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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Invertible completions of $2\times 2$ upper
triangular operator matrices


Authors: Jin Kyu Han, Hong Youl Lee and Woo Young Lee
Journal: Proc. Amer. Math. Soc. 128 (2000), 119-123
MSC (1991): Primary 47A10, 47A55
Published electronically: July 6, 1999
MathSciNet review: 1618686
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Abstract: In this note we prove that if

\begin{equation*}M_{C}=\left (\begin{smallmatrix}A&C\\ 0&B\end{smallmatrix} \right) \end{equation*}

is a $2\times 2$ upper triangular operator matrix acting on the Banach space $X\oplus Y$, then $M_{C}$ is invertible for some $C\in \mathcal{L}(Y,X)$ if and only if $A\in \mathcal{L}(X)$ and $B\in \mathcal{L}(Y)$ satisfy the following conditions:

(i)
$A$ is left invertible;
(ii)
$B$ is right invertible;
(iii)
$X/A(X)\cong B^{-1}(0)$.
Furthermore we show that $\sigma (A)\cup \sigma (B)=\sigma (M_{C})\cup W$, where $W$ is the union of certain of the holes in $\sigma (M_{C})$ which happen to be subsets of $\sigma (A)\cap \sigma (B)$.


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Additional Information

Jin Kyu Han
Affiliation: Department of Mathematics Education, Mokwon University, Daejon 301-719, Korea

Hong Youl Lee
Affiliation: Department of Mathematics, Woosuk University, Wanju-gun, Cheonbuk 565-800, Korea

Woo Young Lee
Affiliation: Department of Mathematics, Sung Kyun Kwan University, Suwon 440-746, Korea
Email: wylee@yurim.skku.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-99-04965-5
Keywords: Spectrum, regular, $2\times 2$ upper triangular operator matrices
Received by editor(s): October 26, 1996
Received by editor(s) in revised form: March 10, 1998
Published electronically: July 6, 1999
Additional Notes: This work was partially supported by BSRI 96-1420 and KOSEF 94-0701-02-01-3.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society