Notes on Infinite Determinants of Hilbert Space Operators

This note represents an approach to the abstract Fredholm theory of trace class (and more generally ~ = {A [ Tr(] A [ ~ ) < oo}) operators on a separable Hi lber t space, ~v{,. There are few new results here but there are a set of new proofs which we feel sheds considerable light on the theory discussed. In part i cular, we would emphasize our proof of Lidskii 's theorem (see Sect. 4): I t was this new proof that motivated our more general discussion here. To help emphasize the differences between our approach and others, we remark on the differences in the definition of the infinite determinant det(1 -IA) for trace class A. First, some notations (formal definitions of algebraic multiplicity, c A rA','tN(A) etc., appear later): Given a compact operator, A, ~ ~ Js~=t ( N ( A ) = 1, 2 ..... or ~ ) is a listing of all the nonzero eigenvalues of A, counted up to algebraic c~ multiplicity and {/~/(A)}~=I, the singular values of A, i.e., eigenvalues of I A L (A 'A)1~ 2 listed so that /~I(A) >//~2(A) >/ -" > /0 . Throughout , the trace of an operator in the trace class is defined by