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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.