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

Summary: When $A\in ℒ\left(ℋ\right)$ and $B\in ℒ\left(𝒦\right)$ are given we denote by ${M}_{C}$ an operator acting on the Hilbert space $ℋ\oplus 𝒦$ of the form

${M}_{C}:=\left(\begin{array}{cc}A& C\\ 0& B\end{array}\right),$

where $C\in ℒ\left(ℒ,ℋ\right)$. In this paper we characterize the boundedness below of ${M}_{C}$. Our characterization is as follows: ${M}_{C}$ is bounded below for some $C\in ℒ\left(𝒦,ℋ\right)$ if and only if $A$ is bounded below and $\alpha \left(B\right)\le \beta \left(A\right)$ if $R\left(B\right)$ is closed; $\beta \left(A\right)=\infty$ if $R\left(B\right)$ is not closed, where $\alpha \left(·\right)$ and $\beta \left(·\right)$ denote the nullity and the deficiency, respectively. In addition, we show that if ${\sigma }_{ap}\left(·\right)$ and ${\sigma }_{d}\left(·\right)$ denote the approximate point spectrum and the defect spectrum, respectively, then the passage from ${\sigma }_{ap}\left(\begin{array}{cc}A& 0\\ 0& B\end{array}\right)$ to ${\sigma }_{ap}\left({M}_{C}\right)$ can be described as follows:

${\sigma }_{ap}\left(\begin{array}{cc}A& 0\\ 0& B\end{array}\right)={\sigma }_{ap}\left({M}_{C}\right)\cup W\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{every}\phantom{\rule{4.pt}{0ex}}C\in ℒ\left(𝒦,ℋ\right),$

where $W$ lies in certain holes in ${\sigma }_{ap}\left(A\right)$, which happens to be subsets of ${\sigma }_{d}\left(A\right)\cap {\sigma }_{ap}\left(B\right)$.

##### MSC:
 47A10 Spectrum and resolvent of linear operators 47A55 Perturbation theory of linear operators 47A66 (Non)quasitriangular, quasidiagonal and nonquasidiagonal operators
##### References:
 [1] [Ap] C. Apostol,The reduced minimum modulus, Michigan Math. J.32 (1985), 279-294. · Zbl 0613.47008 · doi:10.1307/mmj/1029003239 [2] [Do1] R.G. Douglas,Banach Algebra Techniques in Operator Theory, Academic press, New York, 1972. [3] [Do2] R.G. Douglas,Banach Algebra Techniques in the Theory of Toeplitz Operators, CBMS 15, Providence: AMS, 1973. [4] [DJ] H.K. Du and P. Jin,Perturbation of spectrums of 2?2 operator matrices, Proc. Amer. Math. Soc.121 (1994), 761-776. · doi:10.1090/S0002-9939-1994-1185266-2 [5] [GGK1] I. Gohberg, S. Goldberg and M.A. Kaashoek,Classes of Linear Operators, Vol I, OT 49, Birkh?user, Basel, 1990. [6] [GGK2] I. Gohberg, S. Goldberg and M.A. Kaashoek,Classes of Linear Operators, Vol II, OT 63, Birkh?user, Basel, 1993. [7] [Go] S. Goldberg,Unbounded Linear Operators, McGraw-Hill, New York, 1966. [8] [HLL] J.K. Han, H.Y. Lee and W.Y. Lee,Invertible completions of 2?2 upper triangular operator matrices, Proc. Amer. Math. Soc.128 (2000), 119-123. · Zbl 0944.47004 · doi:10.1090/S0002-9939-99-04965-5 [9] [Ha1] R.E. Harte,Invertibility and Singularity for Bounded Linear Operators, Dekker, New York, 1988. [10] [Ha2] R.E. Harte,The ghost of an index theorem, Proc. Amer. Math. Soc.106 (1989), 1031-1033. · doi:10.1090/S0002-9939-1989-0975646-8 [11] [Ha3] R.E. Harte,The ghost of an index theorem II (preprint 1999). [12] [Le] W.Y. Lee,Weyl’s theorem for operator matrices, Int. Eq. Op. Th.32 (1998), 319-331. · Zbl 0923.47001 · doi:10.1007/BF01203773 [13] [Ni] N.K. Nikolskii,Treatise on the Shift Operator, Springer, New York, 1986. [14] [Pe] C.M. Pearcy,Some Recent Developments in Operator Theory, CBMS 36, Providence: AMS, 1978.