On boundaries on the predual of the Lorentz sequence space

Acosta, Marı́a D.; Moraes, Luiza A.; Grados, Luis Romero

doi:10.1016/j.jmaa.2007.02.041

Cited by 8 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this case, we prove the existence of two disjoint and closed numerical boundaries for the space of bounded and uniformly continuous functions from B d * (w,1) to d * (w, 1) which are holomorphic on the open unit ball (Theorem 4.3). The analogous result for the norm in this case appears in [15] and [4]. Section 5 is dedicated to C(K).…”

Section: Introductionmentioning

confidence: 65%

Numerical boundaries for some classical Banach spaces

Acosta

Kim

2009

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

Globevnik gave the definition of boundary for a subspace A ⊂ C b (Ω). This is a subset of Ω that is a norming set for A. We introduce the concept of numerical boundary. For a Banach space X, a subset B ⊂ Π(X) is a numerical boundary for a subspace A ⊂ C b (B X , X) if the numerical radius of f is the supremum of the modulus of all the evaluations of f at B, for every f in A. We give examples of numerical boundaries for the complex spaces X = c 0 , C(K) and d * (w, 1), the predual of the Lorentz sequence space d(w, 1). In all these cases (if K is infinite) we show that there are closed and disjoint numerical boundaries for the space of the functions from B X to X which are uniformly continuous and holomorphic on the open unit ball and there is no minimal closed numerical boundary. In the case of c 0 , we characterize the numerical boundaries for that space of holomorphic functions.

show abstract

Section: Introductionmentioning

confidence: 65%

Numerical boundaries for some classical Banach spaces

Acosta

Kim

2009

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…Later Acosta, Moraes and Romero [2] generalized that characterization proving it for any space d * (w, 1) and obtained another one in terms of the strong peak sets of the unit ball. In this case, there is no Shilov boundary.…”

mentioning

confidence: 89%

Shilov boundary for holomorphic functions on some classical Banach spaces

Acosta

Lourenço

2007

Studia Math.

Self Cite

View full text Add to dashboard Cite

Abstract. Let A ∞ (B X ) be the Banach space of all bounded and continuous functions on the closed unit ball B X of a complex Banach space X and holomorphic on the open unit ball, with sup norm, and let A u (B X ) be the subspace of A ∞ (B X ) of those functions which are uniformly continuous on B X . A subset B ⊂ B X is a boundary for A ∞ (B X ) if f = sup x∈B |f (x)| for every f ∈ A ∞ (B X ). We prove that for X = d(w, 1) (the Lorentz sequence space) and X = C 1 (H), the trace class operators, there is a minimal closed boundary for A ∞ (B X ). On the other hand, for X = S, the Schreier space, and, there is no minimal closed boundary for the corresponding spaces of holomorphic functions.

show abstract

“…In [17] they establish the same result for A b (B Gp ). Recently, Choi and Han, [8], extended the results of the paper of Moraes and Romero Grados to an even larger class of preduals of Lorentz spaces which includes both c 0 and the space G p of Moraes and Romero Grados, [16], while Acosta, Moraes and Romero Grados, [4], give a characterisation of boundaries of preduals of Lorentz spaces in terms of the distance to the set of strong peak points. Acosta, [1], shows that there is noŠilov boundary in the sense of Globevik for A u (B C(K) ) when K is infinite, compact and Hausdorff.…”

Section: Introductionmentioning

confidence: 99%

Geometry and Analytic Boundaries of Marcinkiewicz Sequence Spaces

Boyd

Lassalle

2009

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

We investigate the geometric structure of the unit ball of the Marcinkiewicz sequence space m 0 Ψ , giving characterisations of its real and complex extreme points and of the exposed points in terms of the symbol Ψ. Using our knowledge of the geometry of B m 0 Ψ we then give necessary and sufficient conditions for a subset of B m 0 Ψ to be a boundary for Au(B m 0 Ψ), the algebra of functions which are uniformly continuous on B m 0 Ψ and holomorphic on the interior of B m 0 Ψ. We show that it is possible for the set of peak points of Au(B m 0 Ψ) to be a boundary for Au(B m 0 Ψ) yet for Au(B m 0 Ψ) not to have aŠilov boundary in the sense of Globevnik.

show abstract

On boundaries on the predual of the Lorentz sequence space

Cited by 8 publications

References 10 publications

Numerical boundaries for some classical Banach spaces

Numerical boundaries for some classical Banach spaces

Shilov boundary for holomorphic functions on some classical Banach spaces

Geometry and Analytic Boundaries of Marcinkiewicz Sequence Spaces

Contact Info

Product

Resources

About