Characterization of the closedness of the sum of two shift-invariant spaces

We first present a formula for the supremum cosine angle between two closed subspaces of a separable Hilbert space under the assumption that the ‘generators’ form frames for the subspaces. We then characterize the conditions that the sum of two, not necessarily finitely generated, shift-invariant subspaces of L²(Rd) be closed. If the fibers of the generating sets of the shift-invariant subspaces form frames for the fiber spaces a.e., which is satisfied if the shift-invariant subspaces are finitely generated or if the shifts of the generating sets form frames for the respective subspaces, then the characterization is given in terms of the norms of possibly infinite matrices. In particular, if the shift-invariant subspaces are finitely generated, then the characterization is given wholly in terms of the norms of finite matrices.     

Affiliated subspaces and infiniteness of von Neumann algebras

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.     

An alternative to the Ritz pairs with the EN subspace method

In this note we define EN subspaces by using the Eirola-Nevanlinna algorithm for solving a linear system. We compare this construction with the Arnoldi method for generating Krylov subspaces and computing eigenvalue approximations. Further, we compute Ritz pairs by restricting the updated preconditionerH _k of the EN algorithm to the generated EN subspaces.     

Hypercyclic Subspaces on Fréchet Spaces Without Continuous Norm

Known results about hypercyclic subspaces concern either Fréchet spaces with a continuous norm or the space ω. We fill the gap between these spaces by investigating Fréchet spaces without continuous norm. To this end, we divide hypercyclic subspaces into two types: the hypercyclic subspaces M for which there exists a continuous seminorm p such that ${M \cap {\rm ker} p = \{0\}}$ and the others. For each of these types of hypercyclic subspaces, we establish some criteria. This investigation permits us to generalize several results about hypercyclic subspaces on Fréchet spaces with a continuous norm and about hypercyclic subspaces on ω. In particular, we show that each infinite-dimensional separable Fréchet space supports a mixing operator with a hypercyclic subspace.     

Subspaces of non-commutative spaces

This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a non-commutative space.

     

Banach spaces without minimal subspaces

We prove three new dichotomies for Banach spaces à la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ₁ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability.     

Some results on best L1-approximation of continuous functions

The chief purpose of this paper is to study the question of uniqueness of best L¹-approximations for weighted approximation of continuous vector-valued functions and continuous functions of several variables by finite dimensional subspaces G. A uniqueness result for special subspaces G of the type G=G¹, ×…×G^m is given. Moreover It is studied whether a sufficient condition for uniqueness introduced by DeVore and Strauss and in a more general context by Kroó is also necessary or not and a class of finite dimensional subspaces G for which a positive answer can be given is presented. Finally it is shown that subspaces of linear spline functions of two variables fail to have that sufficient property.     

Unicity of best one-sided L1-approximations for certain classes of spline functions

The chief purpose of this paper is to present conditions ensuring uniqueness of best one-sided L¹-approximations for approximation by finite dimensional subspaces of differentiable functions. Using these results it is shown that uniqueness of such best approximations will hold for a class of generalized spline subspaces and also for spline subspaces satisfying certain boundary conditions. These considerations have an important application to uniqueness of quadrature formulae of “highest possible degree of precision”.     

Accelerating the convergence of the method of alternating projections

The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an ``accelerated' version of this algorithm that was considered earlier by Gubin, Polyak, and Raik (1967) and by Gearhart and Koshy (1989). We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is, in general, not faster than the MAP for more than two subspaces! However, for a ``symmetric' version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for selfadjoint mappings.

     

On perturbations of some constrained subspaces

Perturbation bounds of subspaces, such as eigen-spaces, singular subspaces, and canonical subspaces, have been extensively studied in the literature. In this paper, we study perturbations of some constrained subspaces of 1×2, 2×1, and 2×2 block matrices, in which only one of the sub-matrices can be changed. Such problems rise from the least squares–total least squares problem, the constrained least squares problem, and the constrained total least squares problem.     

Hilbert polynomials and Arveson's curvature invariant

We define and study Hilbert polynomials for certain holomorphic Hilbert spaces. We obtain several estimates for these polynomials and their coefficients. Our estimates inspire us to investigate the connection between the leading coefficients of Hilbert polynomials for invariant subspaces of the symmetric Fock space and Arveson's curvature invariant for coinvariant subspaces. We are able to obtain some formulas relating the curvature invariant with other invariants. In particular, we prove that Arveson's version of the Gauss-Bonnet-Chern formula is true when the invariant subspaces are generated by any polynomials.     

Theory of subdualities

We present a new theory of dual systems of vector spaces that extends the existing notions of reproducing kernel Hilbert spaces and Hilbert subspaces. In this theory, kernels (understood as operators rather than kernel functions) need not be positive or self-adjoint. These dual systems, called subdualities, enjoy many properties similar to those of Hilbert subspaces and include the notions of Hilbert subspaces or Kreîn subspaces as particular cases. Some applications to Green operators or invariant subspaces are given.     

Controlled and conditioned invariant subspaces in linear system theory

The concept of invariance of a subspace under a linear transformation is strongly connected with controllability and observability of linear dynamical systems. In this paper, we definecontrolled andconditioned invariant subspaces as a generalization of the simple invariants, for the purpose of investigating some further structural properties of linear systems. Moreover, we prove some fundamental theorems on which the computation of the above-mentioned subspaces is based. Then, we give two examples of practical application of the previous concepts concerning the determination of the constant output and perfect output controllability subspaces.     

Homogeneity Properties of Some ℓ1-Spaces

Let M be a family of subspaces of a metric space X. The space X is M-homogeneous if any isometry between two subspaces in M can be extended to an isometry of X. In the case when M is the family of all subspaces of X, the space X is said to be fully homogeneous. We show that the space Lⁿ, where L is a fully homogeneous subspace of ℝ, is R-homogeneous where R is the family of subspaces that are products of subsets of L.     

On pseudo-complemented subspaces of Banach spaces

As natural generalizations of complemented subspaces, we introduce pseudo-complemented and strongly pseudo-complemented subspaces of Banach spaces, and we study for such subspaces some of the problems analogous to the classical problems on complemented and quasi-complemented subspaces.     

由给定的子空间构造新的子空间

本给出并证明了若干个子空间的并以及两个子空间的基构成子空间的充要条件，从而本质地揭示了除子空间的交与和是构造新的予空间的方法外，集合的其它运算不能构造新的子空间，最后分析了子空间直和的两种不同定义的优缺点，指出了张禾瑞教材中子空间直和定义推广时应注意的一个问题。     

Covering compacta by discrete and other separated sets

We show that if a space X is the union of not more than κ-many discrete subspaces, where κ is an infinite cardinal, then the same holds for any perfect image of X. It follows that a compact Hausdorff space with no isolated points can never be covered by fewer than continuum many discrete subspaces; this answers a question of I. Juhász and J. van Mill. We also consider coverings by right-separated and left-separated subspaces.     

Resolvent convergence and spectral approximations of sequences of self-adjoint subspaces

This paper studies resolvent convergence and spectral approximations of sequences of self-adjoint subspaces (relations) in complex Hilbert spaces. Concepts of strong resolvent convergence, norm resolvent convergence, spectral inclusion, and spectral exactness are introduced. Fundamental properties of resolvents of subspaces are studied. By applying these properties, several equivalent and sufficient conditions for convergence of sequences of self-adjoint subspaces in the strong and norm resolvent senses are given. It is shown that a sequence of self-adjoint subspaces is spectrally inclusive under the strong resolvent convergence and spectrally exact under the norm resolvent convergence. A sufficient condition is given for spectral exactness of a sequence of self-adjoint subspaces in an open interval lacking essential spectral points. In addition, criteria are established for spectral inclusion and spectral exactness of a sequence of self-adjoint subspaces that are defined on proper closed subspaces.     

Localic subspaces and colimits of localic spaces

A spectral space is localic if it corresponds to a frame under Stone Duality. This class of spaces was introduced by the author (under the name ’locales’) as the topological version of the classical frame theoretic notion of locales, see Johnstone and also Picado and Pultr). The appropriate class of subspaces of a localic space are the localic subspaces. These are, in particular, spectral subspaces. The following main questions are studied (and answered): Given a spectral subspace of a localic space, how can one recognize whether the subspace is even localic? How can one construct all localic subspaces from particularly simple ones? The set of localic subspaces and the set of spectral subspaces are both inverse frames. The set of localic subspaces is known to be the image of an inverse nucleus on the inverse frame of spectral subspaces. How can the inverse nucleus be described explicitly? Are there any special properties distinguishing this particular inverse nucleus from all others? Colimits of spectral spaces and localic spaces are needed as a tool for the comparison of spectral subspaces and localic subspaces.