Integr Equat Oper Th 0378-620X/92/040527-2451.50+0.20/0 Vol. 15 (1992) (c) 1992 Birkh~user Verlag, Basel LOCAL SEMIGROUPS OF ISOMETRIES IN H -SPACES AND RELATED CONTINUATION PROBLEMS FOR n-INDEFINITE TOEPLITZ KERNELS Ram6n Bruzual and Stefania Marcantognini It is shown that every ~-indefinite generalized Toeplltz kernel defined in a bounded interval has a n-indefinite generalized Toeplitz extension to the whole real axis. Some parametrizations of the sets of extensions and a non-uniqueness criterion are also obtained. As a tool, a theory of Local Semlgroups of Operators is carried over to PontrJagyn spaces. 1.INTRODUCTION If X is any non-vold set, an hermitian kernel F defined on X is said to be u-indefinite if for any positive integer n and any xl,...,x n E X, the form (I.i) ~.j=i F ( x , x j ) X ( X1'''''Xn 9 C ) has at most ~ negative squares and exactly ~ for some choice of n s ~ and x,,...,x n e X A positive definite ( p.d. ) kernel can be seen as a ~-indefinite kernel with ~ = 0, more precisely, a kernel F on X is p.d. if the form in (i.i) is positive for any n E ~ and any xi, . . .,x n 9 X 9 Let I = ~ or I = [-a,a] if 0 < a < +~ . A kernel F defined on I is said to be an ordinary Toeplltz kernel if there exists a continuous scalar-valued function f defined on I - I such that F(x,y) = f(x-y) for all x,y E I A well known theorem due to M. G. Krein (1944) says that if F is a p.d. Toeplitz kernel defined on an interval