Weighted projections into closed subspaces

Corach, Gustavo; Fongi, Guillermina; Maestripieri, Alejandra

doi:10.4064/sm216-2-3

Cited by 16 publications

(24 citation statements)

References 21 publications

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“…T is expansive, hence T is left invertible. Accordingly to [15] we say that T is In particular, following the terminology of [12,18,25,26] we say that T is an A-contraction if T is (A, 1)-contractive, i.e. T * AT ≤ A, and T is an A-isometry when T * AT = A.…”

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confidence: 99%

The Cauchy dual and 2-isometric liftings of concave operators

Badea

Suciu

2019

Journal of Mathematical Analysis and Applications

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We present some 2-isometric lifting and extension results for Hilbert space concave operators. For a special class of concave operators we study their Cauchy dual operators and discuss conditions under which these operators are subnormal. In particular, the quasinormality of compressions of such operators is studied. Introduction and preliminariesPreamble. Extensions and liftings are classical notions in Operator Theory. To give some examples, we recall that a linear bounded Hilbert space operator is an isometry if and only if it is the restriction of a unitary operator to an invariant subspace. Also, it is known from the Sz.-Nagy-Foias dilation theory that an operator C is a contraction if and only if it lifts to an isometry V ; that is if and only if its adjoint C * is the restriction of a coisometry V * to an invariant subspace (see [14,27]).In this paper, we prove some 2-isometric lifting and extension results for Hilbert space concave operators, that is for operators satisfying the inequality (1.1) below. A 2-isometry is an operator for which the equality in (1.1) holds true. The notion of Cauchy dual operator for a left invertible operator is more recent, being introduced in 2001 by Shimorin in his seminal study [24] of Wold-type decompositions and wandering subspaces. Here we study the Cauchy dual operators for the special class of concave operators satisfing the condition (1.3) below. Notation and basic definitions. For a complex Hilbert spaces H we denote by B(H) the Banach algebra of all bounded linear operators on H with the unit element I = I H (the identity operator). For T ∈ B(H) the kernel and the (closed) range of T are denoted by N (T ) respectively R(T ). Also, T * ∈ B(H) stands for the adjoint operator of T , and the orthogonal projection in B(H) onto a closed subspace M ⊂ H is denoted by P M . For T ∈ B(H) we consider the operator ∆ T := T * T − I. The operator T is called expansive (respectively contractive) if ∆ T ≥ 0 (respectively ∆ T ≤ 0). If T is a contraction, then D T = −∆ T is the defect operator and D T = R(D T ) is the defect space of T .

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confidence: 99%

The Cauchy dual and 2-isometric liftings of concave operators

Badea

Suciu

2019

Journal of Mathematical Analysis and Applications

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“…When B = I, X 0 is called a W -inverse of A, see [6]. The next theorem shows that there is a close relationship between W -inverses and W -LSS solutions.…”

Section: Preliminariesmentioning

confidence: 94%

“…The notion of compatibility, defined in [8] and developed later in [6,9,10], has its origin in the work of Z. Pasternak-Winiarski [26]. In that work the author studied, for a fixed subspace S, the analiticity of the map W → P W,S which associates to each positive invertible operator W the orthogonal projection onto S under the (equivalent) inner product x, y W = W x, y , for x, y ∈ H. The notion of compatibility appears when W is allowed to be any positive semidefinite operator, not necessarily invertible (and even, a selfadjoint bounded linear operator).…”

Section: Introductionmentioning

confidence: 99%

Weighted Procrustes problems

Contino

Giribet

Maestripieri

2017

Journal of Mathematical Analysis and Applications

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“…To study the existence of solutions of problem (WLSP) for every x ∈ F in the finite dimensional case, Rao and Mitra introduced the notion of W -inverse [26]. Later on, the W -inverse was studied for operators in [9] and [11].…”

Section: Global Solutions Of Weighted Least Squares Problemsmentioning

confidence: 99%

Global solutions of approximation problems in Hilbert spaces

Contino

Lucero

Fongi

2019

Linear and Multilinear Algebra

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We study three well-known minimization problems in Hilbert spaces: the weighted least squares problem and the related problems of abstract splines and smoothing. In each case we analyze the solvability of the problem for every point of the Hilbert space in the corresponding data set, the existence of an operator that maps each data point to its solution in a linear and continuous way and the solvability of the associated operator problem in a fixed p-Schatten norm.Keywords: Abstract spline problems, Schatten p lasses, optimal inverses 2000 MSC: 47A05, 46C05, 41A65

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Weighted projections into closed subspaces

Cited by 16 publications

References 21 publications

The Cauchy dual and 2-isometric liftings of concave operators

The Cauchy dual and 2-isometric liftings of concave operators

Weighted Procrustes problems

Global solutions of approximation problems in Hilbert spaces

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