http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
We investigate various problems associated w ith the structure theory of pairs of com m uting (polynom ially bounded) operators on H ilbert space. In particular, we develop a technique tha t enables us to expand the jo in t spectra of certain pairs of com m uting 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 fo r a single operator T on H ilb ert space and the existence of a common invariant subspace for two com m uting related operators. Thus, the problem of the existence of n o ntrivia l invariant subspaces for a single contraction w ith spectral radius one is reduced to the problem of the existence of n o n triv ia l common invariant subspaces for a p a ir of commuting contractions w ith rich jo in t spectra. We construct an H 00-functional calculus valid fo r certain pairs of commuting polynom ially bounded operators, based on the Briem -Davie-0ksendal construction of such a functional calculus for pairs o f com m uting completely nonunitary contractions, and we use dual algebra techniques in the direction of finding nontriv ia l common invariant subspaces for such pairs. In particular, we show that certain pairs o f absolutely continuous com m uting polynom ially bounded operators w ith rich jo in t spectrum have a large lattice o f common invariant subspaces.