The weak summability dominion of a sequence S of the Hilbert space in relation with the set of linear bounded operators

Abstract

Let H be a separable, real Hilbert space, L(H) the Banach space of all bounded linear operators on H. For a given sequence (xn)n∈N⊆H with xn≠0 for all n∈N let C(xn):={T∈L(H):∑n∈NTxn<∞} and M(xn):={x∈H:∑ n∈N|(xn,x)|<∞}. The author studies injective (i.e. one-to-one, not necessarily invertible) operators, finite rank operators, and completely continuous operators in C(xn). The following results are shown: (1) C(xn) contains an injective operator if and only if M (xn)=H. (2) C(xn) is contained in the set of all finite rank operators on H if and only if the linear subspace M (xn)⊆H is of finite dimension. (3) C(xn) contains operators which are not completely continuous if and only if M(xn) contains an infinite-dimensional closed linear subspace of H. Finally it is proved that whenever all operators in C(xn) are completely continuous, they must necessarily be Hilbert-Schmidt operators.