Pontryagin Spaces and Operator Colligations

Abstract

After a review of reproducing kernel Pontryagin spaces, it is shown in §1.1 that a holomorphic kernel has the same number of negative squares for every region of analyticity. Background on colligations and their characteristic functions is presented in §1.2. Results from operator theory on Julia operators, the indices of a selfadjoint operator, and contractions are discussed in §1.3. An important result in §1.4 gives conditions that the closure of a linear relation is the graph of a continuous operator. In §1.5, the complementation properties of contractively contained spaces are used to show that, in natural situations, sums and differences of reproducing kernels in the indefinite case behave as in the nonnegative case, provided that suitable index conditions hold.