Frames and bases of subspaces in Hilbert spaces

In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {Wi}_i∈I for a Hilbert space H, there exists a Hilbert space K⊇H and an orthonormal basis of subspaces {Ni}_i∈I for K such that Wi=P(Ni), where P is the orthogonal projection of K onto H. We introduce a new definition of atomic resolution of the identity in Hilbert spaces. In particular, we define an atomic resolution operator for an atomic resolution of the identity, which even yield a reconstruction formula.     

Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

Let S be a pre-Hilbert space. We study quasi-splitting subspaces of S and compare the class of such subspaces, denoted by Eq(S), with that of splitting subspaces E(S). In [D. Buhagiar, E. Chetcuti, Quasi splitting subspaces in a pre-Hilbert space, Math. Nachr. 280 (5-6) (2007) 479-484] it is proved that if S has a non-zero finite codimension in its completion, then Eq(S)≠E(S). In the present paper it is shown that if S has a total orthonormal system, then Eq(S)=E(S) implies completeness of S. In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a pre-Hilbert space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system.     

The subspaces and orthonormal bases of symmetry classes of tensors

We present here the dimensions of some subspaces in the symmetry classes of tensors and some methods for constructing orthonormal bases of the symmetry classes of tensors.     

Block bases of the Haar system as complemented subspaces of

It is shown that the span of , where is the Haar system in and the canonical basis of , is well isomorphic to a well complemented subspace of . As a consequence we get that there is a rearrangement of the (initial segments of the) Haar system in , any block basis of which is well isomorphic to a well complemented subspace of .

     

Boundedness of adjoint bases of approximate spectral subspaces and of associated block reduced Resolvents

Block reduced resolvents are often employed in iterative schemes for refining crude approximations of the arithmetic mean of a cluster of eigenvalues and of a basis of the corresponding spectral subspace. We prove that if the bases of approximate spectral subspaces are chosen in such a way that they are bounded and each element of the basis is bounded away from the span of the previously chosen elements, then the corresponding adjoint bases are also bounded. We give an integral representation of the associated block reduced resolvent and show that under such a choice of the bases, the approximate block reduced resolvents are bounded as well. This is crucial in obtaining error estimates for the iterates of several refinement schemes. In the framework of a canonical discretization procedure for finite rank operators, appropriate choices of ises are given for various finite rank approximation methods such as Projection, Sloan, Galerkin, Nyström, Fredholm, Degenerate kernel. If the bases are not chosen appropriately, the error estimates may no longer hold and the iteration scheme may not be numerically stable. Examples are given to illustrate these phenomena     

On closed subspaces with Schauder bases in non-archimedean Fréchet spaces

The main purpose of this paper is to prove that a non-archimedean Fréchet space of countable type is normable (respectively nuclear; reflexive; a Montel space) if and only if any its closed subspace with a Schauder basis is normable (respectively nuclear; reflexive; a Montel space). It is also shown that any Schauder basis in a non-normable non-archimedean Fréchet space has a block basic sequence whose closed linear span is nuclear. It follows that any non-normable non-archimedean Fréchet space contains an infinite-dimensional nuclear closed subspace with a Schauder basis. Moreover, it is proved that a non-archimedean Fréchet space E with a Schauder basis contains an infinite-dimensional complemented nuclear closed subspace with a Schauder basis if and only if any Schauder basis in E has a subsequence whose closed linear span is nuclear.     

Covering of subspaces by subspaces

Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph \(\mathcal{G }_q(n,r)\) by subspaces from the Grassmann graph \(\mathcal{G }_q(n,k)\) , \(k \ge r\) , are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, \(q\) -analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and a recursive construction. New constructions are given for \(q=2\) with \(r=2\) or \(r=3\) . We discuss the density for some of these coverings. Tables for the best known coverings, for \(q=2\) and \(5 \le n \le 10\) , are presented. We present some questions concerning possible constructions of new coverings of smaller size.     

Perturbation analysis of singular subspaces and deflating subspaces

Summary. Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Received July 26, 1994     

Approximating weak Chebyshev subspaces by Chebyshev subspaces

We examine to what extent finite-dimensional spaces defined on locally compact subsets of the line and possessing various weak Chebyshev properties (involving sign changes, zeros, alternation of best approximations, and peak points) can be uniformly approximated by a sequence of spaces having related properties.     

Interior-compact subspaces and differentiation in model subspaces

We discuss relations between Koosis’ theorem on interior-compact subspaces of the space L²(0, ∞) and recent Dyakonov’s results on differentiation in model subspaces of the Hardy class H² in the upper half-plane. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 327, 2005, pp. 17–24.     

Zero-based subspaces and quasi-invariant subspaces of the Bargmann-Fock space

We use the Weierstrass σ-function associated with a lattice in the complex plane to construct finite dimensional zero-based subspaces and quasi-invariant subspaces of given index in the Bargmann-Fock space.     

Realcompact subspaces

We show that every compact space of large enough size has a realcompact subspace of size κ, for κ?c. We also show that an uncountable realcompact space whose pseudocharacter is at most ω₁, has a realcompact subspaces of size ω₁, thus, by continuum hypothesis, every uncountable realcompact space has realcompact subspace of size ω₁.     

多圆盘的加权Bergman空间上的不变子空间和约化子空间

本文主要研究多圆盘的加权Bergman 空间上的不变子空间和约化子空间, 给出了某些解析Toeplitz 算子的极小约化子空间的完全刻画, 以及一类解析Toeplitz 算子Tzi (1≤i≤n) 的不变子空间的Beurling 型定理.