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The boundedness below of 2×2 upper triangular operator matrices. (English) Zbl 0986.47004

Summary: When A() and B(𝒦) are given we denote by M C an operator acting on the Hilbert space 𝒦 of the form

M C :=AC0B,

where C(,). In this paper we characterize the boundedness below of M C . Our characterization is as follows: M C is bounded below for some C(𝒦,) if and only if A is bounded below and α(B)β(A) if R(B) is closed; β(A)= if R(B) is not closed, where α(·) and β(·) denote the nullity and the deficiency, respectively. In addition, we show that if σ ap (·) and σ d (·) denote the approximate point spectrum and the defect spectrum, respectively, then the passage from σ ap A00B to σ ap (M C ) can be described as follows:

σ ap A00B=σ ap (M C )WforeveryC(𝒦,),

where W lies in certain holes in σ ap (A), which happens to be subsets of σ d (A)σ ap (B).


MSC:
47A10Spectrum and resolvent of linear operators
47A55Perturbation theory of linear operators
47A66(Non)quasitriangular, quasidiagonal and nonquasidiagonal operators
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