How good can a nonhereditary family be?

L. N. Dovbysh ; N. K. Nikol'skii ; V. N. Sudakov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 52-69

Résumé

A family of vectors of $\mathfrak X=\{x_n\}_{n\geqslant1}$ 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: $x\in V((x,x'_n)x_n:n\geqslant1)$. In this paper we describe all pairs of spaces $A$, $B$, which contain minimal mutually biorthogonal and complete families $\mathfrak X,\mathfrak X'$ ($V(\mathfrak X)=A$, $V(\mathfrak X')=B$ and $\sup_{n\geqslant1}\|x_n\|\cdot\|x'_n\|+\infty$: 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 $d_n>0$,$\sum_{n\geqslant}d_n^2==\infty$, then there exist an orthonormal basis $\{\varphi_n\}_{n\geqslant1}$ and complete but not hereditarily complete biorthogonal families $\mathfrak X$, $\mathfrak X'$ in $H$, such that $\|x_n-\varphi_n\|\leqslant d_n$, $\|x'_n-\varphi_n\|\leqslant d_n(n\geqslant1)$ 2) if $\omega(n)>0$, $\lim_n\omega(n)=+\infty$, then there exist families of the type described in the preceding assertion for which $|\mathscr P_\sigma|\leqslant c\omega(\operatorname{card}\sigma)$, where $\sigma$ is any finite set of natural numbers and $\mathscr P_\sigma x=\sum_{n\in\sigma}(x,x'_n)x_n$ is the spectral projector corresponding to it. One of the auxiliary assertions is the description of all real collections $\alpha=(\alpha_k)^n_{k=1}$, representable in the form $\alpha_k=q(f_k)$, $1\leqslant k\leqslant n$, where $q$ is a Hilbert seminorm defined in the Euclidean space $E^n$, $\{f_k\}^n_{k=1}$ is a suitable orthonormal basis. This set is the convex hull of all permutations of the eigenvalues $(\lambda_1,\dots,\lambda_n)$ of the seminorm $q$.