Invertible completions of $2\times 2$ upper triangular operator matrices
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- by Jin Kyu Han, Hong Youl Lee and Woo Young Lee PDF
- Proc. Amer. Math. Soc. 128 (2000), 119-123 Request permission
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 $\sigma (A)\cap \sigma (B)$.
References
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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
- MR Author ID: 263789
- Email: wylee@yurim.skku.ac.kr
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 119-123
- MSC (1991): Primary 47A10, 47A55
- DOI: https://doi.org/10.1090/S0002-9939-99-04965-5
- MathSciNet review: 1618686