Symmetric spaces with maximal projection constants

Chalmers, Bruce L.; Lewicki, Grzegorz

doi:10.1016/s0022-1236(02)00080-0

Cited by 31 publications

(12 citation statements)

References 11 publications

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“…Since in the case L ∞ , however, the minimal numerical-radius extensions and the minimal operator-norm extensions coincide, the result follows. (It is also known that isometric copies of the maximal symmetric spaces lie in L 1 (see [4]) and that isometric copies of the maximal (unrestricted) spaces lie in L 1 (see [16]). )…”

Section: Similarly In the Casementioning

confidence: 99%

Minimal numerical-radius extensions of operators

Aksoy

Chalmers

2007

Proc. Amer. Math. Soc.

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Abstract. In this paper we characterize minimal numerical-radius extensions of operators from finite-dimensional subspaces and compare them with minimal operator-norm extensions. We note that in the cases L p , p = 1, ∞, and in the case of self-adjoint extensions in L 2 , the two extensions and their norms are equal.We also show that, in the case of L p , 1 < p < ∞, and more generally in the case of the dual space being strictly convex, if the minimal projections with respect to the operator norm and with respect to the numerical radius have equal norms, then the operator norm is 1. An analogous result is also true for an arbitrary extension. Finally, we provide an example of a projection from l p 3 onto a two-dimensional subspace which is minimal with respect to norm but not with respect to the numerical radius for p = 1, 2, ∞, and we determine the minimal numerical-radius projection in this same situation.

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Section: Similarly In the Casementioning

confidence: 99%

Minimal numerical-radius extensions of operators

Aksoy

Chalmers

2007

Proc. Amer. Math. Soc.

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“…The Chalmers-Metcalf theorem has many applications especially in the case of X = L 1 , e.g., [3][4][5][6][7]13,[21][22][23]35,38,39]. The aim of this paper is to relate some properties of this operator to the uniqueness of minimal projections.…”

Section: Introductionmentioning

confidence: 99%

Chalmers–Metcalf operator and uniqueness of minimal projections

Lewicki

Skrzypek

2007

Journal of Approximation Theory

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We know that not all minimal projections in L p (1 < p < ∞) are unique (see [B. Shekhtman, L. Skrzypek, On the non-uniqueness of minimal projections in L p spaces]). The aim of this paper is examine the connection of the Chalmers-Metcalf operator (introduced in [B.L. Chalmers, F.T. Metcalf, A characterization and equations for minimal projections and extensions, J. Oper. Theory 32 (1994) 31-46]) to the uniqueness of minimal projections. The main theorem of this paper is Theorem 2.2. It relates uniqueness of minimal projections to the invertibility of the Chalmers-Metcalf operator. It is worth mentioning that to a given minimal projection (even unique) we may find many different Chalmers-Metcalf operators, some of them invertible, some not-see Example 2.6. The main application is in Section 3, where we have proven that minimal projections onto symmetric subspaces in smooth Banach spaces are unique (Theorem 3.2). This leads (in Section 4) to the solution of the problem of uniqueness of classical Rademacher projections in L p [0, 1] for 1 < p < ∞.

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“…We note that this characterization has been successfully employed in a variety of settings (see e.g. [2][3][4][5]). …”

Section: Introductionmentioning

confidence: 99%

A characterization and equations for minimal shape-preserving projections

Chalmers

Mupasiri

Prophet

2006

Journal of Approximation Theory

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Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B = B(X, V ) the space of all bounded linear operators from X into V; let P(X, V ) be the set of all projections in B. For a given cone S ⊂ X, we denote by P = P S (X, V ) the set of operators P ∈ P such that P S ⊂ S. When P S = ∅, we characterize those P ∈ P S for which P is minimal. This characterization is then utilized in several applications and examples.

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Symmetric spaces with maximal projection constants

Cited by 31 publications

References 11 publications

Minimal numerical-radius extensions of operators

Minimal numerical-radius extensions of operators

Chalmers–Metcalf operator and uniqueness of minimal projections

A characterization and equations for minimal shape-preserving projections

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