Algebra homomorphisms defined via convoluted semigroups and cosine functions |
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Authors: | Valentin Keyantuo Pedro J Miana |
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Institution: | a University of Puerto Rico, Department of Mathematics, Faculty of Natural Sciences, PO Box 23355, PR 00931, USA b Universidad de Santiago de Chile, Departamento de Matemática, Facultad de Ciencias, Casilla 307, Correo 2, Santiago, Chile c Universidad de Zaragoza, Departamento de Matemáticas & I.U.M.A., 50.009, Zaragoza, Spain |
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Abstract: | Transform methods are used to establish algebra homomorphisms related to convoluted semigroups and convoluted cosine functions. Such families are now basic in the study of the abstract Cauchy problem. The framework they provide is flexible enough to encompass most of the concepts used up to now to treat Cauchy problems of the first- and second-order in general Banach spaces. Starting with the study of the classical Laplace convolution and a cosine convolution, along with associated dual transforms, natural algebra homomorphisms are introduced which capture the convoluted semigroup and cosine function properties. These correspond to extensions of the Cauchy functional equation for semigroups and the abstract d'Alembert equation for the case of cosine operator functions. The algebra homomorphisms obtained provide a way to prove Hille-Yosida type generation theorems for the operator families under consideration. |
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Keywords: | Convolution transform k-Convoluted families Algebra homomorphisms Pseudo-resolvents |
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