Relations between convolution type operators on intervals and on the half-line

This paper is devoted to the question to obtain (algebraic and topologic) e quivalence (after extension) relations between convolution type operators o n unions of intervals and convolution type operators on the half-line. Thes e operators are supposed to act between Bessel potential spaces, H-s,H-p, w hich are the appropriate spaces in several applications. The present approa ch is based upon special properties of convenient projectors, decomposition s and extension operators and the construction of certain homeomorphisms be tween the kernels of the projectors. The main advantage of the method is th at it provides explicit operator matrix identities between the mentioned op erators where the relations are constructed only by bounded invertible oper ators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relatio n and the Bastes-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 - 1/p is not an element of Z. For instance, (generalized) inverses of the operators are explicitly rep resented in terms of operator matrix factorization. Some applications are p resented.