Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces

Badea, Cătălin; Lyubich, Yuri I.

doi:10.4064/sm201-1-2

Cited by 15 publications

(17 citation statements)

References 27 publications

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“…The strong limit T ∞ is a projection of norm one onto the intersection of the ranges of P j . The same result holds [3] if X is uniformly smooth and each projection P j is of norm one. It also holds The first two named authors have been partially supported by ANR Projet Blanc DYNOP.…”

Section: Introductionsupporting

confidence: 57%

The rate of convergence in the method of alternating projections

Badea

Grivaux

Müller

2012

St. Petersburg Math. J.

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A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed.

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Section: Introductionsupporting

confidence: 57%

The rate of convergence in the method of alternating projections

Badea

Grivaux

Müller

2012

St. Petersburg Math. J.

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“…The strong limit T ∞ is a projection of norm one onto the intersection of the ranges of P j . The same result is valid [3] if X is uniformly smooth and each projection P j is of norm one. It is also 2010 Mathematics Subject Classification.…”

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confidence: 57%

“…Applications to products of projections of norm one are given. In particular, the dichotomy (QUC)/(ASC) holds, with several versions of (ASC), for the cases covered by the theorems of von Neumann, Halperin, Bruck-Reich, and those of [3].…”

Section: Theorem 12 (Seementioning

confidence: 99%

Untitled

2012

St. Petersburg Math. J.

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Abstract. The cosine of the Friedrichs angle between two subspaces is generalized to a parameter associated with several closed subspaces of a Hilbert space. This parameter is employed to analyze the rate of convergence in the von Neumann-Halperin method of cyclic alternating projections. General dichotomy theorems are proved, in the Hilbert or Banach space situation, providing conditions under which the alternative QUC/ASC (quick uniform convergence versus arbitrarily slow convergence) holds. Several meanings for ASC are proposed. §1. IntroductionThroughout the paper H is a complex Hilbert space. For a closed linear subspace S of H we denote by S ⊥ its orthogonal complement in H, and by P S the orthogonal projection of H onto S. In this paper N denotes a fixed positive integer greater than or equal to 2.1A. The method of alternating projections. It was proved by J. von Neumann [27, p. 475] that for two closed subspaces M 1 and M 2 of H with intersection M = M 1 ∩ M 2 , the following convergence result holds:; then von Neumann's result says that the iterates T n of T are strongly convergent to T ∞ = P M . The method of constructing the iterates of T by alternately projecting onto one subspace and then onto the other is called the method of alternating projections. This algorithm, and its variations, occur in several fields, pure or applied. We refer to [10, Chapter 9] as a source for more information.A generalization of von Neumann's result to N closed subspacesIn the present paper, the algorithm provided by Halperin's result will be called the method of cyclic alternating projections.A Banach space extension of Halperin's result was proved by Bruck and Reich [9]: if X is a uniformly convex Banach space and P j , 1 ≤ j ≤ N , are N norm one projections in B(X), then the iterates of T = P N . . . P 2 P 1 are strongly convergent. The strong limit T ∞ is a projection of norm one onto the intersection of the ranges of P j . The same result is valid [3] if X is uniformly smooth and each projection P j is of norm one. It is also 2010 Mathematics Subject Classification. Primary 47A05, 47A10, 41A35.

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“…This condition was introduced by I. Halperin in [11]; we refer the reader to [2] and the references therein for more information. In particular, a product of orthogonal projections satisfies (6.1).…”

Section: Examples and Counterexamplesmentioning

confidence: 99%

Classes of contractions and Harnack domination

Badea¹,

Suciu²,

Timotin³

2017

Rev. Mat. Iberoam.

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Several properties of the Harnack domination of linear operators acting onHilbert space with norm less or equal than one are studied. Thus, the maximal elements for this relation are identified as precisely the singular unitary operators, while the minimal elements are shown to be the isometries and the adjoints of isometries. We also show how a large range of properties (e.g. convergence of iterates, peripheral spectrum, ergodic properties) are transfered from a contraction to one that Harnack dominates it.2010 Mathematics Subject Classification. Primary 47A10, 47A45; Secondary 47A20, 47A35, 47B15.In the sequel T, T ′ ∈ B(H) will be linear contractions acting on the complex Hilbert space H; V acting on K and V ′ acting on K ′ will denote the minimal isometric dilations of T and T ′ respectively. N (T ) and R(T ) stand for the kernel and respectively the range of the operator T . We shall denote by I the identity operator on H and by T λ = (T − λI)(I − λT ) −1

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Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces

Cited by 15 publications

References 27 publications

The rate of convergence in the method of alternating projections

The rate of convergence in the method of alternating projections

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Classes of contractions and Harnack domination

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