Essential approximate point spectra and Weyl's theorem for operator matrices

By Xiao-hong Cao, Bin Meng. 2005

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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 a 2 × 2 operator matrix M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 for some C ∈ B ( K , H ) if and only if A is upper semi-Fredholm and { n ( B ) ∞  and n ( A ) + n ( B ) ⩽ d ( A ) + d ( B ) or  n ( B ) = d ( A ) = ∞ ,  if  R ( B )  is closed , d ( A ) = ∞ ,  if  R ( B )  is not closed . We also give the necessary and sufficient conditions for which M C is Weyl or M C is lower semi-Fredholm with nonnegative index for some C ∈ B ( K , H ) . In addition, we explore how Weyl's theorem, Browder's theorem, a-Weyl's theorem, and a-Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.

Subfields: Algebra and Number Theory, Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/6cb0b26d630491cde5ee3ae2c3f76c620b2a9c45