The Kato-type spectrum and local spectral theory

T. L. Miller; V. G. Miller; Michael M. Neumann

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 3, page 831-842
  • ISSN: 0011-4642

Abstract

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Let T ∈ ℒ ( X ) be a bounded operator on a complex Banach space X . If V is an open subset of the complex plane such that λ - T is of Kato-type for each λ ∈ V , then the induced mapping f ( z ) ↦ ( z - T ) f ( z ) has closed range in the Fréchet space of analytic X -valued functions on V . Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of T . Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic sheaves.

How to cite

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Miller, T. L., Miller, V. G., and Neumann, Michael M.. "The Kato-type spectrum and local spectral theory." Czechoslovak Mathematical Journal 57.3 (2007): 831-842. <http://eudml.org/doc/31165>.

@article{Miller2007,
abstract = {Let $T\in \{\mathcal \{L\}\}(X)$ be a bounded operator on a complex Banach space $X$. If $V$ is an open subset of the complex plane such that $\lambda -T$ is of Kato-type for each $\lambda \in V$, then the induced mapping $f(z)\mapsto (z-T)f(z)$ has closed range in the Fréchet space of analytic $X$-valued functions on $V$. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of $T$. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic sheaves.},
author = {Miller, T. L., Miller, V. G., Neumann, Michael M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {decomposable operator; semi-Fredholm operator; semi-regular operator; Kato decomposition; Bishop’s property ($\beta $); property ($\delta $); decomposable operator; semi-Fredholm operator; semi-regular operator; Kato decomposition; Bishop’s property (); property ()},
language = {eng},
number = {3},
pages = {831-842},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Kato-type spectrum and local spectral theory},
url = {http://eudml.org/doc/31165},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Miller, T. L.
AU - Miller, V. G.
AU - Neumann, Michael M.
TI - The Kato-type spectrum and local spectral theory
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 3
SP - 831
EP - 842
AB - Let $T\in {\mathcal {L}}(X)$ be a bounded operator on a complex Banach space $X$. If $V$ is an open subset of the complex plane such that $\lambda -T$ is of Kato-type for each $\lambda \in V$, then the induced mapping $f(z)\mapsto (z-T)f(z)$ has closed range in the Fréchet space of analytic $X$-valued functions on $V$. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to a sharper estimate of Nagy’s spectral residuum of $T$. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic sheaves.
LA - eng
KW - decomposable operator; semi-Fredholm operator; semi-regular operator; Kato decomposition; Bishop’s property ($\beta $); property ($\delta $); decomposable operator; semi-Fredholm operator; semi-regular operator; Kato decomposition; Bishop’s property (); property ()
UR - http://eudml.org/doc/31165
ER -

References

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