Two methods for avoiding hereditary completeness

Abstract

A family of vectors

of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {x_n′;n≥1}((x_n,x_k′)=δ_nk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′_n)x_n:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families

of the same type but already complete in H.. Clearly, in this case

cannot be hereditarily complete. The second method consists in considering linear deformation _n :n⩾1 of the orthogonal bases_n: n⩾1; here A is an unbounded operator of a special type.