How good can a nonhereditary family be?

Abstract

A family of vectors of a Hilbert space H is said to be hereditarily complete, if it has biorthogonal\(\mathfrak{X}'\) (minimally) and any element of H can be reconstructed from its Fourier series:. In this paper we describe all pairs of spaces A, B, which contain minimal mutually biorthogonal and complete families\(\mathfrak{X}, \mathfrak{X}' ( V\left( \mathfrak{X} \right) = A, V\left( {\mathfrak{X}'} \right) = B\) and: for this it is necessary and sufficient that the operator PAPBPA not be completely continuous. This assertion allows one to prove that: 1) if dn > 0,, then there exist an orthonormal basis {ϕn}v⩾1 and complete but not hereditarily complete biorthogonal families\(\mathfrak{X}, \mathfrak{X}'\) in H, such that ∥ Xn-ϕn∥⩽dn, ∥x′n-ϕn∥⩽ dr, (n⩾1), 2) if, then there exist families of the type described in the preceding assertion for which, where σ is any finite set of natural numbers and is the spectral projector corresponding to it. One of the auxiliary assertions is the description of all real collections α=(αk)n k=1, representable in the form, where q is a Hilbert seminorm defined in the Euclidean space En, {fk)n k=1 is a suitable orthonormal basis. This set is the convex hull of all permutations of the eigenvalues (λ1, ..., λn) of the seminorm q.