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Let X be a completely regular Hausdorff space and Cb(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_b(X)$$\end{document} be the space of all bounded continuous functions on X, equipped with the strict topology β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}. We study some important classes of (β,‖·‖E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta ,\Vert \cdot \Vert _E)$$\end{document}-continuous linear operators from Cb(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_b(X)$$\end{document} to a Banach space (E,‖·‖E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(E,\Vert \cdot \Vert _E)$$\end{document}: β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-absolutely summing operators, compact operators and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-nuclear operators. We characterize compact operators and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-nuclear operators in terms of their representing measures. It is shown that dominated operators and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-absolutely summing operators T:Cb(X)→E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T:C_b(X)\rightarrow E$$\end{document} coincide and if, in particular, E has the Radon–Nikodym property, then β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-absolutely summing operators and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorff space.


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 Hausdorff space X. Due to this theorem, different classes of operators on C(X) have been characterized in terms of their representing Radon vector measures.
The aim of this paper is to extend these classical results to the setting, where X is a completely regular Hausdorff k-space.
Throughout the paper, we assume that (X, T) is a completely regular Hausdorff space. By K we denote the family of all compact sets in X. Let Bo denote the -algebra of Borel sets in X.
Let C b (X) (resp. B(Bo)) denote the Banach space of all bounded continuous (resp. bounded Bo-measurable) scalar functions on X, equipped with the topology u of the uniform norm ‖ ⋅ ‖ ∞ . By S(Bo) we denote the space of all Bo-simple scalar functions on X. Let C b (X) � stand for the Banach dual of C b (X).
Following [15,37] and [45,Definition 10.4,p. 137] the strict topology on C b (X) is the locally convex topology determined by the seminorms where w runs over the family W of all bounded functions w ∶ X → [0, ∞) which vanish at infinity, that is, for every >0 there exists K ∈ K such that sup t∈X⧵K w(t) ≤ . Let W 1 ∶= {w ∈ W ∶ 0 ≤ w ≤ X } . For w ∈ W 1 and > 0 let Note that the family {U w ( ) ∶ w ∈ W 1 , > 0} is a local base at 0 for .
The strict topology on C b (X) has been studied intensively (see [15,20,38,41,45]). Note that can be characterized as the finest locally convex Hausdorff topology on C b (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 (C b (X), ) is a generalized DF-space (see [38,Corollary]), and it follows that (C b (X), ) is quasinormable (see [32, p. 422]). If, in particular, X is locally compact (resp. compact), then coincides with the original 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 the Banach space of all scalar Radon measures, equipped with the total variation norm ‖ ‖ ∶= � �(X).
The following characterization of the topological dual of (C b (X), ) will be of importance (see [15,Lemma 4.5]), [20,Theorem 2]. Theorem 1.1 For a linear functional Φ on C b (X) the following statements are equivalent: The following result will be useful (see [41, Recall that a completely regular Hausdorff space (X, T) is a k-space if any subset A of X is closed whenever A ∩ K is compact for all compact sets K in X. In particular, every locally compact Hausdorff space, every metrizable space and every space satisfying the first countability axiom is a k-space (see [14,Chap. 3,§ 3]). From now on, we will assume that (X, T) is a k-space. Then, the space (C b (X), ) is complete (see [15,Theorem 2.4]).
We assume that (E, ‖ ⋅ ‖ E ) is a Banach space. Let B E ′ stand for the closed unit ball in the Banach dual E ′ of E.
Recall that a bounded linear operator T ∶ C b (X) → E is said to be absolutely summing if there exists a constant c > 0 such that for any finite set The infimum of number of c > 0 satisfying (1.1) denoted by ‖T‖ as is called an absolutely summing norm of T.
It is known that a bounded linear operator T ∶ C b (X) → E is absolutely summing if and only if T maps unconditionally convergent series in C b (X) into absolutely convergent series in E (see [9, Definition 1, p. 161 and Proposition 2, p. 162]).
For t ∈ X , let t stand for the point mass measure, that is, (i) T is absolutely summing.

Lemma 1.3 For a bounded linear operator
(ii) There exists c > 0 such that for any set {u 1 , … , u n } in C b (X) , Proof (i)⇒(ii) There exists c > 0 such that for any set {u 1 , … , u n } in C b (X), Note that we have (see [1, p. 205]), Hence, we get, (ii)⇒(i) This is obvious. ◻ The general theory of absolutely summing operators between locally convex spaces was developed by Pietsch [27].
Following [27, 1.2, pp. 23-24], we say that a sequence (u where U w ( ) 0 stands for the polar of U w ( ) with respect to the pairing ⟨C b (X), M(X)⟩ . Then, E w, is a seminorm on 1 w (C b (X), ) and the family {E w, ∶ w ∈ W 1 , > 0} generates the so-called E -topology on 1 w (C b (X), ) (see [27, 1.2.3]). Let F(ℕ) denote the family of all finite sets in ℕ , the set of all natural numbers. By 1 s (C b (X), ) we denote the E-closed subspace of 1 w (C b (X), ) consisting of all -summable sequences in C b (X) (see [27, 1.3]). In view of [27, Theorem 1.3.6] a sequence (u n ) ∈ 1 s (C b (X), ) if and only if the net (s M ) M∈F(ℕ) of partial sums Let 1 (E) stand for the linear space of all absolutely summable sequences in E, [27, 1.4]). According to [27, 2.1], we have Recall that a linear operator T ∶ C b (X) → E is said to be -compact (resp. -weakly compact) if there exists a -neighborhood V of 0 such that T(V) is a relatively norm compact (resp. relatively weakly compact) subset of E.
We will say that an operator T ∶ C b (X) → E is compact (resp. weakly compact) if T is u -compact (resp. u -weakly compact).
Then, the following statements are equivalent: (i) T is weakly compact (resp. compact).
Proof (i)⇒(ii) Assume that (i) holds. Topologies and u have the same bounded sets in C b (X) , so T maps -bounded sets onto relatively weakly compact (resp. norm compact) sets in E. Since the space (C b (X), ) is quasinormable, by the Grothendieck classical result (see [32, p. 429]), we obtain that T is -weakly compact (resp. -compact).

Definition 1.6 A linear operator
Then, we say that T is dominated by .

Proposition 1.7 Every dominated operator
there exist a uniformly bounded and uniformly tight sequence ( n ) in M(X), a bounded sequence (e n ) in E and a sequence ( n ) ∈ 1 such that where the infimum is taken over all sequences ( n ) in M(X), (e n ) in E and ( n ) ∈ 1 such that T admits a representation (1.2).
In [24], the theory of integral representation of continuous operators on C b (X) , equipped with the strict topology has been developed. Making use of the results of [24], we study -absolutely summing operators, compact operators and -nuclear operators T ∶ C b (X) → E . We characterize compact operators and -nuclear operators T ∶ C b (X) → E in terms of their representing measures (see Theorems 4.1 and 5.1 below). It is shown that dominated operators and -absolutely summing Corollary 3.4) and if, in particular, E has the Radon-Nikodym property, then -absolutely summing and -nuclear operators

Integral representation
In this section, we collect basic concepts and facts concerning integral representation of operators on C b (X) that will be useful (see [24] for notation and more details).
Let m ∶ Bo → E be a finitely additive measure. By |m|(A) (resp. ‖m‖(A)) , we denote the variation (resp. the semivariation) of m on A ∈ Bo (see [9, Definition 4, For e � ∈ E � , let Then, where |m e � |(A) stands for the variation of m e ′ on A ∈ Bo.
Recall that a countably additive measure m ∶ Bo → E is called a Radon measure if its semivariation ‖m‖ is regular, i.e., for each A ∈ Bo and > 0 there exist We will need the following result (see [12,§15.6,Proposition 19]).
Assume that m ∶ Bo → E is a finitely additive measure with ‖m‖(X) < ∞ . Then, for every v ∈ B(Bo) , one can define the so-called immediate integral ∫ X v dm ∈ E by where (s n ) is a sequence in S(Bo) such that ‖s n − v‖ ∞ → 0 (see [9, p. 5], [ 3) coincides with the immediate integral defined in (2.1). We have Hence, the corresponding integration operator T m ∶ L 1 (|m|) → E given by Let C b (X) � and C b (X) �� denote the dual and the bidual of (C b (X), ) . Since -bounded subsets of C b (X) are u -bounded, the strong topology Then we can define the biconjugate mapping where m e � ∈ M(X) for every e � ∈ E � . From the general properties of the operator T ′′ it follows that T (C b (X)) ⊂ i E (E) and According to [24,Theorem 4.2], we have the following characterization of ( , The following result will be useful. (ii) Assume that |m|(X) < ∞ . Then m is strongly additive (see [9,Proposition 15,p. 7]) and hence the operator T ∶ B(Bo) → E �� is weakly compact (see [9, Theorem 1, p. 148]). Therefore, in view of (2.5), the operator T ∶ C b (X) → E is weakly compact and by Theorem 2.2, m is a Radon measure. Using Lemma 2.1, we get |m| ∈ M + (X) . ◻

Absolutely summing operators
In this section, we characterize -absolutely summing operators T ∶ C b (X) → E and show that -absolutely summing operators and dominated operators on C b (X) coincide. We will need the following lemma.
, the following statements are equivalent: Proof , M(X))-bounded, and hence it is -bounded. It follows that sup{‖ ∑ i∈M i u i ‖ ∞ ∶ i = ±1 , M ∈ F(ℕ)} < ∞ because u and have the same bounded sets. ◻ The following theorem characterizes -absolutely summing operators Then the following statements are equivalent: Hence, for every n ∈ ℕ , we have and it follows that ∑ ∞ n=1 ‖T(u n )‖ E < ∞ , as desired. (ii)⇒(iii) Assume that (ii) holds and the series ∑ ∞ n=1 u n is unconditionally -convergent in C b (X) . Then ∑ ∞ n=1 � ∫ X u n d � < ∞ for every ∈ M(X) and it follows that ≤ . This means that the partial sums ∑ n i=1 u (i) form a -Cauchy sequence in C b (X) . Since the space (C b (X), ) is complete, we obtain that the series ∑ ∞ n=1 u n is unconditionally -convergent in C b (X) . Hence, we get (iv)⇒(i) Assume that (iv) holds. Let w ∈ W 1 . Then in view of [27, Theo- We will need the following lemma.

Now let > 0 be given. Then there exists a Bo-partition (
(iv) T is absolutely summing.
(iii)⇒(i) Assume that (iii) holds. Then in view of Theorem 3.2, there exists c > 0 such that for every u 1 , … , u n ∈ C b (X) , we have Let (u n ) be a sequence in C b (X) such that sup n ‖u n ‖ ∞ = a < ∞ and supp u n ∩ supp u k = � if n ≠ k . Then, for ∈ M(X) with | |(X) ≤ 1 , we have Then ∑ ∞ n=1 ‖T(u n )‖ E ≤ ca < ∞ , so ‖T(u n )‖ E → 0 and according to Theorem 2.2 T is weakly compact. Hence by Theorem 2.3 m ∶= j E •m ∶ Bo → E is a Radon measure and Now, we shall show that |m|(X) = |m|(X) < ∞ . In fact, let (A i ) n i=1 be a Bo-partition of X and > 0 be given.
Let u ∈ C b (X) and choose a sequence (s n ) in S(Bo) such that ‖u − s n ‖ ∞ → 0 . Hence This means that M (u) = ∫ X u dm .

Corollary 3.6
Let T ∶ C b (X) → E be a -absolutely summing operator and m ∶ Bo → E �� be its representing measure. Then, m ∶= j E •m ∶ Bo → E is a Radon measure with |m| ∈ M + (X) and the following statements hold: (i) Since |m| ∈ M + (X) in view of Proposition 3.5, I is -absolutely summing and ‖I‖ as = ∫ X X d�m� = �m�(X) = �m�(X).
(ii) In view of Theorem 2.3 we have that T(u) = ∫ X u dm for u ∈ C b (X). Thus, we get T = S•I , where by (2.4) ‖S‖ ≤ 1 . ◻

Compact operators
The tensor product ca(Bo) ⊗ E consists of all measures m ∶ Bo → E of the form m = ∑ n i=1 ( i ⊗ e i ) , where i ∈ ca(Bo) and e i ∈ E for i = 1, … , n . Then Now, we can state a characterization of -compact operators T ∶ C b (X)→ E in terms of their representing measures m ∶ Bo → E �� (see [9,Theorem 18,p. 161], [34,Theorem 5.27] if X is compact).

Theorem 4.1
Let T ∶ C b (X) → E be a ( , ‖ ⋅ ‖ E )-continuous linear operator and m ∶ Bo → E �� be its representing measure. Then the following statements are equivalent: Since we obtain that m(Bo) is relatively norm compact in E ′′ .
(ii)⇒(i) Assume that (ii) holds. Since m(Bo) is weakly compact, the corresponding integration operator T ∶ B(Bo) → E �� is weakly compact (see [19,Theorem 7]). Then, in view of (2.5), T is weakly compact, and by Theorem 2.3 m ∶= j E •m ∶ Bo → E is countably additive and m(Bo) is relatively norm compact in E. According to the proof of [34,Theorem 5.18 For each k ∈ ℕ , let T k ∶ C b (X) → E be the finite rank operator defined by T k (u) ∶= ∫ X u dm k . For u ∈ C b (X) , we have and it follows that ‖T k − T‖ → 0 . Hence, T is a compact operator and using Proposition 1.5 we have that T is -compact. ◻

Nuclear operators
We state our main result that characterizes -nuclear operators  (i) T is -nuclear.
Let > 0 be given. Then, there exist sequences (v n ) in L 1 (|m|) and (e n ) in E with lim n ‖v n ‖ 1 = 0 = lim n ‖e n ‖ E and ( n ) ∈ 1 such that and Hence and we obtain that For n ∈ ℕ , let Note that n ∈ M(X) and � n �(X) = ‖v n ‖ 1 . Then we have sup n � n �(X) = sup n ‖v n ‖ 1 < ∞ . To show that the family { n ∶ n ∈ ℕ} is uniformly tight, let > 0 be given. Since ‖v n ‖ 1 → 0 , we can choose n ∈ ℕ such that | n |(X) ≤ for n > n . For n = 1, … , n choose K n ∈ K such that | n |(X⧵K n ) ≤ . Denote K ∶= ⋃ n n=1 K n . Then, | n |(X⧵K) ≤ for every n ∈ ℕ , as desired. Clearly for n ∈ ℕ , we have (see [7,Theorem C.8]), Hence, we have and this means that T is -nuclear. By (5.1) we get (ii)⇒(iii) Assume that (ii) holds, that is, |m|(X) < ∞ and there exists a |m|-Bochner Then, S(u) = T(u) for u ∈ C b (X) and m(A) = S( A ) for A ∈ Bo . Hence, by [9,Lemma 4,p. 62] f is essentially bounded and (iii)⇒(ii) This is obvious.
Thus, (i)⇔(ii)⇔(iii) hold. Moreover, if T is -nuclear and > 0 is given, then there exist a uniformly bounded and uniformly tight sequence ( n ) in M(X), a bounded sequence (e n ) in E and a sequence ( n ) ∈ 1 such that .
n ∫ X u d n e n for u ∈ C b (X)

Remark 5.5
A relationship between vector measures m ∶ Σ → E with a -Bochner integrable derivatives (with respect to a finite measure ) and the nuclearity of the corresponding integration operators T m ∶ L ∞ ( ) → E has been studied by Swartz [42] and Popa [30].

Nuclearity of kernel operators
It is well known that if K is a compact Hausdorff space, ∈ M + (K) and k(⋅, ⋅) ∈ C(K × K) , then the corresponding kernel operator T ∶ C(K) → C(K) between Banach spaces, defined by is nuclear (see [16,Theorem V.22,p. 99] if X = [a, b]). Now as an application of Theorem 5.1, we extend this result to the setting, where X is a k-space and the kernel operator From now on we assume that ∈ M + (X) and k(⋅, ⋅) ∈ C b (K × X) with sup t∈K |k(t, s)| ≥ c for every s ∈ X and some c > 0.
We start with the following lemma.
Let v(s) ∶= Then that is, T is a kernel operator in the sense of Sentilles (see [39,40]) with the kernel and T(u)(t) = (u)(t) for u ∈ C b (X) , t ∈ K. Now, we are ready to state our desire result.