Operator ideal properties of vector measures with finite variation

• Autorzy: Susumu Okada , Werner J. Ricker , Luis Rodríguez-Piazza
• Języki publikacji: EN

Abstrakty

EN

Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the membership of Iₘ in various classical operator ideals(e.g., the compact, p-summing, completely continuous operators). Depending on which operator ideal is under consideration, the geometric nature of the Banach space X may also play a crucial role. Of particular importance in this regard is whether or not X contains an isomorphic copy of the classical sequence space ℓ¹. The compact range property of X is also relevant.

Kategorie tematyczne

• 47L20: Operator ideals
• 28B05: Vector-valued set functions, measures and integrals
• 46B15: Summability and bases
• 47B10: Operators belonging to operator ideals (nuclear, p -summing, in the Schatten-von Neumann classes, etc.)
• 46B42: Banach lattices

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 205
• Numer: 3
• Strony: 215-249

Daty

wydano

2011

Twórcy

autor

Susumu Okada

• 112 Marconi Crescent, Kambah, ACT 2902, Australia

autor

Werner J. Ricker

• Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, D-85072 Eichstätt, Germany

autor

Luis Rodríguez-Piazza

• Facultad de Matemáticas, Universidad de Sevilla, Aptdo 1160, E-41080 Sevilla, Spain

Identyfikatory

DOI

10.4064/sm205-3-2