Products of orthogonal projections

We give a characterization of operators on a separable Hilbert space of norm less than one that can be represented as products of orthogonal projections and give an estimate on the number of factors. We also describe the norm closure of the set of all products of orthogonal projections.

     

Boundary scalar controllability in projections for the Navier-Stokes system

We consider the following controllability problem for the Navier-Stokes equations: ensure, by dynamically varying the boundary values of the total fluid head, that the solution at a given time has a given projection onto some finite-dimensional subspace. We suggest a feedback control construction and obtain a priori estimates for the solution of the closed-loop system, which are then used to prove the boundary scalar controllability.     

Multicommutators and multianticommutators of orthogonal projections

We establish sufficient and necessary conditions for an n?×?n Hermitian matrix A to be a multicommutator or a multianticommutator of orthoprojections.     

The product of affine orthogonal projections

In the case of a finite number of subspaces in a given Hilbert space, by a theorem of J. von Neumann, the iteration of the product of projectors is always convergent. In a finite dimensional Hilbert space, this theorem has been generalized for affine subspaces. In this paper we construct an example which shows that this result does not hold in the infinite dimensional case.     

On perturbation bounds for orthogonal projections

In this paper, we present some new perturbation bounds for the orthogonal projections onto the column and row spaces of a matrix, which improve some existing results. Numerical examples are presented to illustrate our results.     

Bessel sequences as projections of orthogonal systems

We prove generalizations of the Schur and Olevskii theorems on the continuation of systems of functions from an interval I to orthogonal systems on an interval J, I ? J. Only Bessel systems in L ²(I) are projections of orthogonal systems from the wider space L ²(J). This fact allows us to use a certain method for transferring the classical theorems on the almost everywhere convergence of orthogonal series (the Men’shov-Rademacher, Paley-Zygmund, and Garcia theorems) to series in Bessel systems. The projection of a complete orthogonal system from L ²(J) onto L ²)(I) is a tight frame, but not a basis.     

Products of orthogonal projections and polar decompositions

We characterize the sets X of all products PQ, and Y of all products PQP, where P,Q run over all orthogonal projections and we solve the problems argmin{‖P-Q‖:(P,Q)∈Z}, for Z=X or Y. We also determine the polar decompositions and Moore-Penrose pseudoinverses of elements of X.     

Products of orthogonal projections as Carleman operators

In this paper, we consider the product of two orthogonal projectionsP andQ on a separable, infinite dimensional Hilbert spaceH. For the operatorQP, there holds the dichotomy:QP is either a Carleman operator or a semi-Fredholm operator with finite defect. Both cases are characterized in terms of the dimensions of the ranges and null spaces ofP andQ and some of their intersections. This extends the case, whereP andQ are the special projections onto the subspaces of time- and band-limited functions inL ²() resp., first considered by Slepian, Pollak and Landau.     

Polytopes which are orthogonal projections of regular simplexes

We consider the polytopes which are certain orthogonal projections of k-dimensional regular simplexes in k-dimensional Euclidean space R ^k . We call such polytopes -polytopes. Every sufficiently symmetric polytope, such as a regular polytope, a quasi-regular polyhedron, etc., belongs to this class. We denote by P _m,n all n-dimensional -polytopes with m vertices. We show that there is a one-to-one correspondence between the elements of P _m,n and those of P _m,m–n–1 and that this correspondence preserves the symmetry of -polytopes. Using this duality, we determine some of the P _m,n 's. We also show that a -polytope is an orthogonal projection of a cross polytope if and only if it has central symmetry.     

Subspaces,angles and pairs of orthogonal projections

We give a new proof for the Wedin theorem on the simultaneous unitary similarity transformation of two orthogonal projections and show that it is equivalent to Halmos' theorem on the unitary equivalence of projection pairs. As a consequence of these theorems, we derive several results on pairs of orthogonal projections, relative subspace positions and oblique projections as well.     

Discrete orthogonal projections on multiple knot periodic splines

This paper establishes properties of discrete orthogonal projections on periodic spline spaces of order r, with knots that are equally spaced and of arbitrary multiplicity Mr. The discrete orthogonal projection is expressed in terms of a quadrature rule formed by mapping a fixed J-point rule to each sub-interval. The results include stability with respect to discrete and continuous norms, convergence, commutator and superapproximation properties. A key role is played by a novel basis for the spline space of multiplicity M, which reduces to a familiar basis when M=1.     

The Drazin inverses of products and differences of orthogonal projections

In this note, the Drazin inverses of products and differences of orthogonal projections on a Hilbert space are established.     

Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

The proper orthogonal decomposition(POD)and the singular value decomposition(SVD) are used to study the finite difference scheme(FDS)for the nonstationary Navier-Stokes equations. Ensembles of data are compiled from the transient solutions computed from the discrete equation system derived from the FDS for the nonstationary Navier-Stokes equations.The optimal orthogonal bases are reconstructed by the elements of the ensemble with POD and SVD.Combining the above procedures with a Galerkin projection approach yields a new optimizing FDS model with lower dimensions and a high accuracy for the nonstationary Navier-Stokes equations.The errors between POD approximate solutions and FDS solutions are analyzed.It is shown by considering the results obtained for numerical simulations of cavity flows that the error between POD approximate solution and FDS solution is consistent with theoretical results.Moreover,it is also shown that this validates the feasibility and efficiency of POD method.     

Algebras of singular integral operators generated by three orthogonal projections

The structure of theC ^*-algebra with identity generated by two orthogonal projections is well understood. All irreducible representations of this algebra are either two-dimensional or one-dimensional. The situation becomes unpredictable in the case of theC ^*-algebra generated by three orthogonal projections. Even in the more specific case when two of the projections commute, the algebra under consideration may have infinite dimensional irreducible representations.In this paper we produce three concrete realizations of the algebra generated by three orthogonal projections in that specific case. It turns out that these algebras have quite different structure.This work was partially supported by CONACYT Project 4069-E9404, México.     

The strong closure of $\sigma$-complete Boolean algebras of projections

A result of T. A. Gillespie implies that the strong operator closure of any abstractly s\sigma -complete Boolean algebra of projections in a Banach space X which does not contain a copy of c₀ is Bade complete. It is shown that the same conclusion is valid for another (extensive) class of Banach spaces X, namely those which are weakly compactly generated. As a consequence, it follows that a Boolean algebra of projections in a separable Banach space is abstractly s\sigma -complete iff it is abstractly complete. It is also shown that a Banach space X has the property that the strong closure of every abstractly complete Boolean algebra of projections in X is Bade complete iff X does not contain a copy of l^￥\ell ^\infty \!.     

On the spectra of sums of orthogonal projections with applications to parallel computing

Many parallel iterative algorithms for solving symmetric, positive definite problems proceed by solving in each iteration, a number of independent systems on subspaces. The convergence of such methods is determined by the spectrum of the sums of orthogonal projections on those subspaces, while the convergence of a related sequential method is determined by the spectrum of the product of complementary projections. We study spectral properties of sums of orthogonal projections and in the case of two projections, characterize the spectrum of the sum completely in terms of the spectrum of the product.This work was supported in part by the Norwegian Research Council for Science and the Humanities under grant D.01.08.054 and by The Royal Norwegian Council for Scientific and Industrial Research under grant IT2.28.28484; also supported in part by the Air Force Office of Scientific Research under grant AFOSR-86-0126 and by the National Science Foundation under grant DMS-8704169.