Bases in Banach Spaces I

Singer, Ivan

doi:10.1007/978-3-642-51633-7

Cited by 416 publications

(250 citation statements)

References 0 publications

Supporting

Mentioning

243

Contrasting

Unclassified

Order By: Relevance

“…(See, for example, Remark 15.3. (a) of [16].) Ricker [12] showed, using this fact, that on a GDP-space, every well-bounded operator of type (B) has a finite spectrum and hence is a scalar-type operator with real spectrum.…”

Section: Rmentioning

confidence: 98%

Compact well-bounded operators

Qingping¹,

Doust²

2001

Glasgow Math. J.

View full text Add to dashboard Cite

Abstract. Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.1991 Mathematics Subject Classification. Primary 47B40. Secondary 34L10, 47A60, 47B07.1. Introduction. Well-bounded operators are defined as those which possess a functional calculus for the absolutely continuous functions on some compact interval ½a; b of the real line. Well-bounded operators were introduced by Smart [17] and Ringrose [14] in order to provide a theory for Banach space operators that was similar to the successful theory of self-adjoint operators on Hilbert space, but which included operators whose spectral expansions may only converge conditionally.On a general Banach space the integral representation theorems which one obtains for these operators are much less satisfactory than those obtained for selfadjoint operators. Nonetheless, even on an arbitrary Banach space, every compact well-bounded operator can be written in the form

show abstract

Section: Rmentioning

confidence: 98%

Compact well-bounded operators

Qingping¹,

Doust²

2001

Glasgow Math. J.

View full text Add to dashboard Cite

show abstract

“…(1) follows from Lemma 2.1 and from the fact that the standard base in l 2 is unconditional (see [19]). (2) and (3) follow from the fact that…”

Section: Moreover If Either Of the Integrals Exists Then It Equalsmentioning

confidence: 99%

Lineability of non-differentiable Pettis primitives

2014

View full text Add to dashboard Cite

Abstract. Let X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0,1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND. IntroductionThroughout this note X is an infinite dimensional Banach space. For X-valued functions there are essentially two distinct notions of integration: the Bochner integral and the Pettis integral. The latter one includes properly the Bochner integral and preserves some of its good properties, e.g, countable additivity and absolute continuity of the indefinite integral, and convergence theorems. For X-valued functions on [0, 1] the Bochner indefinite integral is almost everywhere differentiable. As Pettis himself pointed out [17], the same property is not enjoyed by the Pettis integral. An interesting problem left open in [17] was whether the indefinite Pettis integral of a strongly measurable Pettis integrable function f is almost everywhere weakly differentiable (that is, does there exist a set E ⊂ [0, 1] of full measure such that t 0 x * f is differentiable to x * f for each t ∈ E and for each x * in the dual of X). Dilworth and Girardi [5]. They exhibited that every infinite dimensional Banach-space admits a Pettis integrable function from [0, 1] into X whose primitive is nowhere weakly differentiable. Their proof is rather flexible and gives the impression that there are many such functions. How does one make such a statement precise? One possibility is to show that in the space of all strongly measurable function from [0, 1] into X, the set of Pettis integrable functions with nowhere weakly differentiable primitive is a dense G δ set. This was done by Popa [20] in 2000. The topology one uses in this setting is the topology generated by the Pettis norm. The shortcoming of this method is that the Pettis norm is not complete [4]. Hence, proving that the set of Pettis integrable functions with nowhere weakly differentiable primitive is a dense G δ set loses some of its significance.An alternate notion of bigness in Banach space was introduced by Gurarity [11] and followed up in. This notion of bigness is of algebraic nature. If X is a Banach space then a subset M of X is lineable if M ∪{0} contains an infinite dimensional vector space. If, moreover, this infinite dimensional vector space is closed in the norm topology, then M is said to be spaceable. During the last twenty years many classical, pathological subsets of Banach spaces have been shown to be lineable or spaceable. What is surprising is that most of these sets are far from being vector spaces. For example, Gurarity [12] showed that the space of continuous nowhere differentiable functions on [0, 1] is lineable. The spaceability of this set was ...

show abstract

“…On démontre ici que ce résultat est encore vrai si B est un sous-espace homogène au sens de Silow de L^Tc) où TT est le tore à une dimension. L'espace H 1 de Hardy, par exemple, est un tel espace dont on ne sait pas [4] Démonstration. -a) On démontre d'abord cette proposition sur chaque sous-espace F^.…”

Section: Fonctionnelles Analytiques Sub Certains Espaces De Banach Paunclassified

Fonctionnelles analytiques sur certains espaces de Banach

Cœuré¹

1971

Annales De L’institut Fourier

View full text Add to dashboard Cite

Bases in Banach Spaces I

Cited by 416 publications

References 0 publications

Compact well-bounded operators

Compact well-bounded operators

Lineability of non-differentiable Pettis primitives

Fonctionnelles analytiques sur certains espaces de Banach

Contact Info

Product

Resources

About