Joint Functional Calculus for Definitizable Tuples of Self-Adjoint Krein Space Operators

Abstract

The indefiniteness of Krein spaces gives rise to substantial complications. For instance, bounded self-adjoint linear Krein space operators are not well-behaved enough to allow for an appropriate analogue of the Spectral Theorem. To overcome this, classical literature imposes the additional assumption of definitizability. In the present work, we extend the notion of definitizability to tuples of pairwise commuting bounded self-adjoint operators and formulate the Spectral Theorem, expressed as a joint functional calculus, for definitizable tuples of Krein space operators. The definitizability of a tuple is a significantly weaker assumption than requiring each operator in the tuple to be definitizable. The constructed functional calculus will produce the zero operator if applied to a function that vanishes on the joint spectrum of the respective operator tuple. Moreover, while the construction of the functional calculus is based on the choice of generators of the smallest ideal containing all definitizing polynomials of the respective operator tuple, it will be shown that the resulting functional calculus is not affected by that choice. Finally, the functional calculus will be compatible with the functional calculus of subtuples via the canonical projection.