Indefinite least-squares problems and pseudo-regularity

Giribet, Juan I.; Maestripieri, Alejandra; Peria, Francisco Dardo Martinez

doi:10.1016/j.jmaa.2015.05.015

Cited by 6 publications

(11 citation statements)

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“…The following propositions can be found in [19,Lemma 3.4 The following result, taken from [5, Chapter 1, §1], deals with indefinite inner product spaces which are not necessarily Krein spaces. It will be one of the main tools to study the problem posed in (1.1).…”

Section: Preliminariesmentioning

confidence: 99%

“…Indefinite least-squares problems have been thoroughly studied before, see e.g. [8,18,19] and the references therein. It is well-known that, if ρ = 0, there exists a solution to (3.3) if and only if R(L) is nonnegative in [⊥] , see [7,Thm.…”

Section: Indefinite Smoothing Splinesmentioning

confidence: 99%

“…The advantage of the above regularized problem is that it can be restated as an indefinite least-squares problem. These problems have been thoroughly studied before, both in finite-dimensional spaces [20,21,30,13,22,27,8] and in infinite dimensional Krein spaces [7,18,19]. Although the existence of solutions to Problem 1 as well as descriptions of the set of solutions can be derived from the corresponding indefinite least-squares problem, it is desirable to reformulate these conditions in terms of the original data T , V , ρ and z 0 .…”

Section: Introductionmentioning

confidence: 99%

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Indefinite Abstract Splines with a Quadratic Constraint

Zerbo

Maestripieri

Peria

2020

J Optim Theory Appl

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Section: Preliminariesmentioning

confidence: 99%

Section: Indefinite Smoothing Splinesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Indefinite Abstract Splines with a Quadratic Constraint

Zerbo

Maestripieri

Peria

2020

J Optim Theory Appl

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“…Furthermore, if R(C) ⊆ R(B) [⊥] , a converse of Theorem 6.8 holds: if X 1 is a solution of problem (6.1) then X 1 = DC, where D ∈ L(H) is a solution of (6.6). In fact, let X 1 be a solution of problem (6.1), then by similar arguments as those in [12,Theorem 3.5], there exists P ′ ∈ Q N (B # B) such that…”

Section: Itmentioning

confidence: 92%

“…In [12,13] least squares problems in the indefinite metric setting were studied. From those references we recall the definition of indefinite least squares solution.…”

Section: Introductionmentioning

confidence: 99%

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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