# Generalized distribution semi-groups of bounded linear operators

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## Abstract

Generalized distribution semi-groups of bounded linear operators ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze ERIC LARSSON Generalized distribution semi-groups of bounded linear operators Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 21, no 2 (1967), p. 137-159. <http://www.numdam.org/item?id=ASNSP_1967_3_21_2_137_0> © Scuola Normale Superiore, Pisa, 1967, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ GENERALIZED DISTRIBUTION SEMI-GROUPS OF BOUNDED LINEAR OPERATORS by ERIC LARSSON Introduction. Let B be a Banach space the algebra of all bounded linear operators from B to B. Set R+ = ~t E R ; t &#x3E; 0) and denote by °0 ~I~T) the set of all continuous functions with compact support in R+. An ordinary semi-group of bounded linear operators from B to itself is a mapping L from R+ to E (B) satisfying when and a suitable continuity condition, usually when and To get a natural generalization of these semi-groups we consider the bounded linear operators defined by when 99 E Co ;R+) and a E B. L has the following properties : Pervenuto alla Redazione 118 Giugno 1966. 138 and the norm in ~ (B) tends to 0 when 99 iiniformly 990 with the support in a fixed compact subset of R+ . We observe that (1) and (2) correspond to the last two properties. This leads to the following generalization. Let F be a topological con- volution algebra of functions with support in R+. By a F (distri

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