Weighted least squares solutions of the equation AXB−C= 0

Contino, Maximiliano; Giribet, Juan I.; Maestripieri, Alejandra

doi:10.1016/j.laa.2016.12.028

Cited by 4 publications

(5 citation statements)

References 20 publications

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“…For a treatment of the weighted least square solutions of the operator equation DXB − C = 0 considering the seminorm • p,A , the reader is referred to [13].…”

Section: Reconstruction Operators By Means Of Restricted Weighted Inversesmentioning

confidence: 99%

Sampling and Reconstruction by Means of Weighted Inverses

Arias

Gonzalez

2020

J Fourier Anal Appl

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In this article, we address the problem of reconstructing an element in a Hilbert space from its samples by means of a weighted least square approximation. We show how this problem is linked with the notions of weighted inverses, weighted projections and an angle condition known as compatibility. In addition, we study perfect reconstruction operators and their relationship with the previous problem. Finally, since the reconstructions through these approaches may not be unique, we propose different criteria for choosing an optimal one among all of them.

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“…For a treatment of the weighted least square solutions of the operator equation DXB − C = 0 considering the seminorm • p,A , the reader is referred to [13].…”

Section: Reconstruction Operators By Means Of Restricted Weighted Inversesmentioning

confidence: 99%

Sampling and Reconstruction by Means of Weighted Inverses

Arias

Gonzalez

2020

J Fourier Anal Appl

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“…Conversely, suppose W /R(A) = min− ≤ X∈L(H) F (X). Then, there exists X 0 ∈ L(H), such that (AX 0 B − I) * W (AX 0 B − I) = min− ≤ X∈L(H) F (X) = W /R(A) .Then, by Lemma 4.2, the pair (W, R(A)) is compatible.ii) ⇔ iii) and ii) ⇔ iv) : follow from the fact that N (B) = N (A * W ) and[10, Theorem 4.3].…”

mentioning

confidence: 92%

“…Moreover, also in [10] it was proved the the minimum of the previous set exists if and only if the pair (W, R(A)) is compatible.…”

Section: Introductionmentioning

confidence: 98%

“…Given W ∈ L(H) + and A, B ∈ L(H) with closed ranges such that N (B) = N (A * W ), in [10] it was shown that, considering the order induced by the cone of positive operators,…”

Section: Introductionmentioning

confidence: 99%

“…Also in[10, Theorem 3.2], it was proved that the minimum of this set exists if and only if the pair (W, R(A)) is compatible.In this section, we study a similar problem considering the minus order: for W ∈ L(H) + and A, B ∈ CR(H), analyze the existence ofmin− ≤ X∈L(H) (AXB − I) * W (AXB − I).…”

mentioning

confidence: 99%

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Shorted operators and minus order

Contino

Giribet

Maestripieri

2018

Linear and Multilinear Algebra

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Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W ∈ L(H) a positive operator. Given a closed subspace S of H, we characterize the shorted operator W /S of W to S as the maximum and as the infimum of certain sets, for the minus order − ≤. Also, given A ∈ L(H) with closed range, we study the following operator approximation problem considering the minus order:We show that, under certain conditions, the shorted operator of W /R(A) is the minimum of this problem and we characterize the set of solutions. KEYWORDS Shorted operators; minus order; oblique projections ≤ B (where the symbol − ≤ stands for the minus order of operators) if the Dixmier angle between R(A) and R(B − A) and the Dixmier angle between R(A * ) and R(B * − A * ) are less than 1, where R(T ) stands for the range of the operator T. Independently, in [31]Šemrl gave another characterization of the minus order, he showed that A − ≤ B if and only if there are bounded (oblique) projections, i.e. idempotents, P and Q such that A = P B and A * = QB * . In [18], a new characterization of the minus order for operators acting on Hilbert spaces was given in terms of the so called range additivity property. Namely, it was proved that A − ≤ B is equivalent to the range of B being the direct sum of ranges of A and B − A and the range of B * being the direct sum of ranges of A * and B * − A * , which generalizes previous results presented in [31]. This plays an equivalent role to the rank additivity characterization when A and B are matrices [21, 26]. In [5] the notion of shorted operator appears in relation with the minus order. Given a closed subspace S of H and W ∈ L(H) a positive operator, in 1947, Krein [23], proved the existence of a maximum (with respect to the order induced by the cone of positive operators) of the set M(W, S) = {X ∈ L(H) : 0 ≤ X ≤ W and R(X) ⊆ S ⊥ }.Krein used this extremal operator in his theory of extension of symmetric operators. Years later, Anderson and Trapp [2] studying the same problem, called this maximum the shorted operator of W to S (in the following denoted W /S ) and showed interesting properties of this operator and its connections with electrical circuit theory. The shorted operator have shown to be useful in many applications [5], [32].The pair (W, S) is said to be compatible if there exists a bounded linear (not necessarily selfadjoint) projection Q onto S such that W Q is selfadjoint. Thus, ifthen (W, S) is compatible if and only if P(W, S) is not empty. In [15], it was shown that there exists a strong relationship between compatibility, the projections of P(W, S) and the shorted operator W /S . Later, in [5], the notion of shorted operator was generalized to that of bilateral shorted operator, for an operator W ∈ L(H) (not necessarly positive) and a pair of closed subspaces. The proposed definition comes from the notion of weak complementability, which is a refinement of a finite dimensional notion due to T. Ando [3]. In that paper, it was proven that given an operator W ∈ L(H) positive an...

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Operator Least Squares Problems and Moore–Penrose Inverses in Krein Spaces

Contino

Maestripieri

Palacios

2018

Integr. Equ. Oper. Theory

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A Krein space H and bounded linear operators B, C on H are given. Then, some min and max problems about the operators (BX − C) # (BX − C), where X runs over the space of all bounded linear operators on H, are discussed. In each case, a complete answer to the problem, including solvability conditions and characterization of the solutions, is presented. Also, an adequate decomposition of B is considered and the min-max problem is addressed. As a by-product the Moore-Penrose inverse of B is characterized as the only solution of a variational problem. Other generalized inverses are described in a similar fashion as well.Mathematics Subject Classification (2010). 47A58, 47B50, 41A65.Keywords. Operator approximation, Krein spaces, Moore-Penrose inverse. 1 L(H) stands for the space of all the bounded linear operators from H to H. Vol. 99 (9999) Operator LSP and Moore-Penrose inverses 3 characterized as the indefinite inverses of B. The results about this indefinite minimization problem and their counterparts for the symmetric maximization problem are applied in Section 5 where B is factorized as B = B + + B − and the min-max problem min X∈L(H) max Y ∈L(H)is addressed. Section 6 contains the main results about the Moore-Penrose inverse and the generalized inverses of a given Krein space operator. PreliminariesIn the following all Hilbert spaces are complex and separable. If H and K are Hilbert spaces, L(H, K) stands for the space of all the bounded linear operators from H to K and CR(H, K) for the subset of L(H, K) comprising all the operators with closed ranges. When H = K we write, for short, L(H) and CR(H). The range and null space of any given A ∈ L(H, K) are denoted by R(A) and N (A), respectively. The direct sum of two closed subspaces M and N of H is represented by M+N . If H is decomposed as H = M+N , the projection onto M with null space N is denoted P M//N and abbreviated P M when N = M ⊥ . In general, Q is used to indicate the subset of all the oblique projections in L(H), namely, Q := {Q ∈ L(H) : Q 2 = Q}.The following is a well-known result about range inclusion and factorizations of operators. We will refer to it along the paper. Theorem 2.1 (Douglas' Theorem [9]). Let Y, Z ∈ L(H). Then R(Z) ⊆ R(Y ) if and only if there exists D ∈ L(H) such that Z = Y D. Krein SpacesAlthough familiarity with operator theory on Krein spaces is presumed, we hereafter include some basic notions. Standard references on Krein spaces and operators on them are [1,4,5]. We also refer to [10,11] as authoritative accounts of the subject.Consider a linear space H with an indefinite metric, i.e., a sesquilinear Hermitian form [ , ]. A vector x ∈ H is said to be positive if [ x, x ] > 0. A subspace S of H is positive if every x ∈ S, x = 0, is a positive vector. Negative, nonnegative, nonpositive and neutral vectors and subspaces are defined likewise.We say that two closed subspaces M and N are orthogonal, and we write M [⊥] N , if [ m, n ] = 0 for every m ∈ M and n ∈ N . We denote the orthogonal direct sum of two closed subspaces M and ...

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Weighted least squares solutions of the equation AXB−C= 0

Cited by 4 publications

References 20 publications

Sampling and Reconstruction by Means of Weighted Inverses

Sampling and Reconstruction by Means of Weighted Inverses

Shorted operators and minus order

Operator Least Squares Problems and Moore–Penrose Inverses in Krein Spaces

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