The single-valued extension property and spectral manifolds

Sun, Shan Li

doi:10.1090/S0002-9939-1993-1156474-0

The single-valued extension property and spectral manifolds
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by Shan Li Sun

Proc. Amer. Math. Soc. 118 (1993), 77-87

DOI: https://doi.org/10.1090/S0002-9939-1993-1156474-0

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Abstract:

We discuss the relation between the single-valued extension property (that is, Dunford’s property (A)) and spectral manifolds ${X_T}(F)$ of a bounded linear operator. In particular, we prove that Dunford’s property (C) implies the property (A). We also prove that if $T \in B(X)$ has the property $({\beta ^{\ast }})$ introduced by Fong, then $X_{{T^{\ast }}}^{\ast }(F) = {X_T}{(\mathbb {C}\backslash F)^ \bot }$ for every closed set $F$ in the complex plane $\mathbb {C}$.

References

Errett Bishop, A duality theorem for an arbitrary operator, Pacific J. Math. 9 (1959), 379–397. MR 117562
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
C. K. Fong, Decomposability into spectral manifolds and Bishop’s property $(\beta )$, Northeast. Math. J. 5 (1989), no. 4, 391–394. MR 1053518
Ştefan Frunză, A duality theorem for decomposable operators, Rev. Roumaine Math. Pures Appl. 16 (1971), 1055–1058. MR 301552
Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
Shan Li Sun, Decomposability of weighted shift operators and hyponormal operators on Hilbert spaces, Chinese Ann. Math. Ser. A 5 (1984), no. 5, 575–584 (Chinese). An English summary appears in Chinese Ann. Math. Ser. B 5 (1984), no. 4, 742. MR 794925