THE TRACE IN BANACH JORDAN PAIRS

Abstract

The algebraic trace form (as defined by O. Loos) of an element (x, y) of a (complex) Banach Jordan pair V, where x or y is in the socle, is equal to the sum of the products of all spectral values and their multiplicity. The trace form is calculated for two examples, the Banach Jordan pair of bounded linear operators between two Banach spaces, and the Banach Jordan pair of a quadratic form. Using analytic multifunctions, it is also shown that the complement of the socle of a Banach Jordan pair V is either dense or empty. In the last case, V has finite capacity.