Weyl spectra of operator matrices. (English) Zbl 0965.47011

Lee, Woo Young

Proc. Am. Math. Soc. 129, No. 1, 131-138 (2001).

The author studies Fredholm properties of the operator matrix \(M_C = \left(\begin{smallmatrix} A & C \\ 0 & B\end{smallmatrix} \right)\) for operators in Hilbert spaces. Let \(\omega (T)\) denote the Weyl spectrum of an operator \(T\), i.e., the set of all complex \(\lambda\) such that \(\lambda -T\) is not a Fredholm operator of index zero. The main result of the paper reads as follows:
\(\omega (A) \cup \omega (B) = \omega (M_C) \cup \kappa\), where \(\kappa\) is the union of some holes in \(\omega (M_C)\) which are contained in \(\omega (A) \cap \omega (B)\).
This implies that the Weyl spectral radius of \(M_C\) is independent of \(C\).

Reviewer: Karl-Heinz Förster (Berlin)

MSC:

47A53	(Semi-) Fredholm operators; index theories
47A55	Perturbation theory of linear operators

Keywords:

Weyl spectrum; Weyl’s theorem; operator matrices; Fredholm properties; Weyl spectral radius

References:

[1]	S. K. Berberian, An extension of Weyl’s theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273 – 279. · Zbl 0175.13603
[2]	S. K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 (1970/1971), 529 – 544. · Zbl 0203.13401
[3]	Bernard Chevreau, On the spectral picture of an operator, J. Operator Theory 4 (1980), no. 1, 119 – 132. · Zbl 0465.47015
[4]	L. A. Coburn, Weyl’s theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285 – 288. · Zbl 0173.42904
[5]	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. · Zbl 0814.47016
[6]	Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek, Classes of linear operators. Vol. I, Operator Theory: Advances and Applications, vol. 49, Birkhäuser Verlag, Basel, 1990. · Zbl 0745.47002
[7]	Robin Harte, Fredholm, Weyl and Browder theory, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 151 – 176. · Zbl 0567.47001
[8]	Robin Harte, Invertibility and singularity for bounded linear operators, Monographs and Textbooks in Pure and Applied Mathematics, vol. 109, Marcel Dekker, Inc., New York, 1988. · Zbl 0636.47001
[9]	Robin Harte and Woo Young Lee, The punctured neighbourhood theorem for incomplete spaces, J. Operator Theory 30 (1993), no. 2, 217 – 226. · Zbl 0873.47007
[10]	Robin Harte and Woo Young Lee, An index formula for chains, Studia Math. 116 (1995), no. 3, 283 – 294. · Zbl 0870.46049
[11]	Robin Harte and Woo Young Lee, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. 349 (1997), no. 5, 2115 – 2124. · Zbl 0873.47001
[12]	Kirti K. Oberai, On the Weyl spectrum, Illinois J. Math. 18 (1974), 208 – 212. · Zbl 0277.47002
[13]	Carl M. Pearcy, Some recent developments in operator theory, American Mathematical Society, Providence, R.I., 1978. Regional Conference Series in Mathematics, No. 36. · Zbl 0444.47001
[14]	Joseph G. Stampfli, Hyponormal operators, Pacific J. Math. 12 (1962), 1453 – 1458. · Zbl 0129.08701
[15]	H. Weyl, Über beschränkte quadratische Formen, deren Differenz vollsteig ist, Rend. Circ. Mat. Palermo 27 (1909), 373-392. · JFM 40.0395.01
[16]	Kung Wei Yang, Index of Fredholm operators, Proc. Amer. Math. Soc. 41 (1973), 329 – 330. · Zbl 0248.47017
[17]	Kung Wei Yang, The generalized Fredholm operators, Trans. Amer. Math. Soc. 216 (1976), 313 – 326. · Zbl 0297.47027