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Weighted Projections and Riesz Frames

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Weighted Projections and Riesz Frames Weighted projections and Riesz frames Jorge Antezana ∗ Gustavo Corach y Mariano Ruiz z Demetrio Stojanoff x Abstract Let be a (separable) Hilbert space and ek k 1 a fixed orthonormal basis of H f g ≥ . Motivated by many works on scaled projections, angles of subspaces and oblique H projections, we define and study the notion of compatibility between a subspace and the abelian algebra of diagonal operators in the given basis. This is used to refine previous work on scaled projections, and to obtain a new characterization of Riesz frames. Keywords: scaled projection, weighted projection, compatibility, angles, frames, Riesz frames. AMS classification: Primary 47A30, 47B15 1 Introduction Weighted projections (also called scaled projections) play a relevant role in a variety of least- square problems. Just as a sample of their applications, let us mention that they have been used in optimization (feasibility theory, interior point methods), statistics (linear regression, weighted estimation), signal processing (noise reduction). Associated to the projections, weighted pseudoinverses appear in applications. t 1 t Frequently, the weighted pseudoinverses take the forms (ADA )− AD,(ADA )yAD, 1 (ADA∗)− AD or (ADA∗)yAD, according to the hypothesis of invertibility or if the problem involves real or complex matrices. Analogous formulas hold for the corresponding weighted projections. In general D is a positive definite matrix and A is a full column rank matrix. ∗Depto. de Matem´atica, FCE-UNLP, La Plata, Argentina and IAM-CONICET (e-mail: an- [email protected]) yDepto. de Matem´atica, FI-UBA, Buenos Aires, Argentina and IAM-CONICET (e-mail: gco- rach@fi.uba.ar). Partially supported by CONICET (PIP 2083/00), Universidad de Buenos Aires (UBACYT X050) and ANPCYT (PICT03-09521) zDepto. de Matem´atica, FCE-UNLP, La Plata, Argentina and IAM-CONICET (e-mail: [email protected]) xDepto. de Matem´atica,FCE-UNLP, 1 y 50 (1900), La Plata, Argentina and IAM-CONICET (e-mail: [email protected]). Partially supported CONICET (PIP 2083/00), Universidad de La PLata (UNLP 11 X350) and ANPCYT (PICT03-09521) 1 In a series of papers, Stewart [31], O’Leary [28], Ben-Tal and Taboulle [4], Hanke and Neumann [22], Forsgren [18], Gonzaga and Lara [21], Forsgren and Sporre [19], Wei [35], [34], [33], [32] have studied and computed quantities of the type sup γ(D; A) ; D Γ k k 2 where Γ denotes a certain subset of positive definite invertible matrices and γ(D; A) is any of the weighted pseudoinverses mentioned above. The reader is referred to the papers by Forsgren and Sporre [18], [19] for excellent surveys on the history and motivations of the problem of estimating the supremum above. It should be said, however, that the references mentioned above only deal with the finite dimensional context. In order to deal with in- creasing dimensions or arbitrary large data sets, we also present the problem in an infinite dimensional Hilbert space. Moreover, we present a different approach to this theory, valid also in the finite dimen- sional context, based on techniques and results on generalized selfadjoint projections. Recall that, if D is a selfadjoint operator on a complex (finite or infinite dimensional) Hilbert space , another operator C on is called D-selfadjoint if C is Hermitian with respect to the HermitianH sesquilinear formH ξ; η = Dξ; η (ξ; η ); h iD h i 2 H i.e. if DC = C∗D. We say that a closed subespace of is compatible with D (or that the pair (D; ) is compatible) if there exists an D-selfadjointS H projection Q in with image . It is wellS known [10] that in finite dimensional spaces, every subspaceH is compatible withS any positive semidefinite operator D. In infinite dimensional spaces this is not longer true; however, every (closed) subspace is compatible with any positive invertible operator and, in general, compatibility can be characterized in terms of angles between certain closed subspaces of , e. g., the angle between and (D )?. If the pairH (D; ) is compatible, theS set of DS-selfadjoint projections onto may be S S infinite; anyway, a distinguished one denoted by PD; , can be defined and computed (see S [10] or section 2.2 below). The study of weighted projections in the finite dimensional case from the point of view of D-selfadjoint projections let us obtain simpler proofs of some known results; another advantage is that these proofs can be easily extended to more general settings which are also important in applications, such as projections with complex weights and the infinite dimensional case. Moreover, this approach makes clear the relation among the quantities that have appeared in different works on weighted projection (usually, operator norms, vector norms and angles). A well known result due to Ben-Tal and Teboulle states that the solutions of the weighted least squares problems lies in the convex hull of solutions to some non-singular squares subsystems. We refer the reader to Ben-Tal and Teboulle’s paper [4], or [18], [35] for the following formulation: let A be an m n matrix of full rank. Denote by J(A) the set of all × orthogonal diagonal projections of m m such that QA : Cn R(Q) is bijective. Then, for every m m positive diagonal matrix×D, ! × 2 1 det(DQ) det(AQ) 1 ∗ − ∗ − A(A DA) A D = j j 2 A(QA) Q (1) det(DP ) det(AP ) Q J(A) P J(A) ! 2X 2 j j P 2 where AQ (resp. DQ) is QA (resp. QD) considered as a square submatrix of A (resp. D). In section 3 we show that, for every D + and Q J(A), if we denote = R(A), the following identities hold: 2 D 2 S 1 1 A(A∗DA)− A∗D = PD; and A(QA)− Q = PQ; ; S S where PD; and PQ; denote the distinguished projections onto which are D-selfadjoint S S and Q-selfadjoint, respectively. Then, Ben-Tal and Teboulle’s formulaS (1) can be rewritten in the following way: for every D +, 2 D PD; co PQ; : Q J(A) : S 2 f S 2 g This implies, in particular, that sup PD; max PQ; : The same inequality was + k S k ≤ Q J(A) k S k D n 2 proved independently by O’Leary in2D [28], while the reverse inequality was initially proved by Stewart [31]. A slight generalization of Stewart’s result is proved in this section. Another application of the projections techniques provides an easy proof of a result of Gonzaga and Lara [21] about scaled projections, even for complex weights. In section 4, we extend the notion of compatibility of a closed subspace, with respect to certain subsets of L( )+. Given Γ L( )+ and a closed subspace , we say that is compatible with Γ if (D;H) is compatible⊆ forH every D Γ and it satisfies Stewart’sS condition:S S 2 sup PD; < : D Γ k S k 1 2 For a fixed orthonormal basis = en n N of , we denote by the diagonal algebra with B f g 2 H D respect to , i.e. D if Den = λnen (n N) for a bounded sequence (λn) of complex numbers. NowB we consider2 D compatibility of 2with respect to S 1. +, the set of positive invertible elements of (i.e. all λ > ", for some " > 0). D D n 2. ( ), the set of projections in (i.e. all λ = 0 or 1). P D D n 3. ( ), the set of elements in ( ) with finite rank. P0 D P D 4. 0; ( ), the set of elements Q 0( ) such that R(Q) = 0 . P S D 2 P D \S f g We show that, for a closed subspace , compatibility with any of these sets is equivalent and, in this case, we say that is compatibleS with the basis (or -compatible). This notion is very restrictive.S Nevertheless, the class of subspacesB B which are compatible with a given basis has its own interest. Indeed, as we show in section 5, if dim ? = , then is -compatibleB if and only if the class of frames whose preframe operators (inS terms1 of the basisS B) have nullspace , consists of Riesz frames (see section 5 for definitions or Casazza [5], ChristensenB [6], [7] for modernS treatments of Riesz frame theory and applications). We completely characterize compatible subspaces with in terms of Friedrichs angles (see Definition 2.1) and we obtain an analogous of Stewart-O’LearyB identity. We summarize the main results of this paper. Let be a closed subspace of . For J N, denote by J S H ⊆ H the closed span of the set en : n J and PJ the orthogonal projector onto J . In case that J = 1; : : : ; n , we denotef 2andg P instead of and P . Then H f g Hn n HJ J 3 1. the following conditions are equivalent: (a) is compatible with +. S D (b) sup c [ ; J ]: J N < 1, were c [ ; ] denotes the Friedrichs angle between thef closedS H subspaces⊆ gand . T M T M (c) sup c [ ; J ]: J N and J is finite < 1. f S H ⊆ g (d) all pairs (PJ ; ) are compatible and sup PPJ ; : J N < . S fk S k ⊆ g 1 In this case 1=2 + 2 − sup PD; : D = sup PPJ ; : J N = 1 sup c [ ;R(Q)] : k S k 2 D k S k ⊆ − Q ( ) S n o n o 2P D 2. is compatible with + if and only if S D (a) = n N n and S [ 2 S\H (b) for every n N, the subspace n is compatible with and there exists M > 0 2 S\H B such that sup PPJ ; n : J N M for every n N.
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