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Characterizations of Continuous Operators on with the Strict Topology

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Characterizations of Continuous Operators on with the Strict Topology Tusi Mathematical Ann. Funct. Anal. (2021) 12:28 Research https://doi.org/10.1007/s43034-021-00112-1 Group ORIGINAL PAPER Characterizations of continuous operators on Cb(X) with the strict topology Marian Nowak1 · Juliusz Stochmal2 Received: 2 September 2020 / Accepted: 7 January 2021 / Published online: 16 February 2021 © The Author(s) 2021 Abstract C X Let X be a completely regular Hausdorf space and b( ) be the space of all bounded continuous functions on X, equipped with the strict topology . We study some ⋅ C X important classes of (, ‖ ‖E)-continuous linear operators from b( ) to a Banach E ⋅ space ( , ‖ ‖E) : -absolutely summing operators, compact operators and -nuclear operators. We characterize compact operators and -nuclear operators in terms of their representing measures. It is shown that dominated operators and -absolutely T C X → E summing operators ∶ b( ) coincide and if, in particular, E has the Radon– Nikodym property, then -absolutely summing operators and -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concern- ing the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorf space. Keywords Spaces of bounded continuous functions · k-spaces · Radon vector measures · Strict topologies · Absolutely summing operators · Dominated operators · Nuclear operators · Compact operators · Generalized DF-spaces · Projective tensor product Mathematics Subject Classifcation 46G10 · 28A32 · 47B10 Communicated by Raymond Mortini. * Juliusz Stochmal [email protected] Marian Nowak [email protected] 1 Institute of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Gora, Poland 2 Institute of Mathematics, Kazimierz Wielki University, ul. Powstańców Wielkopolskich 2, 85-090 Bydgoszcz, Poland Vol.:(0123456789) 28 Page 2 of 26 M. Nowak and J. Stochmal 1 Introduction and preliminaries The Riesz representation theorem plays a crucial role in the study of operators on the Banach space C(X) of continuous functions on a compact Hausdorf space X. Due to this theorem, diferent classes of operators on C(X) have been character- ized in terms of their representing Radon vector measures. Absolutely summing operators between Banach spaces have been the object of several studies (see [1, pp. 209–233] and [5, 8, 11, 27, 28, 31, 34]). It originates in the fundamental paper of Grothendieck [17] from 1953. Grothendieck’s ine- quality has equivalent formulation using the theory of absolutely summing opera- tors (see [1, Theorem 8.3.1] and [4, 22]). In the multilinear case, it is also con- nected with the Bohnenblust–Hille and the Hardy–Littlewood inequalities (see [2]). There is a vast literature on absolutely summing operators from the Banach space C(X) to a Banach space E (see [1], [9, Chap. VI], [11, 34, 43]). The concept of nuclearity in Banach spaces is due to Grothendieck [17, 18] and Ruston [33] and has the origin in Schwartz’s kernel theorem [18]. Many authors have studied nuclear operators between locally convex spaces (see [21, §17.3], [37, Chap. 3, §7], [46, p. 289]) and Banach spaces (see [9, Chap. VI], [11, 16] [46, p. 279]). If F is a Banach space, nuclear operators from the Banach space C(X, F) of F-valued continuous functions on a compact Hausdorf space X to E have been studied intensively by Popa [29], Saab [35], Saab and Smith [36]. In particular, a characterization of nuclear operators from C(X) to E in terms of their representing measures can be found in [9, Theorem 4, pp. 173–174], [34, Propo- sition 5.30], [43, Proposition 1.2]. The interplay between absolutely summing operators, dominated operators of Dinculeanu (see [12, §19], [13, §1]) and nuclear operators T ∶ C(X) → E has been an interesting issue in operator theory. Pietsch [27, 2.3.4, Proposition, p. 41] proved that dominated operators and absolutely summing operators on the Banach space C(X) coincide. It is known that if in particular, E has the Radon–Nikodym property, then absolutely summing and nuclear operators T ∶ C(X) → E coincide (see [9, Corollary 5, p. 174]). Moreover, Uhl [44, Theorem 1] showed that if, E has the Radon–Nikodym property, then every dominated operator T ∶ C(X) → E is compact. The aim of this paper is to extend these classical results to the setting, where X is a completely regular Hausdorf k-space. Throughout the paper, we assume that (X, T) is a completely regular Hausdorf space. By K we denote the family of all compact sets in X. Let Bo denote the -algebra of Borel sets in X. B Let Cb(X) (resp. B( o)) denote the Banach space of all bounded continuous (resp. bounded Bo-measurable) scalar functions on X, equipped with the topology ⋅ S B B u of the uniform norm ‖ ‖∞ . By ( o) we denote the space of all o-simple sca- � lar functions on X. Let Cb(X) stand for the Banach dual of Cb(X). Following [15, 37] and [45, Defnition 10.4, p. 137] the strict topology on Cb(X) is the locally convex topology determined by the seminorms Characterizations of continuous operators on Cb(X)... Page 3 of 26 28 pw(u) ∶= sup w(t)u(t) for u ∈ Cb(X), t∈X where w runs over the family W of all bounded functions w ∶ X → [0, ∞) which van- K ish at infnity, that is, for every �>0 there exists K ∈ such that supt∈X⧵K w(t) ≤ . W W W Let 1 ∶= {w ∈ ∶ 0 ≤ w ≤ X} . For w ∈ 1 and �>0 let Uw() ∶= {u ∈ Cb(X)∶pw(u) ≤ }. W Note that the family {Uw(�)∶w ∈ 1, �>0} is a local base at 0 for . The strict topology on Cb(X) has been studied intensively (see [15, 20, 38, 41, 45]). Note that can be characterized as the fnest locally convex Haus- dorf topology on Cb(X) that coincides with the compact-open topology c on u -bounded sets (see [41, Theorem 2.4]). The topologies and u have the same bounded sets. This means that (Cb(X), ) is a generalized DF-space (see [38, Cor- ollary]), and it follows that (Cb(X), ) is quasinormable (see [32, p. 422]). If, in particular, X is locally compact (resp. compact), then coincides with the origi- nal strict topology of Buck [6] (resp. = u). Recall that a countably additive scalar measure on Bo is said to be a Radon measure if its variation is regular, that is, for every A ∈ Bo and �>0 there exist K ∈ K and O ∈ T with K ⊂ A ⊂ O such that (O⧵K) ≤ . Let M(X) denote the Banach space of all scalar Radon measures, equipped with the total variation norm ‖‖ ∶= ��(X). The following characterization of the topological dual of (Cb(X), ) will be of importance (see [15, Lemma 4.5]), [20, Theorem 2]. Theorem 1.1 For a linear functional Φ on Cb(X) the following statements are equivalent: (i) Φ is -continuous. (ii) There exists a unique ∈ M(X) such that Φ(u)=Φ(u)= ud for u ∈ Cb(X) X � ⋅ and ‖Φ‖ = ��(X) for ∈ M(X) (here ‖ ‖ denotes the conjugate norm in � Cb(X) ). The following result will be useful (see [41, Theorem 5.1]). Theorem 1.2 For a subset M of M(X) the following statements are equivalent: M (i) sup�∈M �(X) < ∞ and is uniformly tight, that is, for each �>0 there K exists K ∈ such that sup∈M (X ⧵ K) ≤ . M (ii) The family {Φ ∶ ∈ } is -equicontinuous. Recall that a completely regular Hausdorf space (X, T) is a k-space if any sub- set A of X is closed whenever A ∩ K is compact for all compact sets K in X. In 28 Page 4 of 26 M. Nowak and J. Stochmal particular, every locally compact Hausdorf space, every metrizable space and every space satisfying the frst countability axiom is a k-space (see [14, Chap. 3, § 3]). T From now on, we will assume that (X, ) is a k-space. Then, the space (Cb(X), ) is complete (see [15, Theorem 2.4]). ⋅ We assume that (E, ‖ ‖E) is a Banach space. Let BE stand for the closed unit ball in the Banach dual E of E. → Recall that a bounded linear operator T ∶ Cb(X) E is said to be absolutely sum- ming if there exists a constant c > 0 such that for any fnite set {u1, … , un} in Cb(X), � � �n �n ‖T(u )‖ c sup �Φ(u )� ∶Φ∈B � . i E ≤ i Cb(X) (1.1) i=1 i=1 The infmum of number of c > 0 satisfying (1.1) denoted by ‖T‖as is called an abso- lutely summing norm of T. → It is known that a bounded linear operator T ∶ Cb(X) E is absolutely summing if and only if T maps unconditionally convergent series in Cb(X) into absolutely con- vergent series in E (see [9, Defnition 1, p. 161 and Proposition 2, p. 162]). For t ∈ X , let t stand for the point mass measure, that is, t(A) ∶= A(t) B + for A ∈ o . Then t ∈ M (X) and ∫X u dt = u(t) for u ∈ Cb(X) . Clearly, ‖t‖ = t(X)=1. → Lemma 1.3 For a bounded linear operator T ∶ Cb(X) E , the following statements are equivalent: (i) T is absolutely summing. (ii) There exists c > 0 such that for any set {u1, … , un} in Cb(X) , � � �n �n � � � � ‖T(ui)‖E ≤ c sup � ui d� ∶ ∈ M(X), ��(X) ≤ 1 . i=1 i=1 � �X � ⇒ Proof (i) (ii) There exists c > 0 such that for any set {u1, … , un} in Cb(X), � � �n �n ‖T(u )‖ c sup �Φ(u )� ∶Φ∈B � .
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