Abstract
Let \(C_b(X)\) be the Banach lattice of all bounded continuous real-valued functions on a completely regular Hausdorff space X and \(\beta \) denote the natural strict topology on \(C_b(X)\). For a Banach space \((E,\Vert \cdot \Vert _E)\), a linear operator \(T:C_b(X)\rightarrow E\) is said to be tight if \(\Vert T(u_\alpha )\Vert _E\rightarrow 0\) whenever \((u_\alpha )\) is a uniformly bounded net in \(C_b(X)\) such that \(u_\alpha \rightarrow 0\) uniformly on all compact sets in X. It is shown that a linear operator \(T:C_b(X)\rightarrow E\) is nuclear tight if and only if T is a nuclear operator between the locally convex space \((C_b(X),\beta )\) and a Banach space E and if and only if T is Bochner representable, that is, there exist a positive Radon measure \(\mu \) on X and a E-valued \(\mu \)-Bochner integrable function g on X so that \(T(u)=\int _X u(x)g(x)d\mu \) for all \(u\in C_b(X)\).
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1 Introduction and Preliminaries
Nuclear operators on the Banach space C(X) of continuous functions on a compact Hausdorff space X have been studied intensively (see [5, 8, 11, 16,17,18, 23]). In particular, due to Tong [23, Theorem 1.2] a linear operator T from C(X) to a Banach space E is nuclear if and only if T is Bochner representable (see also [5, Theorem 4, pp. 173–174], [18, Proposition 5.30]).
The main aim of the present paper is to extend and generalize this study to the setting of X being a completely regular Hausdorff space (see Theorems 2.2, 3.2 and Corollary 3.3).
Throughout the paper we assume that X is a completely regular Hausdorff space. Let \({\mathcal B}o\) (resp. \({\mathcal B}\)) denote the \(\sigma \)-algebra of all Borel sets (resp., the algebra of all Baire sets) in X.
Let \(C_b(X)\) (resp. \(B({\mathcal B}o)\)) denote the Banach lattice of all bounded continuous (resp. bounded \({\mathcal B}o\)-measurable) real-valued functions on X, equipped with the uniform norm \(\Vert \cdot \Vert _\infty \). Let \(C_b(X)'\) and \(C_b(X)''\) stand for the dual and the bidual of \(C_b(X)\), respectively.
Recall that a countably additive real-valued measure \(\lambda \) on \({\mathcal B}o\) is called a Radon measure if for every \(A\in {\mathcal B}o\) and \(\varepsilon >0\), there exist a compact set K and an open set O in X with \(K\subset A\subset O\) such that \(|\lambda |(O\smallsetminus K)\le \varepsilon \)
By \(rca({\mathcal B}o)\) we denote the Banach lattice of all real-valued Radon measures \(\lambda \), equipped with the norm \(\Vert \lambda \Vert :=|\lambda |(X)\).
From now on we assume that \((E,\Vert \cdot \Vert _E)\) is a real Banach space.
For \(\mu \in rca({\mathcal B}o)^+\), let \(L^1(\mu ,E)\) denote the Banach space of \(\mu \)-equivalence classes of all E-valued Bochner integrable functions g on X, equipped with the norm \(\Vert g\Vert _1:=\int _X\Vert g(x)\Vert _E\,d\mu \).
Following [23] we say that a linear operator \(T:C_b(X)\rightarrow E\) is Bochner representable if there exist a measure \(\mu \in rca({\mathcal B}o)^+\) and a function \(g\in L^1(\mu ,E)\) so that
Recall that a linear operator \(T:C_b(X)\rightarrow E\) is said to be tight if \(\Vert T(u_\alpha )\Vert _E\rightarrow 0\) whenever \((u_\alpha )\) is a uniformly bounded net in \(C_b(X)\) such that \(u_\alpha \rightarrow 0\) uniformly on all compact sets in X (see [15, Definition 1.1]). Note that if X is compact, then every bounded operator \(T:C_b(X)\rightarrow E\) is tight.
The concept of nuclear operators between Banach spaces is due to Grothendieck [9, 10] (see also [26, p. 279], [16, Chap. 3], [17, 5, Chap. 6], [8, Chap. 5], [18, 22, 24]).
Recall (see [26, p. 279], [22]) that a linear operator \(T:C_b(X)\rightarrow E\) is said to be nuclear if there exist a bounded sequence \((\Psi _n)\) in \(C_b(X)'\), a bounded sequence \((e_n)\) in E and a sequence \((\alpha _n)\in \ell ^1\) so that
Then the space \({\mathcal N}(C_b(X),E)\) of all nuclear operators \(T:C_b(X)\rightarrow E\) is a Banach space, equipped with the nuclear norm defined by
where the infimum is taken over all sequences \((\Psi _n)\) in \(C_b(X)'\), \((e_n)\) in E and \((\alpha _n)\in \ell ^1\) such that T admits a representation (1.1) (see [16, Proposition, p. 51]).
2 Nuclear Tight Operators on \({\varvec{C_b(X)}}\)
For terminology concerning vector measures, we refer the reader to [5, 6, 18].
Let M(X) denote the Banach lattice of all finitely additive bounded real-valued measures \(\nu \) on \({\mathcal B}\) (inner regular by zero-sets), equipped with the norm \(\Vert \nu \Vert :=|\nu |(X)\). Due to the Alexandrov representation theorem (see [25, Theorem 5.1]) \(C_b(X)'\) can be identified with M(X) through the lattice isomorphism
where \(\Psi _\nu (u)=\int _X u(x)\,d\nu \) for all \(u\in C_b(X)\) and \(\Vert \Psi _\mu \Vert =|\nu |(X)\).
By \(B({\mathcal B})\) we denote the Banach lattice (with the uniform norm \(\Vert \cdot \Vert _\infty \)) of all bounded real functions on X that are uniform limits of sequences of \({\mathcal B}\)-simple functions on X. Then \(C_b(X)\subset B({\mathcal B})\) (see [1, Lemma 1.2]), and one can embed isometrically \(B({\mathcal B})\) in \(C_b(X)''\) by the mapping \(\pi :B({\mathcal B})\rightarrow C_b(X)''\) , where for \(v\in B({\mathcal B})\),
Let \(E'\) and \(E''\) denote the dual and the bidual of a Banach space E. Let \(i:E\rightarrow E''\) stand for the canonical isometry, that is, \(i(e)(e')=e'(e)\) for all \(e\in E\) and \(e'\in E'\). Let \(j:i(E)\rightarrow E\) denote the left inverse of i, that is, \(j(i(e))=e\) for \(e\in E\).
Assume that \(T:C_b(X)\rightarrow E\) is a wekly compact bounded linear operator. Let \(T':E'\rightarrow C_b(X)'\) and \(T'':C_b(X)''\rightarrow E''\) denote the conjugate and biconjugate of T, respectively. Then by the Gantmacher type theorem (see [2, Theorem 17.2] we have \(T''(C_b(X)'')\subset i(E)\). Let us put
Then \(m:{\mathcal B}\rightarrow E\) is called the representing measure of T and
The strict topology \(\beta \) on \(C_b(X)\) has been studied intensively (see [4, 7, 12, 21, 25]). \(\beta \) can be characterized as the finest locally convex Hausdorff topology on \(C_b(X)\) which coincides with the compact-open topology \(\tau _c\) on all \(\Vert \cdot \Vert _\infty \)-bounded sets in \(C_b(X)\) (see [21, Theorem 2.4], [20]). This means that \((C_b(X),\beta )\) is a generalized DF-space (see [20, Corollary]). Then \(\beta \) is weaker than the \(\Vert \cdot \Vert _\infty \)-norm topology on \(C_b(X)\), and if, in particular, X is compact, then these topologies coincide.
Due to [7, Lemma 4.5] the dual space \((C_b(X),\beta )'\) of \((C_b(X),\beta )\) can be identified with \(rca({\mathcal B}o)\) through the isomorphic isometry
where \(\Phi _\lambda (u)=\int _X u(x)\,d\lambda \) for all \(u\in C_b(X)\) and \(\Vert \Phi _\lambda \Vert = |\lambda |(X)\).
The following characterization of weakly compact tight operators \(T:C_b(X)\!\rightarrow \!E\) will be useful (see [14, Theorem 5.5]).
Theorem 2.1
Assume that T is a weakly compact operator and \(m:{\mathcal B}\rightarrow E\) is its representing measure. Then the following statements are equivalent:
-
(i)
T is tight.
-
(ii)
T is \((\beta ,\Vert \cdot \Vert _E)\)-continuous.
-
(iii)
m can be uniquely extended to a Radon measure \({\widetilde{m}}:{\mathcal B}o\rightarrow E\), that is, \({\widetilde{m}}\) is countably additive and for every \(A\in {\mathcal B}o\) and \(\varepsilon >0,\) there exist a compact set K and an open set O in X with \(K\subset A\subset O\) such that \(\Vert {\widetilde{m}}\Vert (O\smallsetminus K)\le \varepsilon \).
Then \(T(u)=\int _X u(x)\,dm=\int _X u(x)\,d{\widetilde{m}}\) for all \(u\in C_b(X).\)
From now on \(|{\widetilde{m}}|(A)\) stands for the variation of the measure \({\widetilde{m}}\) on a set \(A\in {\mathcal B}o\).
The following result establishes the relationship between nuclear tight operators \(T:C_b(X)\rightarrow E\) and their representing measures \({\widetilde{m}}:{\mathcal B}o\rightarrow E\).
Theorem 2.2
Assume that \(T:C_b(X)\rightarrow E\) is a weakly compact tight operator and \(m:{\mathcal B}\rightarrow E\) is its representing measure. Then the following statements are equivalent:
-
(i)
T is nuclear.
-
(ii)
\(|{\widetilde{m}}|\in rca({\mathcal B}o)^+\) and the measure \({\widetilde{m}}:{\mathcal B}o\rightarrow E\) has a \(|{\widetilde{m}}|\)-Bochner integrable derivative.
-
(iii)
\(|{\widetilde{m}}|\in rca ({\mathcal B}o)^+\) and T is Bochner representable with respect to \(|{\widetilde{m}}|\).
In this case, \(\Vert T\Vert _{nuc}=|{\widetilde{m}}|(X)\).
Proof
(i)\(\Leftrightarrow \)(ii) See [15, Theorem 3.3].
(ii)\(\Rightarrow \)(iii) Assume that (ii) holds, that is, there exists a function \(f\in L^1(|{\widetilde{m}}|,E)\) such that
Let \(u\in C_b(X)\). Since \(u\in B({\mathcal B}o)\), there exists a sequence \((s_n)\) of \({\mathcal B}o\)-simple functions on X such that \(\Vert u-s_n\Vert _\infty \rightarrow 0\). Then for every \(n\in {\mathbb {N}}\),
and hence we have
Since we have that
we get \(T(u)=\int _X u(x)\,f(x)\,d|{\widetilde{m}}|\), as desired.
(iii)\(\Rightarrow \)(ii) This is obvious.
In view of [15, Theorem 3.3] we have that \(\Vert T\Vert _{nuc}=|{\widetilde{m}}|(X)\). \(\square \)
Grothendieck carried over the concept of nuclear operators to locally convex spaces (see [26, p. 289], [19, Chap. 3, Sect. 7], [13, Sect. 17.3], [24, Sect. 47]).
We will need the following result (see [21, Theorem 5.1]).
Theorem 2.3
For a subset \({\mathcal M}\) of \(rca({\mathcal B}o)\) the following statements are equivalent:
-
(i)
\(\{\Phi _\lambda :\lambda \in {\mathcal M}\}\) is \(\beta \)-equicontinuous.
-
(ii)
\(\sup _{\lambda \in {\mathcal M}}\Vert \lambda \Vert <\infty \) and \({\mathcal M}\) is uniformly tight, i.e., for every \(\varepsilon >0,\) there exists a compact set K in X such that \(\sup _{\lambda \in {\mathcal M}}|\lambda |(X\smallsetminus K)\le \varepsilon \).
Following [19, Chap. 3, Sect. 7] (see also [24, Sect. 47], [13, 17.3, p. 379]) and using Theorem 2.3 we have the following definition.
Definition 2.1
A linear operator \(T:C_b(X)\rightarrow E\) is said to be a nuclear operator between the locally convex space \((C_b(X),\beta )\) and a Banach space E, if there exist a bounded uniformly tight sequence \((\lambda _n)\) in \(rca({\mathcal B}o)\), a bounded sequence \((e_n)\) in E and a sequence \((\alpha _n)\in \ell ^1\) so that
Then T is \((\beta ,\Vert \cdot \Vert _E)\)-compact, that is, T(V) is relatively norm compact in E for some \(\beta \)-neighborhood V of zero in \(C_b(X)\) (see [19, Chap. 3, Sect. 7, Corollary 1]). Hence T is \((\beta ,\Vert \cdot \Vert _E)\)-continuous. Let us put
where the infimum is taken over all sequences \((\lambda _n)\) in \(rca({\mathcal B}o)\), \((e_n)\) in E and sequences \((\alpha _n)\in \ell ^1\) such that T admits a representation (2.2).
Now we can state the following result.
Corollary 2.4
For a linear subset operator \(T:C_b(X)\rightarrow E,\) the following statements are equivalent:
-
(i)
T is a nuclear tight operator.
-
(ii)
T is a nuclear operator between the locally convex space \((C_b(X),\beta )\) and a Banach space E.
In this case, \(\Vert T\Vert _{nuc}=\Vert T\Vert _{\beta -nuc}\).
Proof
(i)\(\Rightarrow \)(ii) Assume that (i) holds. Then in view of Theorem 2.2\(|{\widetilde{m}}|\in rca({\mathcal B}o)^+\) and there exists a function \(f\in L^1(|{\widetilde{m}}|,E)\) so that
and
Let \(L^1(|{\widetilde{m}}|){\hat{\otimes }}_\gamma E\) denote the projective tensor product of \(L^1(|{\widetilde{m}}|)\) and E, equipped with the completed norm \(\gamma \) (see [5, p. 227], [18, p. 17]). Note that for \(w\in \) \(L^1(|{\widetilde{m}}|)\,{\hat{\otimes }}_\gamma \,E\), we have
where the infimum is taken over all sequences \((v_n)\) in \(L^1(|{\widetilde{m}}|)\) and \((e_n)\) in E with \(\lim _n\Vert v_n\Vert _1=0=\lim _n\Vert e_n\Vert _E\) and \((\alpha _n)\in \ell ^1\) such that \(w=\sum ^\infty _{n=1} \alpha _n v_n\otimes \,e_n\) in \(\gamma \)-norm (see [18, Proposition 2.8, pp. 21–22]).
It is known that \(L^1(|{\widetilde{m}}|){\hat{\otimes }}_\gamma E\) is isometrically isomorphic to the Banach space \((L^1(|{\widetilde{m}}|,E),\Vert \cdot \Vert _1)\) by the isometry \(J:L^1(|{\widetilde{m}}|)\,{\hat{\otimes }}_\gamma \,E\rightarrow L^1(|{\widetilde{m}}|,E)\), defined by
(see [5, Example 10, p. 228], [18, Example 2.19, p. 29]).
Let \(\varepsilon >0\) be given. Then there exist sequences \((v_n)\) in \(L^1(|{\widetilde{m}}|)\) and \((e_n)\) in E with \(\lim _n\Vert v_n\Vert _1=0=\lim _n\Vert e_n\Vert _E\) and \((\alpha _n)\in \ell ^1\) so that
and
Thus this follows that
and hence
For \(n\in {\mathbb {N}}\), let \(\lambda _n(A):=\int _A v_n(x)\,d|{\widetilde{m}}|\) for all \(A\in {\mathcal B}o\). Then we have
(see [3, Theorem 8C, p. 380]). Note that \(\lambda _n\in rca({\mathcal B}o)\) and \(|\lambda _n|(X)=\Vert v_n\Vert _1\). We shall show that the family \(\{\lambda _n:n\in {\mathbb {N}}\}\) is uniformly tight.
Indeed, let \(\varepsilon >0\) be given. Since \(\lim \Vert v_n\Vert _1=0\), we can choose \(n_\varepsilon \in {\mathbb {N}}\) such that \(|\lambda _n|(X)\le \varepsilon \) for \(n>n_\varepsilon \). For \(n=1,\dots ,n_\varepsilon \), choose a compact set \(K_n\) in X such that \(|\lambda _n|(X\smallsetminus K_n)\le \varepsilon \). Denote \(K=\bigcup ^{n_\varepsilon }_{n=1}K_n\). Then \(|\lambda _n|(X\smallsetminus K)\le \varepsilon \) for all \(n\in {\mathbb {N}}\), as desired.
Since \(T(u)=\sum ^\infty _{n=1}\alpha _n(\int _X u(x)\,d\lambda _n)e_n\) for all \(u\in C_b(X)\), we see that T is a nuclear operator between \((C_b(X),\beta )\) and a Banach space E, and in view of (2.3) we get
Since \(\Vert T\Vert _{nuc}\le \Vert T\Vert _{\beta -nuc}\), we have that \(\Vert T\Vert _{nuc}=\Vert T\Vert _{\beta -nuc}\).
(ii)\(\Rightarrow \)(i) Assume that (ii) holds. Then T is nuclear and \((\beta ,\Vert \cdot \Vert _E)\)-continuous. Hence by Theorem 2.1T is tight. \(\square \)
As a consequence of Corollary 2.4, we get
Corollary 2.5
Assume that \(T:C_b(X)\rightarrow E\) is a nuclear tight operator and \(m:{\mathcal B}\rightarrow E\) is its representing measure. Then the mapping
is a nuclear operator and \(\Vert T^*\Vert _{nuc}=\Vert T\Vert _{nuc}=|{\widetilde{m}}|(X).\)
Proof
Let \(\varepsilon >0\) be given. According to Corollary 2.4, there exist bounded sequences \((\lambda _n)\) in \(rca({\mathcal B}o)\) and \((e_n)\) in E and a sequence \((\alpha _n)\in \ell ^1\) so that
and
Thus it follows that for each \(e'\in E'\), we have
By Theorem 2.1 for each \(e'\in E'\), we have
where \(e'\circ {\widetilde{m}}\in rca({\mathcal B}o)\). Hence we have
This means that \(T^*\) is a nuclear operator and in view of (2.4) we get
Now we shall show that
Indeed, let \(\varepsilon >0\) be given. Since \(T^*\) is a nuclear operator, there exist a bounded sequence \((e''_n)\) in \(E''\), a bounded sequence \((\lambda _n)\) in \(rca({\mathcal B}o)\) and \((\alpha _n)\in \ell ^1\) so that
and
Then for \(A\in {\mathcal B}o\), we have
By the Hahn-Banach theorem for every \(A\in {\mathcal B}o\), there exists \(e'_A\in E'\) with \(\Vert e'_A\Vert _{E'}=1\) such that \(\Vert {\widetilde{m}}(A)\Vert _E=|(e'_A\circ {\widetilde{m}})(A)|\). Hence, if \(\Pi \) is a finite \({\mathcal B}o\)-partition of X, then using (2.5) we get
Since \(\varepsilon >0\) is arbitrary, we get \(|{\widetilde{m}}|(X)\le \Vert T^*\Vert _{nuc}\) and hence \(\Vert T^*\Vert _{nuc}=|{\widetilde{m}}|(X)=\Vert T\Vert _{nuc}\). \(\square \)
3 Bochner Representable Operators on \({C_b(X)}\)
Making use of [6, Sect. 2, F, Theorem 30, p. 32], we have the following
Lemma 3.1
For \(\mu \in rca({\mathcal B}o)^+\) and \(g\in L^1(\mu ,E)\), let us put
and \(f_g(x):=g(x)/\Vert g(x)\Vert _E\) if \(g(x)\ne 0\) and \(f_g(x):=0\) if \(g(x)=0.\)
Then
In particular, \(\int _A f_g(x)\,d\lambda =\int _A g(x)\,d\mu \) for all \(A\in {\mathcal B}o.\)
Now we can characterize Bochner representable operators \(T:C_b(X)\rightarrow E\).
Theorem 3.2
If \(T:C_b(X)\rightarrow E\) is a Bochner representable operator, then T is a nuclear tight operator.
Proof
There exist \(\mu \in rca({\mathcal B}o)^+\) and \(g\in L^1(\mu ,E)\) so that
Define
and let
Then
and
(see [5, Theorem 13, p. 6 and Theorem 4, p. 46]). Since \(|m_S|\) is a \(\mu \)-absolutely continuous measure, we obtain that \(|m_S|\in rca({\mathcal B}o)^+\) and hence \(m_S:{\mathcal B}o\rightarrow E\) is a Radon measure. Thus, it follows that \(S:B({\mathcal B}o)\rightarrow E\) is weakly compact (see [5, Theorem 1, p. 148]), and hence \(T:C_b(X)\rightarrow E\) is weakly compact and
To show that T is a tight operator, assume that \((u_\alpha )\) is a net in \(C_b(X)\) such that \(\sup _\alpha \Vert u_\alpha \Vert _\infty =a<\infty \) and \(u_\alpha \rightarrow 0\) uniformly on all compact sets in X.
Let \(\varepsilon >0\) be given and choose \(\delta >0\) such that
Choose a compact subset \(K_\varepsilon \) of X such that \(\mu (X\smallsetminus K_\varepsilon ) \le \delta \). Then there exists \(\alpha _0\) such that
Hence for \(\alpha \ge \alpha _0\), we get
This means that T is a tight operator, as desired. Then by Theorem 2.1 and (3.1) we get
and hence for each \(e'\in E'\), we have
Since \(e'\circ \,{\widetilde{m}}\in rca({\mathcal B}o)\) and \(e'\circ m_S\in rca({\mathcal B}o)\), in view of (2.1) we get \(e'\circ \,{\widetilde{m}}=e'\circ \,m_S\) for every \(e'\in E'\). Thus this follows that \({\widetilde{m}}=m_S\), that is, for all \(A\in {\mathcal B}o\),
By Lemma 3.1\(f_g\in L^1(|{\widetilde{m}}|,E)\) and for all \(A\in {\mathcal B}o\),
According to Theorem 2.2 this means that T is a nuclear operator. \(\square \)
Now we can state an extension of Tong’s result (see [23, Theorem 1.2]) to the setting of completely regular Hausdorff spaces.
Corollary 3.3
For a linear operator \(T:C_b(X)\rightarrow E\), the following statements are equivalent:
-
(i)
T is nuclear and tight.
-
(ii)
T is Bochner representable.
-
(iii)
T is a nuclear operator between the locally convex space \((C_b(X),\beta )\) and a Banach space E.
Proof
(i)\(\Rightarrow \)(ii) This follows from Theorem 2.2.
(ii)\(\Rightarrow \)(i) This follows from Theorem 3.2.
(i)\(\Leftrightarrow \)(iii) See Corollary 2.4. \(\square \)
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Nowak, M. Bochner Representable Operators on Spaces of Continuous Functions. Results Math 77, 205 (2022). https://doi.org/10.1007/s00025-022-01727-z
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DOI: https://doi.org/10.1007/s00025-022-01727-z
Keywords
- Spaces of bounded continuous functions
- radon vector measures
- tight operators
- nuclear operators
- strict topology