Semi-Browder operators
by
Vladimir Rakočević
University of Nis

Recall that an operator T in B(X) is semi-Fredholm if R(T) is closed and at least one of \alpha(T) and \beta(T) is finite. Let \Phi₊(X)    (\Phi_-(X)) denote the set of upper (lower) semi-Fredholm operators, i.e., the set of semi-Fredholm operators with \alpha(T) < \infty    (\beta(T) < \infty). It is well known that \Phi₊(X) and \Phi_-(X) are open semigroups in B(X) ([1, 3]). Recall that a(T) ( d(T) ), the ascent (descent) of T in B(X), is the smallest non-negative integer n such that N(Tⁿ) = N(Tⁿ⁺¹)  (R(Tⁿ) = R(Tⁿ⁺¹)). If no such n exists, than a(T) = \infty ( d(T) = \infty ). An operator T is called upper semi-Browder if T in \Phi₊(X) and a(T) < \infty; T is called lower semi-Browder if T in \Phi_-(X) and d(T) < \infty [3, Definition 7.9.1]. Let \Cal B₊(X)    (\Cal B_-(X)) denote the set of upper (lower) semi-Browder operators. An operator in a Banach space is called semi-Browder if it is upper semi-Browder or lower semi-Browder. Semi-Browder operators were studied by many authors; see e.g. ( [2, 3, 4, 5, 6, 8, 9, 10, 11, 12]). The name was introduced in [3]. An operator T is Browder if it is both upper semi-Browder and lower semi-Browder [3, Definition 7.7.1]. Let \Cal B(X) denote the set of Browder operators, i.e., \Cal B(X)=\Cal B₊(X) \cap \Cal B_-(X). Let us recall that \Cal B₊(X) and \Cal B_-(X) are open subsets in B(X) [10, Satz 4], but not stable under finite-rank perturbations [1, pp. 13-14].

The set of upper (lower) semi-Browder operators and Browder operators define, respectively, the corresponding spectra, i.e., for T in B(X) set

\sigmaab(T) = {\lambda in C: T - \lambdaI not in \Cal B+(X)}, \sigmadb(T) = {\lambda in C: T - \lambdaI not in \Cal B-(X) }, \sigmaeb(T) = {\lambda in C: T - \lambdaI not in \Cal B(X)}.

We survey some recent results on the perurbations of the semi-Browder operators, semi-Browder operator spectra and some applications to semiregular (essentialy semiregular) operators ([6, 7, 9]). Let us point up that the notation of semi-Browder operators were extended by V. Kordula, V. Müller and V. Rakocevi\' c [5] to n-tuples of commuting operators.

References

1.

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2.

S. Grabiner, Ascent, descent, and compact perturbations, Proc. Amer. Math. Soc., 71(1978), 79-80.

3.

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4.

M. A. Kaashoek and D. C. Lay, Ascent, descent, and commuting perturbations, Trans. Amer. Math. Soc., 186 (1972), 35-47.

5.

V. Kordula, V. Müller V. Rakocevi\' c On the semi-Browder spectrum, Studia Math., 123 (1997), 1-13.

6.

H. Kroh and P. Volkmann, Störungssätze für Semifredholmoperatoren, Math. Z. 148(1976), 295-297.

7.

V. Müller, On the regular spectrum J. Operator Theory, 31 (1994), 363-380.

8.

V. Rakocevi\' c, Approximate point spectrum and commuting compact perturbations, Glasgow Math. J. 28 (1986), 193-198.

9.

V. Rakocevi\' c, Generalized spectrum and commuting compact perturbations, Proc. Edinb. Math. Soc. 36(1993), 197-209.

10.

V. Rakocevi\' c, Semi-Fredholm operators with finite ascent or descent and perturbations, Proc. Amer. Math. Soc. 123 (1995), 3823-3828.

11.

V. Rakocevi\' c, Semi-Browders operators and perturbations, Studia Math., 122 (1997), 131-137.

12.

J. Zem\' anek, Compressions and the Weyl-Browder spectra, Proc. R. Irish Acad. 86A (1986), 57-62.

Date received: June 19, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-78.