Acta Math. Hungar., 114 (1–2) (2007), 13–36. DOI: 10.1007/s10474-006-0526-6 SEMIGROUPS OF LINEAR OPERATORS ON p-FR ´ ECHET SPACES, 0 <p< 1 S. G. GAL 1 and J. A. GOLDSTEIN 2 1 University of Oradea, Department of Mathematics, Str. Armatei Romˆ ane 5, 410087 Oradea, Romania e-mail: galso@uoradea.ro 2 The University of Memphis, Department of Mathematical Sciences, Memphis, TN, 38152, U.S.A. e-mail: jgoldste@memphis.edu (Received May 13, 2005; revised May 2, 2006; accepted August 28, 2006) Abstract. We develop the beginning of a theory of semigroups of linear op- erators on p-Fr´ echet spaces, 0 <p< 1 (which are non-locally convex F -spaces), and give some applications. 1. Introduction It is well known that an F -space ( X, +, ·, ‖·‖ ) is a linear space (over the field K = R or K = C) such that ‖x + y‖ ≦ ‖x‖ + ‖y‖ for all x, y ∈ X , ‖x‖ = 0 if and only if x = 0, ‖λx‖ ≦ ‖x‖, for all scalars λ with |λ| ≦ 1, x ∈ X , and with respect to the metric d(x, y)= ‖x − y‖, X is a complete metric space (see e.g. [8, p. 52] or [12]). In addition, if there exists 0 <p< 1 with ‖λx‖ = |λ| p ‖x‖, for all λ ∈ K, x ∈ X , then ‖·‖ will be called a p-norm and X will be called p-Fr´ echet space. (This is only a slight abuse of terminology. Note that e.g. in [1] these spaces are called p-Banach spaces.) It is known that the F -spaces are not necessarily locally convex spaces. Three classical examples of p-Fr´ echet spaces, non-locally convex, are the Key words and phrases: p-Fr´ echet space 0 <p< 1, semigroups of linear operators, Cauchy problem. 2000 Mathematics Subject Classification: 47D03, 47D06, 47D09, 47D60. 0236–5294/$ 20.00 c  2007 Akad´ emiai Kiad´o, Budapest