Essential Spectra of 2 × 2 Block Operator Matrices

Aref Jeribi
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Aref Jeribi: University of Sfax, Department of Mathematics

Chapter Chapter 10 in Spectral Theory and Applications of Linear Operators and Block Operator Matrices, 2015, pp 327-374 from Springer

Abstract: Abstract Let X and Y be Banach spaces. In the product space X × Y, we consider an operator formally defined by a matrix 10.0.1 L 0 : = A B C D . $$\displaystyle{ L_{0}:= \left (\begin{array}{*{10}c} A&B\\ C &D \end{array} \right ). }$$ In general, the operators occurring in the representation of L 0 are unbounded. The operator A acts on the Banach space X and has the domain 𝒟 ( A ) $$\mathcal{D}(A)$$ , D is defined on 𝒟 ( D ) $$\mathcal{D}(D)$$ and acts on the Banach space Y, and the intertwining operator B (resp. C) is defined on the domain 𝒟 ( B ) $$\mathcal{D}(B)$$ (resp. 𝒟 ( C ) $$\mathcal{D}(C)$$ ) and acts from Y into X (resp. from X into Y ). One of the problems in the study of such operators is that in general L 0 is not closed or even closable, even if its entries are closed.

Keywords: Block Operator Matrix; Essential Spectrum; Banach Space; Product Space; Fredholm Perturbation (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-17566-9_10

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