Denseness of the generalized eigenvectors of an H-S discrete operator

Abstract

Let T be a closed densely defined linear operator in a Hilbert space H, and assume there exists ξ 0 ϵ p( T) such that R ξ0 ( T) is a Hilbert-Schmidt operator. The operator T is a special type of discrete operator, a so-called H-S discrete operator, which is shown to be a Fredholm operator in H with Fredholm set equal to the whole complex plane. Let σ( T) = { λ i } i = 1 ∞where the operator T λ i has ascent m i , let P i , i = 1, 2,…, be the projection of H onto the generalized eigenspace N [[T λ i] m i] along R [[T λ i] m i], and let S ∞ and M ∞ be the subspaces of H consisting of all x ϵ H such that x = ∑ i = 1 ∞ P i x and such that P i x = 0 for i = 1, 2,…, respectively. Sufficient conditions are introduced which guarantee that S ∝ = H and M ∞ = {0} . These conditions require that ∥ R λ ( T)∥ be bounded on certain rays in the complex plane and ∥ R λ ( T)∥ → 0 as λ → ∞ on at least one of the rays, but specific decay rates for ∥ R λ ( T)∥ are not necessary.