Relations between convolution type operators on intervals and on the half-line (original) (raw)
2000, Integral Equations and Operator Theory
Dedicated to Ant6nio Ferreira dos Santos on the occasion of his 60th birthday. This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces, H ~,p, which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is that it provides explicit operator matrix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of properties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders: s -lip r 7Z. For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented. = c H ,P: supp c (1.5) H"P(f~) = raY "'p. (1.6) *This research was supported by Junta Nacional de Investigaq~m Cientffica e TecnolSgica (Portugal) and the Bundesminister fiir Forschung und Technologie (Germany) within the project Singular Operators -new features and applications, and by a PRAXIS XXI project under the title Factorization of Operators and Applications to Mathematical Physics.
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