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The essential spectrum of Toeplitz tuples with symbols in $H^{∞} + C$

100%

Eschmeier J.

Studia Mathematica

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2013

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tom 219

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nr 3

237-246

EN

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 $T_{f} = (T_{f₁}, ..., T_{fₘ}) ∈ L(H²(σ))^{m}$ 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) to systems of Toeplitz operators.

2

Fredholm spectrum and growth of cohomology groups

100%

Eschmeier J.

Studia Mathematica

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2008

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tom 186

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nr 3

237-249

EN

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 $Reg(σ_{F}(T))$ 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.

3

A commutant lifting theorem on analytic polyhedra

63%

Ambrozie C., Eschmeier J.

Banach Center Publications

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2005

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tom 67

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nr 1

83-108

EN

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 for Schur class functions. Our methods apply in particular to the unit ball, the unit polydisc and the classical symmetric domains of types I, II and III.

Ograniczanie wyników

2 Studia Mathematica

1 Banach Center Publications

3 Eschmeier J.

1 Ambrozie C.

1 2013

1 2008

1 2005

The essential spectrum of Toeplitz tuples with symbols in $H^{∞} + C$

Fredholm spectrum and growth of cohomology groups

A commutant lifting theorem on analytic polyhedra