Fredholm spectrum and growth of cohomology groups

Let T\in L(E)^{n} be a commuting tuple of bounded linear operators on a complex Banach space E and let \sigma_{F}(T)=\sigma(T)\setminus\sigma_{e}(T) be the non-essential spectrum of T. We show that, for each connected component M of the manifold \mbox{Reg}(\sigma_{F}(T)) of all smooth points of \sigma_{F}(T), there is a number p\in\{0,...,n\} such that, for each point z\in M, the dimensions of the cohomology groups H{}^{p}((z-T)^{k},E) grow at least like the sequence (k^{d})_{k\geq1} with d = dimM

On the Hilbert-Samuel multiplicity of Fredholm tuples

For commuting tuples R\in L(Z)^{n} of Banach-space operators that arise as quotients of lower semi-Fredholm systems T\in L(X)^{n} with constant cohomology dimension \textrm{dim}H^{n}(z-T,X) near the origin 0\in\mathbb{C}^{n}, we show that the Hilbert-Samuel multiplicity of R calculates the rank of the cohomology sheaf 0\in\mathbb{C}^{n}(z-R,\mathcal{O}_{\mathbb{C}^{n}}^{Z}) at z=0

On the reflexivity of multivariable isometries

Let A\subsetC(K) be a unital closed subalgebra of the algebra of all continuous functions on a compact set K in \mathbb{C}^{n}. We define the notion of an A-isometry and show that, under a suitable regularity condition needed to apply Aleksandrov\u27s work on the inner function problem, every A-isometry T\in L(\mathcal{H})^{n} is reflexive. This result applies to commuting isometries, spherical isometries, and more general, to all subnormal tuples with normal spectrum contained in the Bergman-Shilov boundary of a strictly pseudoconvex or bounded symmetric domain

Samuel multiplicity for several commuting operators

It is shown that the Samuel multiplicity of a lower semi-Fredholm tuple T\in L(X)^{n} of commuting bounded operators on a complex Banach space X coincides with the generic dimension of the last cohomolgoy groups H^{n}(z-T,X) of its Koszul complex near z=0. As applications we show that the algebraic and analytic Samuel multiplicities of T coincide and that the Samuel multiplicity is additive for closed invariant subspaces of the symmetric Fock space