contractions, and we are able thereby to generalize some well-known results for the case of one contraction. We also present a connection between solving the invariant subspace problem for a single operator T on Hilbert space and the existence of a common invariant subspace for two commuting related operators. Thus, the problem of the existence of nontrivial invariant subspaces for a single contraction with spectral radius one is reduced to the problem of the existence of nontrivial common invariant subspaces for a pair of commuting contractions with rich joint spectra.; We construct an H{dollar}spinfty{dollar}-functional calculus valid for certain pairs of commuting polynomially bounded operators, based on the Briem-Davie-Oksendal construction of such a functional calculus for pairs of commuting completely nonunitary contractions, and we use dual algebra techniques in the direction of finding non-trivial common invariant subspaces for such pairs. In particular, we show that certain pairs of absolutely continuous commuting polynomially bounded operators with rich joint spectrum have a large lattice of common invariant subspaces.