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Fredholm spectrum and growth of cohomology groups

Jörg Eschmeier — 2008

Studia Mathematica

Let T ∈ L(E)ⁿ be a commuting tuple of bounded linear operators on a complex Banach space E and let σ F ( T ) = σ ( T ) σ e ( T ) be the non-essential spectrum of T. We show that, for each connected component M of the manifold R e g ( σ F ( T ) ) of all smooth points of σ F ( T ) , there is a number p ∈ 0, ..., n such that, for each point z ∈ M, the dimensions of the cohomology groups H p ( ( z - T ) k , E ) grow at least like the sequence ( k d ) k 1 with d = dim M.

The essential spectrum of Toeplitz tuples with symbols in H + C

Jörg Eschmeier — 2013

Studia Mathematica

Let H²(D) be the Hardy space on a bounded strictly pseudoconvex domain D ⊂ ℂⁿ with smooth boundary. Using Gelfand theory and a spectral mapping theorem of Andersson and Sandberg (2003) for Toeplitz tuples with H -symbol, we show that a Toeplitz tuple T f = ( T f , . . . , T f ) L ( H ² ( σ ) ) m with symbols f i H + C is Fredholm if and only if the Poisson-Szegö extension of f is bounded away from zero near the boundary of D. Corresponding results are obtained for the case of Bergman spaces. Thus we extend results of McDonald (1977) and Jewell (1980)...

A commutant lifting theorem on analytic polyhedra

Calin AmbrozieJörg Eschmeier — 2005

Banach Center Publications

In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in ℂⁿ is proved. Let T be the compression of the multiplication tuple M z to a *-invariant closed subspace of the underlying functional Hilbert space. Our main result characterizes those operators in the commutant of T which possess a lifting to a multiplier with Schur class symbol. As an application we obtain interpolation results of Nevanlinna-Pick and Carathéodory-Fejér type...

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