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M. Gürdal 《Expositiones Mathematicae》2009,27(2):153-160
In the present paper we consider the Volterra integration operator V on the Wiener algebra W(D) of analytic functions on the unit disc D of the complex plane C. A complex number λ is called an extended eigenvalue of V if there exists a nonzero operator A satisfying the equation AV=λVA. We prove that the set of all extended eigenvalues of V is precisely the set C?{0}, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V. The similar result for some weighted shift operator on ?p spaces is also obtained. 相似文献
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We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables x1,x2,…∈D. The setting is based on a reproducing kernel k for functions on D, a family of non-negative weights γu, where u varies over all finite subsets of N, and a probability measure ρ on D. We consider the weighted superposition K=∑uγuku of finite tensor products ku of k. Under mild assumptions we show that K is a reproducing kernel on a properly chosen domain in the sequence space DN, and that the reproducing kernel Hilbert space H(K) is the orthogonal sum of the spaces H(γuku). Integration on H(K) can be defined in two ways, via a canonical representer or with respect to the product measure ρN on DN. We relate both approaches and provide sufficient conditions for the two approaches to coincide. 相似文献