Characterization of Bessel sequences

Let H be a separable Hilbert space, L(H) be the algebra of all bounded linear operators of H and Bess(H) be the set of all Bessel sequences of H. Fixed an orthonormal basis E = {ek}kÎN of H, a bijection aE: Bess(H) ? L(H) can be defined. The aim of this paper is to characterize a-1E (A) for different classes of operators A Í L(H). In particular, we characterize the Bessel sequences associated to injective operators, compact operators and Schatten p-classes