Spectral decomposition of a Hilbert space by a Fredholm operator
Given a Fredholm operator with countable spectrum in a Hilbert space, the authors characterize the possibility of decomposing the Hilbert space into a direct sum of all generalized eigenspaces and the intersection of the ranges of the corresponding iterated operators. This is a generalization of the well-known spectral decomposition with respect to finitely many poles. The main result says that such a decomposition is possible iff the sums of the spectral projections are uniformly bounded. The result is illustrated by a differential operator in \(L_ 2[0,1]\).
Fredholm operator with countable spectrum in a Hilbert space
decomposing the Hilbert space into a direct sum of all generalized eigenspaces
scientific article; zbMATH DE number 4082344