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 P_AP_BP_A not be completely continuous. This assertion allows one to prove that: 1) if d_n > 0,

, then there exist an orthonormal basis {ϕ_n}_v⩾1 and complete but not hereditarily complete biorthogonal families$\mathfrak{X}, \mathfrak{X}'$ in H, such that ∥ X_n-ϕ_n∥⩽d_n, ∥x′_n-ϕ_n∥⩽ d_r, (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=(α_k)ⁿ _k=1, representable in the form

, where q is a Hilbert seminorm defined in the Euclidean space Eⁿ, {f_k)ⁿ _k=1 is a suitable orthonormal basis. This set is the convex hull of all permutations of the eigenvalues (λ₁, ..., λ_n) of the seminorm q.