Polar decompositions and related classes of operators in spaces (n-ary product)(kappa)

Polar decompositions with respect to an indefinite inner product are studied for bounded linear operators acting on a (n-ary product) (kappa) space. Criteria are given for existence of various forms of the polar decompositions, under the conditions that the range of a given operator X is closed and that zero is not an irregular critical point of the selfadjoint operator X[*]X. Both real and complex spaces (n-ary product)(kappa) are considered. Relevant classes of operators having a selfadjoint (in the sense of the indefinite inner product) square root, or a selfadjoint logarithm, are characterized.