Skip to main content
Log in

Uniformly summing sets of operators on spaces of vector–valued continuous functions

  • Original Paper
  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract.

Let X, Y be Banach spaces. We say that a set \(\user2{\mathbb{M}}\, \subset \,\prod\nolimits_p {(X,\,Y)}\) is uniformly p–summing if the series \(\sum\limits_n {||\,T\,x_n \,||^p }\) is uniformly convergent for \(T\, \in \,\user2{\mathbb{M}}\) whenever (x n ) belongs to \(\ell _w^p \,(X)\). We consider uniformly summing sets of operators defined on a \(\user1{\mathcal{C}}\,(\Omega ,\,X)\)-space and prove, in case X does not contain a copy of c0, that \(\user2{\mathbb{M}}\) is uniformly summing iff \(\user2{\mathbb{M}}^\# \, = \,\{ T^\# :\user1{\mathcal{C}}(\Omega )\, \to \,\prod\nolimits_1 {(X,\,Y)\,:} \,T \in \user2{\mathbb{M}}\}\) is, where Tx)  =  (T#φ) x for all \(\varphi \, \in \,\user1{\mathcal{C}}(\Omega )\) and xX. We also characterize the sets \(\user2{\mathbb{M}}\) with the property that \(\user2{\mathbb{M}}^\#\) is uniformly summing viewed in \(\prod\nolimits_1 {(\user1{\mathcal{C}}(\Omega ),\,\user1{\mathcal{L}}(X,\,Y))}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. M. Delgado.

Additional information

Received: 1 July 2005

Rights and permissions

Reprints and permissions

About this article

Cite this article

Delgado, J.M., Piñeiro, C. Uniformly summing sets of operators on spaces of vector–valued continuous functions. Arch. Math. 87, 141–153 (2006). https://doi.org/10.1007/s00013-005-1648-8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-005-1648-8

Mathematics Subject Classification (2000).

Navigation