Index formulae for subspaces of Kreîn spaces

Abstract

For a subspaceS of a Kreîn spaceK and an arbitrary fundamental decompositionK=K ⁻[+]K ⁺ ofK, we prove the index formula

$$\kappa ^ - \left( \mathcal{S} \right) + \dim \left( {\mathcal{S}^ \bot \cap \mathcal{K}^ + } \right) = \kappa ^ + \left( {\mathcal{S}^ \bot } \right) + \dim \left( {\mathcal{S} \cap \mathcal{K}^ - } \right)$$

where κ^±(S) stands for the positive/negative signature ofS. The difference dim(S∩K ⁻)−dim(S ^⊥∩K ⁺), provided it is well defined, is called the index ofS. The formula turns out to unify other known index formulac for operators or subspaces in a Kreîn space.