A note on the spectrum of an upper triangular operator matrix

Barraa, Mohamed; Boumazgour, Mohamed

doi:10.1090/S0002-9939-03-06862-X

A note on the spectrum of an upper triangular operator matrix
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by Mohamed Barraa and Mohamed Boumazgour PDF

Proc. Amer. Math. Soc. 131 (2003), 3083-3088 Request permission

Abstract:

Let $M_C= [\begin {smallmatrix} A& C 0 & B \end {smallmatrix}]$ be a $2\times 2$ upper triangular operator matrix acting on the Banach space $E\oplus F$. We investigate the set of the operators $C$ for which $\sigma (M_C)=\sigma (A)\cup \sigma (B)$, where $\sigma (.)$ denotes the spectrum.

References

Hong Ke Du and Pan Jin, Perturbation of spectrums of $2\times 2$ operator matrices, Proc. Amer. Math. Soc. 121 (1994), no. 3, 761–766. MR 1185266, DOI 10.1090/S0002-9939-1994-1185266-2
Lawrence A. Fialkow, A note on the range of the operator $X\rightarrow AX-XB$, Illinois J. Math. 25 (1981), no. 1, 112–124. MR 602902
Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952
Jin Kyu Han, Hong Youl Lee, and Woo Young Lee, Invertible completions of $2\times 2$ upper triangular operator matrices, Proc. Amer. Math. Soc. 128 (2000), no. 1, 119–123. MR 1618686, DOI 10.1090/S0002-9939-99-04965-5