H-infinity-FUNCTIONAL CALCULUS FOR COMMUTING FAMILIES OF RITT OPERATORS AND SECTORIAL OPERATORS

We introduce and investigate H-infinity-functional calculus for commuting finite families of Ritt operators on Banach space X. We show that if either X is a Banach lattice or X or X* has property (alpha) , then a commuting d-tuple (T-1, ..., T-d) of Ritt operators on X has an H-infinity joint functional calculus if and only if each T-k admits an H-infinity functional calculus. Next for p is an element of (1, infinity) , we characterize commuting d-tuple of Ritt operators on L-p(Omega) which admit an H-infinity joint functional calculus, by a joint dilation property. We also obtain a similar characterisation for operators acting on a UMD Banach space with property (alpha). Then we study commuting d -tuples (T-1, ..., T-d) of Ritt operators on Hilbert space. In particular we show that if parallel to T-k parallel to <= 1 for every k = 1, ..., d, then (T-1, ..., T-d) satisfies a multivariable analogue of von Neumann's inequality. Further we show analogues of most of the above results for commuting finite families of sectorial operators.