A note of duality theorem for decomposable operators. (English) Zbl 0615.47030
Let X be a complex Banach space, \(X^*\) the conjugate space of X, T a bounded linear operator on X, \(T^*\) the conjugate operator of T. In this paper the author proves if \(T^*\) has the single valued extension property, F is a closed subset of the complex plane, then \(X^*_{T^*}(F)\) is weak star closed whenever it is closed in the strong topology. Furthermore the author gives a very simple proof of the duality theorem for decomposable operators on Banach spaces.
MSC:
| 47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |
Keywords:
single valued extension property; duality theorem for decomposable operators on Banach spacesReferences:
| [1] | St. Frunză, A duality theorem for decomposable operators,Rev. Roum. Math. Pures Appl.,16 (1971) 1055–1058. · Zbl 0227.47007 |
| [2] | Erdelyi, I. and Lange, R., Spectral decompositions on Banach spaces,Springer, 1977. · Zbl 0381.47001 |
| [3] | Dunford, N. and Schwartz, J., Linear operators, Part 1,Interscience, New York, 1971. · Zbl 0243.47001 |
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