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The intersection of essential approximate point spectra of operator matrices

Authors
Journal
Journal of Mathematical Analysis and Applications
0022-247X
Publisher
Elsevier
Publication Date
Volume
323
Issue
2
Identifiers
DOI: 10.1016/j.jmaa.2005.11.032
Keywords
  • Essential Approximate Point Spectrum
  • [Formula Omitted]Operator Matrix
  • Perturbations Of Spectra

Abstract

Abstract When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite-dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, it is shown that there exists some operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 if and only if there exists some left invertible operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 . A necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for some C ∈ Inv ( K , H ) is given, where Inv ( K , H ) denotes the set of all the invertible operators of B ( K , H ) . In addition, we give a necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for all C ∈ Inv ( K , H ) .

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