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Essential Spectra of 2 × 2 Block Operator Matrices

In: Spectral Theory and Applications of Linear Operators and Block Operator Matrices

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  • Aref Jeribi

    (University of Sfax, Department of Mathematics)

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.

Suggested Citation

  • Aref Jeribi, 2015. "Essential Spectra of 2 × 2 Block Operator Matrices," Springer Books, in: Spectral Theory and Applications of Linear Operators and Block Operator Matrices, edition 127, chapter 0, pages 327-374, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-17566-9_10
    DOI: 10.1007/978-3-319-17566-9_10
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