Let X be a completely regular Hausdorff space and let E,·E and (F,·F) be Banach spaces. Let Cb(X,E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology βσ. We study the relationship between important classes of (βσ,·F)-continuous linear operators T:Cb(X,E)→F (strongly bounded, unconditionally converging, weakly completely continuous, completely continuous, weakly compact, nuclear, and strictly singular) and the corresponding operator measures given by Riesz representing theorems. Some applications concerning the coincidence among these classes of operators are derived.
1. Introduction and Terminology
Throughout the paper let (E,·E) and (F,·F) be real Banach spaces and let E′ and F′ denote the Banach duals of E and F, respectively. By BF′ and BE we denote the closed unit ball in F′ and E, respectively. By L(E,F) we denote the space of all bounded linear operators from E to F. Given a locally convex space (L,ξ) by L,ξ′ or Lξ′ we will denote its topological dual. We denote by σ(L,K) the weak topology on L with respect to a dual pair L,K. Let F(N) stand for the collection of all finite subsets of the set N of all natural numbers.
Assume that X is a completely regular Hausdorff space. By Z (resp., P) we will denote the family of all zero sets (resp., of cozero sets) in X, respectively. Let Cb(X,E) stand for the Banach space of all bounded continuous functions f:X→E, equipped with the uniform norm ·. We write Cb(X) instead of Cb(X,R). By CbX,E′ we denote the Banach dual of Cb(X,E). For f∈Cb(X,E) let f~(t)=f(t)E for t∈X.
Let B (resp., Ba) stand for the algebra (resp., σ-algebra) of Baire sets in X, respectively. Let B(B,E) (resp., B(Ba,E)) stand for the Banach space of all totally B-measurable (resp., totally Ba-measurable) functions f:X→E (see [1, 2]).
The strict topology βσ (called also a superstrict topology and denoted by β1) on Cb(X) and Cb(X,E) is of importance in the topological measure theory (see [3–9] for definitions and more details). Cb(X,E)βσ′ is a closed subspace of the Banach space CbX,E′ and βσ-bounded sets in Cb(X,E) are ·-bounded. It is known that Cb(X)⊗E is βσ-dense in Cb(X,E) if one of the following conditions holds (see [6, Theorems 5.1 and 5.2]):
Xhas a σ-compact dense subset (e.g., X separable).
X is a D-space (see [10]).
E is a D-space.
Remark 1.
Throughout the paper we will assume that Cb(X)⊗E is βσ-dense in Cb(X,E).
For X being a locally compact Hausdorff space, by Co(X,E) we denote the Banach space of all continuous functions f:X→E tending to zero at infinity, equipped with the uniform norm. If X is a compact Hausdorff space, then βσ coincides with the uniform norm topology on Cb(X,E). In this case we write simply C(X,E) instead of Cb(X,E).
Let M(X) stand for the Banach lattice of all Baire measures on B, provided with the norm ν=ν(X) (= the total variation of ν). Due to the Alexandrov representation theorem CbX′ can be identified with M(X) through the lattice isomorphism M(X)∋ν↦φν∈CbX′, where φν(u)=∫Xudν for u∈Cb(X), and φν=ν (see [4, Theorem 5.1]).
By M(X,E′) we denote the set of all finitely additive measures μ:B→E′ with the following properties:
for each x∈E, the function μx:B→R defined by μx(A)=μ(A)(x) belongs to M(X);
μ(X)<∞, where μ(A) stands for the variation of μ on A∈B.
Let Crc(X,E) denote the Banach space of all continuous functions h:X→E such that h(X) is a relatively compact set in E, equipped with the uniform norm ·. Then Cb(X)⊗E⊂Crc(X,E)⊂B(B,E). In view of [11, Theorem 2.5] CrcX,E′ can be identified with MX,E′ through the linear mapping M(X,E′)∋μ↦Φμ∈CrcX,E′, where Φμ(h)=∫Xhdμ for h∈Crc(X,E) and Φμ=μ(X). Then one can embed B(B,E) into CrcX,E′′ by the mapping π:B(B,E)→CrcX,E′′, where, for g∈B(B,E),
(1)π(g)(Φμ)≔∫Xgdμforμ∈MX,E′.
Assume that T:Cb(X,E)→F is a bounded linear operator. Then we can define the corresponding operator measure m:B→LE,F′′ (called the representing measure of T) by setting
(2)mAx≔T|CrcX,E′′∘π(1A⊗x)forA∈B,x∈E.
Here TCrcX,E′′:CrcX,E′′→F′′ stand for the biconjugate of TCrc(X,E). Then m~(X)<∞, where the semivariation m~(A) of m on A∈B is defined by m~(A)≔supΣm(Ai)(xi)F′′, where the supremum is taken over all finite B-partitions (Ai) of A and xi∈BE for each i. For y′∈F′ let us put
(3)my′(A)(x)≔(m(A)(x))(y′)forA∈B,x∈E.
Let my′(A) stand for the variation of my′ on A. Then (see [1, §4, Proposition 5])
(4)m~(A)=supmy′A:y′∈BF′.
By MX,LE,F′′ we denote the set of all operator measures m:B→LE,F′′ such that m~(X)<∞ and my′∈MX,E′ for each y′∈F′.
Let iF:F→F′′ denote the canonical embedding; that is, iF(y)(y′)=y′(y) for y∈F, y′∈F′. Moreover, let jF:iF(F)→F stand for the left inverse of iF; that is, jF∘iF=idF.
For x∈E define
(5)Txu≔Tu⊗xforu∈CbX,mx(A)≔m(A)(x)forA∈B.
The following Bartle-Dunfor-Schwartz type theorem will be useful (see [12, Theorem 2], [13, Theorem 5, pages 153-154]).
Theorem 2.
Let T:Cb(X,E)→F be a bounded linear operator and M(X,L(E,F′′)) be its representing measure. Then for each x∈E the following statements are equivalent:
Tx:Cb(X)→F is weakly compact.
m(A)(x)∈iF(F) for each A∈B and {jF(m(A)(x)):A∈B} is a relatively weakly compact set in F.
mx:B→F′′ is strongly bounded.
Following [14–16] we have the following definition.
Definition 3.
A bounded linear operator T:Cb(X,E)→F is said to be strongly bounded if its representing measure m∈M(X,L(E,F′′)) is strongly bounded; that is, m~(An)→0 whenever (An) is a pairwise disjoint sequence in B.
Note that m∈M(X,L(E,F′′)) is strongly bounded if and only if the family {my′:y′∈BF′} is uniformly strongly additive.
For each x∈E, mx(A)F′′≤m~(A)xE for A∈B. It follows that if T:Cb(X,E)→F is strongly bounded, then Tx:Cb(X)→F is weakly compact, and hence m(A)(x)∈iF(F) for A∈B (see Theorem 2).
For X being a compact Hausdorff space (resp., a locally compact Hausdorff space) different classes of bounded linera operators T:Cb(X,E)→F (resp., T:C0(X,E)→F) have been studied intensively; see [14–33]. The study of the relationship between operators T:C(X,E)→F (resp., T:C0(X,E)→F) and their representing operator-valued measures is a central problem in the theory. The main aim of the present paper is to extend to “the completely regular setting” some classical results concerning various classes of bounded operators T:C(X,E)→F (resp., T:C0(X,E)→F), where X is a compact Hausdorff space (resp., a locally compact Hausdorff space). In [12] using the device of embedding the space B(B,E) into CrcX,E′′ we establish general Riesz representation theorems for (βσ,·F)-continuous linear operators T:Cb(X,E)→F with respect to the representing measures m:B→L(E,F′′) (see Theorems 6 and 8 below). In Section 3 we show that if T:Cb(X,E)→F is (βσ,·F)-continuous and strongly bounded, then its representing measure m:B→L(E,F′′) has its values in L(E,F) and possesses a unique extension m¯:Ba→L(E,F) that is variationally semiregular; that is, the set {|m¯y′|:y′∈BF′} is uniformly countably additive (see Theorem 11 below). In Sections 4–9 we study the folowing classes of (βσ,·F)-continuous linear operators T:Cb(X,E)→F: unconditionally converging, weakly completely continuous, completely continuous, weakly compact, nuclear, and strongly singular. We show that if a (βσ,·F)-continuous linear operator T:Cb(X,E)→F belongs to any of these classes of operators, then T is strongly bounded and, for each A∈Ba, the operator m¯(A):E→F shares the property of T (see Theorems 17, 23, 26, 29, 34, and 36 below). We derive some applications concerning to the coincidence among these classes of (βσ,·)-continuous operators (see Corollary 13, Theorems 18 and 19, Corollary 27, Theorem 29).
2. Integral Representation of Continuous Operators on Cb(X,E)
The space of all σ-aditive members of M(X) will be denoted by Mσ(X) (see [3, 4]). Then CbX,βσ′=φν:ν∈MσX. Let
(6)MσX,E′≔μ∈MX,E′:μx∈MσXforeachx∈E.
Then μ∈Mσ(X) if μ∈MσX,E′ (see [5, Proposition 3.9]).
For the integration theory of functions f∈Cb(X,E) with respect to μ∈MσX,E′ we refer the reader to [6, page 197], [5]. The following result will be of importance (see [6, Theorem 5.3]).
Theorem 4.
The following statements hold:
for Φ∈CbX,E′ the following conditions are equivalent:
Φ is βσ-continuous;
there exists a unique μ∈MσX,E′ such that
(7)Φ(f)=Φμ(f)=∫Xfdμforf∈Cb(X,E),
and Φμ=|μ|(X);
for μ∈MσX,E′, ∫Xfdμ≤∫Xf~dμ for f∈Cb(X,E).
In view of [9, Corollary 5] we have the following characterization of convergence in (Cb(X,E),σ(Cb(X,E),Cb(X,E)βσ′)).
Theorem 5.
For a sequence (fn) in Cb(X,E) the following statements are equivalent:
fn→0 for σCbX,E,MσX,E′;
supnfn<∞ and fn(t)→0 in σE,E′ for each t∈X.
The following theorem gives a characterization of (βσ,·F)-continuous operators T:Cb(X,E)→F in terms of the corresponding operator measures m:B→LE,F′′ (see [12, Theorem 9 and Corollary 7]).
Theorem 6.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous linear operator and m∈MX,LE,F′′ be the representing measure of T. Then the following statements hold.
m∈MσX,LE,F′′.
For each y′∈F′, y′(T(f))=∫Xfdmy′ for f∈Cb(X,E).
For each f∈Cb(X,E) and A∈B there exists a unique vector in F′′, denoted by ∫Afdm, such that (∫Afdm)(y′)=∫Afdmy′ for each y′∈F′.
For each A∈B, the mapping Cb(X,E)∋f↦∫Afdm∈F′′ is a (βσ,·F′′)-continuous linear operator.
For f∈Cb(X,E), ∫Xfdm∈iF(F) and T(f)=jF(∫Xfdm).
T=m~(X).
For U∈P and y′∈F′; we have
(8)my′U=sup∫Uhdmy′:h∈Cb(X)⊗E,∫Uhdmy′h=1,supph⊂U.
Following [34] by Mσ(Ba) (= ca(Ba)), we denote the space of all bounded countably additive, real-valued, regular (with respect to zero sets) measures on Ba.
We define Mσ(Ba,E′) to be the set of all measures μ:Ba→E′ such that the following two conditions are satisfied.
For each x∈E, the function μx:Ba→R defined by μx(A)=μ(A)(x) for A∈Ba, belongs to Mσ(Ba).
μ(X)<∞, where, for each A∈Ba, we define |μ|(A)=sup∑μAixi, where the supremum is taken over all finite Ba-partitions (Ai) of A and all finite collections xi∈BE.
It is known that if μ∈MσBa,E′, then |μ|∈Mσ(Ba) (see [34, Lemma 2.1]).
The following result will be of importance (see [34, Theorem 2.5]).
Theorem 7.
Let μ∈MσX,E′. Then μ possesses a unique extension μ¯∈MσBa,E′ and |μ¯|(X)=|μ|(X).
From Theorem 7 and [13, Corollary 10, page 4] it follows that if μ∈MσX,E′, then |μ¯|(A)=|μ|(A) for A∈B.
By Mσ(X,L(E,F)) we will denote the space of all operator measures m:B→L(E,F) such that m~(X)<∞ and my′∈Mσ(X,E′) for each y′∈F′. By Mσ(Ba,L(E,F)) we will denote the space of all operator measures m:Ba→L(E,F) with m~(X)<∞ such that my′∈Mσ(Ba,E′) for each y′∈F′.
The following theorem characterizes (βσ,·F)-continuous linear operators T:Cb(X,E)→F such that Tx:Cb(X)→F are weakly compact for each x∈E (see [12, Theorem 14 and Lemma 11]).
Theorem 8.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous linear operator such that Tx:Cb(X)→F is weakly compact for each x∈E, and let m∈Mσ(X,L(E,F′′)) be the representing measure of T. Then the following statements hold.
m(A)(x)∈iF(F) for each A∈B, x∈E and the measure mF:B→L(E,F) defined by mF(A)(x)≔jF(m(A)(x)) for A∈B, x∈E, belongs to Mσ(X,L(E,F)) and possesses a unique extension m¯∈Mσ(Ba,L(E,F)) with m¯~(X)=m~(X) which is countably additive both in the strong operator topology and the weak star operator topology. Moreover, m¯y′=my′¯ for y′∈F′.
For every f∈Cb(X,E) and A∈Ba there exists a unique vector in F, denoted by ∫Afdm¯, such that for each y′∈F′, y′(∫Afdm¯)=∫Afdm¯y′ and
(9)∫Afdm¯y′≤∫Afdm¯y′.
For each A∈Ba, the mapping TA:Cb(X,E)→F defined by TA(f)=∫Afdm¯ is a (βσ,·F)-continuous linear operator.
T(f)=TX(f)=∫Xfdm¯ for f∈Cb(X,E).
Remark 9.
As a consequence of Theorem 8 (for F=R) we have
(10)CbX,E,βσ′=Φμ:μ∈MσBa,E′,
where for μ∈Mσ(Ba,E′), Φμ(f)=∫Xfdμ for f∈Cb(X,E) and Φμ=μ(X).
3. Strongly Bounded Operators on Cb(X,E)
Making use of [35, Theorem 8] we can state the following analogue (for Baire measures on a completely regular Hausdorff space) of the celebrated Dieudonné-Grothendieck’s criterion on weak compactness in the space of Borel measures on a compact Hausdorff space (see [36, Theorem 2], [37, Theorem 14, pages 98–103]), which will play a crucial role in the study of different classes of operators on Cb(X,E).
By Ts we denote the topology of simple convergence in ca(Ba). Then Ts is generated by the family {pA:A∈Ba} of seminorms, where pA(ν)=|ν(A)| for ν∈ca(Ba).
A completely regular Hausdorff space X is said to be an z-space if a subset which meets every zero-set in a zero-set must be a zero-set. One can note that every metrizable space is a z-space.
From now on we will assume that X is a z-space.
Theorem 10.
Assume that M is a subset of ca+(Ba) such that supν∈Mν(X)<∞. Then the following statements are equivalent.
M is relatively Ts-compact subset of ca(Ba).
M is uniformly countably additive, that is, supν∈Mν(An)→0 whenever An↓∅, (An)⊂Ba.
M is uniformly strongly additive, that is, supν∈Mν(An)→0 whenever (An) is pairwise disjoint in Ba.
supν∈Mν(Un)→0 for every pairwise disjoint sequence (Un) in P.
Proof.
(i)⇔(ii) See [38, Theorem 7].
(ii)⇔(iii) See [37, Theorem 10, pages 88-89].
(iv)⇔(i) See [35, Theorem 8].
Now we can state a characterization of (βσ,·F)-continuous strongly bounded operators T:Cb(X,E)→F.
Theorem 11.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous linear operator and let m∈MσX,LE,F′′ be its representing measure. Then the following statements are equivalent.
For each x∈E, Tx:Cb(X)→F is weakly compact and m¯ is variationally semiregular; that is, supm¯y′An:y′∈BF′→0 whenever An↓∅, (An)⊂Ba.
T is strongly bounded.
T(fn)→0 whenever (fn) is a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 in E for each t∈X.
T(fn)→0 whenever (fn) is a uniformly bounded sequence in Cb(X,E) such that suppfn∩suppfm=∅ for n≠m.
Proof.
(i)⇔(ii) It follows from Theorem 8 and [12, Theorem 16].
(ii)⇒(iii) It follows from [12, Theorem 17].
(iii)⇒(iv) It is obvious.
(iv)⇒(i) Assume that (iv) holds. First we shall show that for each x∈E, Tx:Cb(X)→F is weakly compact. Assume on the contrary that Txo:Cb(X)→F is not weakly compact for some xo∈E. This means that mxo:B→F′′ is not strongly bounded. Since for A∈B, mx0(A)F′′=supmx0,y′A:y′∈BF′, we obtain that the family mxo,y′:y′∈BF′ is not uniformly strongly additive. Hence the family mxo,y′¯:y′∈BF′ is not uniformly countably additive. It follows that the family mxo,y′¯:y′∈BF′ is not uniformly countably additive. In view of Theorem 10 there exist ε0>0, a sequence (yn′) in BF′ and a pairwise disjoint sequence (Un) in P such that for n∈N, mxo,y′(Un)≥ε0. Note that
(11)mxo,yn′Un=sup∫Unudmxo,yn′:u∈CbX,∫Unudmxo,yn′u=1,suppu⊂Un.
Hence there exists a sequence (un) in Cb(X) such that un=1, suppun⊂Un and
(12)yn′Tx0un=∫Xundmx0,yn′=∫Unundmxo,yn′≥mxo,yn′(Un)-ε02≥ε02.
Let fn=un⊗x0 for n∈N. Then suppfn∩suppfn=∅ for n≠m and by (iv), T(fn)F→0, which contradics (12). This means that Tx:Cb(X)→F is weakly compact for each x∈E, as desired.
In view of Theorem 8m can be uniquely extended to a measure m¯:Ba→L(E,F). Assume that m¯ is not variationally semiregular. Then by Theorem 10 there exist ε0>0, a pairwise disjoint sequence (Un) in P and a sequence (yn′) in BF′ such that myn′(Un)>ε0. Hence by Theorem 7 there exists a sequence (hn) in Cb(X)⊗E and hn=1 with supphn⊂Un for n∈N such that
(13)∫Unhndmyn′≥myn′(Un)-ε02>ε02.
Then, for n∈N,
(14)ThnF=supy′Thn:y′∈BF′=sup∫Xhndmy′:y′∈BF′=sup∫Unhndmy′:y′∈BF′≥∫Unhndmyn′>ε02.
On the other hand, since supphn∩supphm=∅ for n≠m, by (iv), T(hn)F→0. This contradiction establishes that (i) holds.
Corollary 12.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous and strongly bounded linear operator and let m∈Mσ(X,L(E,F′′)) be its representing measure. Then the set m¯y′:y′∈BF′ is uniformly regular on Ba; that is, for each A∈Ba and ɛ>0, there exist Z∈Z with Z⊂A and U∈P with A⊂U such that
(15)sup|m¯y′|(B):B∈Ba,B⊂U∖Z,y′∈BF′≤ɛ.
Proof.
In view of Theorem 11 the family m¯y′:y′∈BF′ is uniformly countably additive. Let λ∈ca+(Ba) be a control measure for m¯y′:y′∈BF′ and let A∈Ba and ɛ>0 be given. Then there is δ>0 such that supm¯y′B:y′∈BF′≤ɛ whenever B∈Ba and λ(B)≤δ. By the regularity of λ there exists Z∈Z with Z⊂A and U∈P with A⊂U such that λ(U∖Z)≤δ. Hence we get sup{|m¯y′|(B):B∈Ba, B⊂U∖Z,y′∈BF′}≤ɛ.
Corollary 13.
Assume that T:Cb(X,E)→F is a (βσ,·F)-continuous linear operator and F contains no isomorphic copy of c0. Then T is strongly bounded.
Proof.
Let m∈Mσ(X,L(E,F′′)) stand for the representing measure of T. We shall first show that Tx:Cb(X)→F is weakly compact for each x∈E. Assume on the contrary that Tx0:Cb(X)→F is not weakly compact for some x0∈E. Then by the proof of implication (iv)⇒(i) of Theorem 11 there exist ε0>0, a sequence (yn′) in BF′, and a pairwise disjoint sequence (Un) in P such that mx0,yn′(Un)≥ε0 for n∈N. By the Rosenthall lemma (see [13, Lemma 1, page 18]) the sequence (Un) in P and (yn′) in BF′ can be chosen such that for n∈N,
(16)mx0,yn′Un≥ε0,mx0,yn′⋃m≠nUm<ε02.
Since, for n∈N,
(17)mx0,yn′Un=sup∫Unudmx0,yn′:u∈Cb(X),∫Unudmx0,yn′u=1withsuppu⊂Un,
there exists a sequence (un) in Cb(X) such that un=1 with suppun⊂Un and
(18)yn′Tx0un=∫Xundmx0,yn′=∫Unundmx0,yn′>ε0.
Let Y={∑n=1∞anun:(an)∈c0}. We see that Y is an isomorphic copy of c0. Assume that u=∑n=1∞anun for some sequence (an) in c0. Then for n∈N we have
(19)yn′Tx0u=∫Xudmx0,yn′=an∫Unundmx0,yn′+∫⋃m≠nUmudmx0,yn′≥anε0-∫⋃m≠nUmudmx0,yn′≥anε0-mx0,yn′⋃m≠nUmu≥anε0-ε02u.
But u=supnan, so
(20)Tx0(u)F≥supnyn′Tx0u≥ε0u-ε02u=ε02u.
This means that Tx0:Cb(X)→F is an isomorphism on Y, so F contains an isomorphic copy of c0, which contradicts our assumption on F. This means that Tx is weakly compact for each x∈E. Hence in view of Theorem 8m¯:Ba→L(E,F) is countably additive in the weak star operator topology and by [19, Remark 7, page 923] and Theorem 11 we derive that T is strongly bounded, as desired.
Remark 14.
If X is a compact Hausdorff space, the equivalence (ii)⇔(iii) of Theorem 11 was obtained by Brooks and Lewis (see [16, Theorem 2.1]).
Let L∞(Ba,E) stand for the Banach space of all bounded strongly Ba-measurable functions g:X→E, equipped with the uniform norm ·. Assume that m:B→L(E,F) with m~(X)<∞ is variationally semiregular. Then every g∈L∞(Ba,E) is m-integrable (see [39, Definition 2, page 523 and Theorem 5, page 524]) and ∫Xgndm→0 whenever (gn) is a uniformly bounded sequence in L∞(Ba,E) converging pointwise to 0 (see [40, Proposition 2.2]).
Note that if f∈Cb(X,E) then y′∘f is Ba-measurable. Hence if E is assumed to be separable then f is strongly Ba-measurable; that is, f∈L∞(Ba,E) (see [2, Proposition 21, page 9]).
Recall that a function g:X→E′ is weak*-measurable if for each x∈E the function X∋t↦〈x,g(t)〉∈R is Ba-measurable. For λ∈ca+(Ba) by Lw*1(λ,E′) we denote the vector space of all weak*-measurable functions g:X→E′ for which there exists u∈L1(λ) such that g(t)E′≤u(t)λ-a.e. on X (see [41, page 26]).
Following [40] we can distinguish an important class of operators on L∞(Ba,E).
Definition 15.
A bounded linear operator S:L∞(Ba,E)→F is said to be σ-smooth if S(gn)→0 whenever (gn) is a uniformly bounded sequence in L∞(Ba,E) such that gn(t)→0 for each t∈X.
Proposition 16.
Assume that E is separable. Let T:Cb(X,E)→F be a (βσ,·F)-continuous and strongly bounded linear operator, and let m∈Mσ(X,L(E,F′′)) be its representing measure. Then for each y′∈F′ there exists gy′∈Lw*1(λ,E′) such that
(21)y′Tf=∫Xf,gy′dλforf∈Cb(X,E),
where λ∈ca+(Ba) is a control measure for m¯y′:y′∈BF′.
Proof.
Since E is supposed to be separable, Cb(X,E)⊂L∞(Ba,F). Moreover, since m¯:Ba→L(E,F) is variationally semiregular (see Theorem 11), the corresponding integration operator Sm¯:L∞(Ba,E)→F is σ-smooth and for y′∈F′ we have (see [40, Proposition 2.2])
(22)y′(Sm¯(f))=∫Xfdm¯y′=y′(T(f))∀f∈Cb(X,E).
It follows that Sm¯(f)=T(f) for each f∈Cb(X,E).
Let y′∈F′. Then y′∘Sm¯ is a σ-smooth functional on L∞(Ba,E), and m¯y′ is λ-absolutely continuous; that is, m¯y′∈cabvλ(Ba,E′). According to the Radon-Nikodym type theorem (see [41, Theorem 1.5.3]) there exists a weak*-measurable function gy′:X→E′ which satisfies the following conditions.
The function X∋t↦gy′(t)E′∈R is Ba-measurable and λ-integrable; that is, gy′(·)E′∈L1(λ).
For every x∈E and A∈Ba,
(23)m¯y′Ax=∫Ax,gy′dλ,m¯y′(A)=∫Agy′(·)E′dλ.
It follows that gy′∈Lw*1(λ,E′). Note that for every s=∑i=1n(1Ai⊗xi)∈S(Ba,E) the mapping s,gy′:X∋t↦st,gy′t∈R is Ba-measurable and using (2) we get
(24)y′∘Sm¯s=∫Xsdm¯y′=∑i=1nm¯y′(Ai)(xi)=∑i=1n∫X1Ai⊗xi,gy′dλ=∫X∑i=1n1Ai⊗xi,gy′dλ=∫X∑i=1n1Ai⊗xi,gy′dλ=∫Xs,gy′dλ.
Now let f∈Cb(X,E)⊂L∞(Ba,E). Then there exists a sequence (sn) in S(Ba,E) such that sn(t)-f(t)E→0 and sn(t)E≤f(t)E for each t∈X and n∈N (see [2, Theorem 1.6, page 4]). Then the mapping 〈f,gy′〉:X∋t↦〈f(t),gy′(t)〉∈R is Ba-measurable. Using the Lebesgue dominated convergence theorem we have
(25)∫Xsn,gy′dλ-∫Xf,gy′dλ=∫Xsn-f,gy′dλ≤∫Xsn-f,gy′dλ≤∫X(sn-f)(t)E·gy′(t)E′dλ⟶0.
It follows that
(26)y′Tf=(y′∘Sm¯)(f)=limn(y′∘Sm¯)(sn)=limn∫Xsn,gy′dλ=∫Xf,gy′dλ.
4. Unconditionally Converging Operators on Cb(X,E)
Recall that a series ∑i=1∞zi in a Banach space G is called weakly unconditionally Cauchy (wuc) if, for each z′∈G′, ∑i=1∞z′zi<∞. We say that a bounded linear operator T:G→F is unconditionally converging if, for every weakly unconditionally Cauchy series ∑i=1∞zi in G, the series ∑i=1∞T(zi) converges unconditionally in a Banach space F.
If X is a compact Hausdorff space, Swartz [33] proved that every unconditionally converging operator T:C(X,E)→F is strongly bounded. Dobrakov (see [28, Theorem 3]) showed that if X is a locally compact Hausdorff space, then every unconditionally converging operator T:C0(X,E)→F is strongly bounded and for every Borel set A in X, the operator m(A):E→F is unconditionally converging. Moreover, Brooks and Lewis [27, Theorem 5.2] showed that if E contains no isomorphic copy of c0, then every strongly bounded operator T:C0(X,E)→F is unconditionally converging. We will extend these results to the setting when T:Cb(X,E)→F is a (βσ,·F)-continuous linear operator and X is a completely regular Hausdorff space.
Theorem 17.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous and unconditionally converging linear operator, and m∈Mσ(X,L(E,F′′)) stand for the representing measure of T. Then the following statements hold.
T is strongly bounded.
For each A∈Ba, m¯(A):E→F is an unconditionally converging operator.
Proof.
(i) Assume that (fn) is a uniformly bounded sequence in Cb(X,E) such that suppfn∩suppfm=∅ for n≠m. Then {∑n∈Mfn:M∈F(N)} is bounded in Cb(X,E) and, since T is unconditionally converging, we obtain that T(fn)→0. Hence by Theorem 11T is strongly bounded.
(ii) Let A∈Ba and assume that ∑n=1∞xn is wuc in E. Then sup{∑i∈MxiE:M∈F(N)}≤r. In view of Theorem 11m¯y′:y′∈BF′ is uniformly countably additive and let λ∈ca+(Ba) stand for the control measure of m¯y′:y′∈BF′ (see Corollary 12). Let ɛ>0 be given. Then there is δ>0 such that sup{|m¯y′|(B):y′∈BF′}≤ɛ/r whenever B∈Ba, λ(B)≤δ. Then there exist Z∈Z with Z⊂A and U∈P with A⊂U such that λ(U∖Z)≤δ. Hence
(27)supmy′U∖Z:y′∈BF′≤ɛ.
Then one can choose u∈Cb(X) with 0≤u≤1X, u|Z≡1, and u|X∖U≡0. Define Tu(x):=T(u⊗x) for x∈E. We shall show that Tu:E→F is unconditionally converging. Indeed, for M∈F(N), ∑i∈M(u⊗xi)≤∑i∈MxiE≤r. Hence the series ∑n=1∞T(u⊗xn) is unconditionally convergent; that is, Tu is unconditionally converging, as desired. Then for each x∈BE, we have
(28)Tu(x)-m¯(A)(x)F=∫Xu-1A⊗xdm¯F=supy′∫Xu-1A⊗xdm¯:y′∈BF′≤sup∫Xu-1Adm¯y′:y′∈BF′≤sup∫U∖Z1Xdmy′:y′∈BF′≤supmy′U∖Z:y′∈BF′≤ɛ.
Hence Tu-m¯(A)≤ɛ and since the class of all unconditionally converging operators from E to F is a closed linear subspace of (L(E,F),·) (see [28, page 20]), we derive that m¯(A) is unconditionally converging.
Theorem 18.
Assume that E is separable and contains no isomorphic copy of c0. Then for a (βσ,·F)-continuous linear operator T:Cb(X,E)→F the following statements are equivalent.
T is unconditionally converging.
T is strongly bounded.
Proof.
(i)⇒(ii) See Theorem 17.
(ii)⇒(i) See [12, Corollary 18].
Recall that a subset P of a Banach space G is said to be weakly precompact if every bounded sequence (zn) in P contains a subsequence (zkn) so that z′(zkn) converges for each z′∈G′. An operator T:G→F is said to be weakly precompact if T(BG) is weakly precompact in a Banach space F.
Abbott et al. [17, Theorem 2.8] discussed the relationship between strongly bounded and unconditionally converging operators T:C(X,E)→F whenever X is a compact Hausdorff space. They showed that if E′ contains no isomorphic copy of l1 and E′ has the RNP, then the classes of strongly bounded and unconditionally converging operators T:C(X,E)→F coincide. Now we state an analogue of Theorem 2.8 of [17] for (βσ,·F)-continuous linear operator T:Cb(X,E)→F, where X is a completely regular Hausdorff space.
Theorem 19.
Assume that E′ contains no isomorphic copy of l1 and E′ has the RNP. Then for a (βσ,·F)-continuous linear operator T:Cb(X,E)→F the following statements are equivalent.
T′:F′→Cb(X,E)′ is weakly precompact.
T is unconditionally converging.
T is strongly bounded.
Proof.
(i)⇒(ii) See [17, Theorem 2.7].
(ii)⇒(iii) See Theorem 17.
(iii)⇒(i) Assume that T is strongly bounded. Since {y′∘T:y′∈BF′}⊂Cb(X,E)βσ′, we have to show that {y′∘T:y′∈BF′} is a weakly precompact subset of the Banach space (Cb(X,E)βσ′,·). By Theorem 11{|m¯y′|:y′∈BF′} is uniformly countably additive, and let λ∈ca+(Ba) be a control measure for m¯y′:y′∈BF′. Since E′ is supposed to have the RNP, for each y′∈BF′ there exists gy′∈L1(λ,E′) such that m¯y′(A)=∫Agy′dλ and m¯y′(A)=∫Agy′(·)E′dλ for A∈Ba. It follows that gy′·E′:y′∈BF′ is a uniformly integrable subset of L1(λ) and since E′ contains no isomorphic copy l1, {gy′:y′∈BF′} is a weakly precompact subset of L1(λ,E′) (see [42]). Since {m¯y′:y′∈BF′}⊂cabvλ(Ba,E′) (= the Banach space of all λ-continuous members of cabv(Ba,E′)) and the Radon-Nikodym theorem establishes the isometry between cabvλ(Ba,E′) and L1(λ,E′), we obtain that {y′∘T:y′∈BF′} is a weakly precompact subset of Cb(X,E)βσ′ because (y′∘T)(f)=∫Xfdm¯y′ for f∈Cb(X,E).
5. Weakly Completely Continuous Operators on Cb(X,E)
Recall that a bounded linear operator T from a Banach space G to a Banach space F is said to be a Dieudonné operator if T maps σ(G,G′)-Cauchy sequences in G into weakly convergent sequences in F.
If X is a compact Hausdorff space, then Dieudonné operators from the Banach space C(X,E) to F were studied by Bombal and Cembranos [23] and Abbott et al. (see [17, Theorems 3.1, 3.5 and Theorem, page 334].
Definition 20.
A bounded linear operator T:Cb(X,E)→F is said to be weakly completely continuous if T(fn) is σ(F,F′)-convergent in F whenever (fn) is a uniformly boundd sequence in Cb(X,E) such that (fn(t)) is a σ(E,E′)-Cauchy sequence in E for each t∈X.
Proposition 21.
Let T:Cb(X,E)→F be a bounded linear operator. Then the following statements are equivalent.
T is weakly completely continuous.
T maps σ(Cb(X,E),Cb(X,E)βσ′)-Cauchy sequences in Cb(X,E) onto σ(F,F′)-convergent sequences in F.
Proof.
(i)⇒(ii) Assume that T is weakly completely continuous and (fn) is a σ(Cb(X,E),Cb(X,E)βσ′)-Cauchy sequence in Cb(X,E). Then for each t∈X, (fn(t)) is a σ(E,E′)-Cauchy sequence in E because Φt,x′∈Cb(X,E)βσ′, where Φt,x′(f)=x′(f(t)) for f∈Cb(X,E). Since (fn) is βσ-bounded, we get supfn<∞. It follows that (T(fn)) is σ(F,F′)-convergent.
(ii)⇒(i) Assume that (ii) holds and (fn) is a uniformly bounded sequence in Cb(X,E) such that (fn(t)) is a σ(E,E′)-Cauchy sequence in E for each t∈X. We shall show that (fn) is a σ(Cb(X,E),Cb(X,E)βσ′)-Cauchy sequence. Assume on the contrary that (fn) is not a σ(Cb(X,E),Cb(X,E)βσ′)-Cauchy sequence. Then there exist Φ0∈Cb(X,E)βσ′ and ε0>0 and a subsequence (gn) of (fn) satisfying Φ0g2n-g2n≥ε0 for n∈N. Since g2n(t)-g2n-1(t)→0 for each t∈X, by Theorem 5g2n-g2n-1→0 for σ(Cb(X,E), Cb(X,E)βσ′). Hence Φ0(g2n-g2n-1)→0. This contradiction establishes that (fn) is a σ(Cb(X,E), Cb(X,E)βσ′)-Cauchy sequence, and it follows that a sequence (T(fn)) is σ(F,F′)-convergent in F.
From Proposition 21 it follows that every weakly completely continuous operator T:Cb(X,E)→F is a Dieudonné operator. As a consequence, we get the following result (see [37, Problem 8, page 54]).
Corollary 22.
Assume that T:Cb(X,E)→F is a weakly completely continuous operator. Then T is unconditionally converging.
Theorem 23.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous and weakly completely continuous linear operator and m∈Mσ(X,L(E,F′′)) stand for its representing measure. Then the following statements hold.
T is strongly bounded.
For each A∈Ba, m¯(A):E→F is a Dieudonné operator.
Proof.
(i) It follows from Corollary 22 and Theorem 17.
(ii) Let A∈Ba and assume that (xn) is a σ(E,E′)-Cauchy sequence in E. Hence supnxnE<∞. Since T is strongly bounded, arguing as in the proof of Theorem 17 for a given ɛ>0 there exist Z∈Z with Z⊂A and U∈P with A⊂U such that
(29)supmy′U∖Z:y′∈BF′≤ɛ.
Then we can choose u∈Cb(X) with 0≤u≤1X, u|Z≡1, and u|X∖U≡0. Define Tu(x):=T(u⊗x) for x∈E. We shall show that Tu:E→F is a Dieudonné operator. Let hn=u⊗xn for n∈N. Then supnhn≤supnxnE<∞ and (hn(t)) is a σ(E,E′)-Cauchy sequence in E for each t∈X. Hence (T(hn)) is σ(F,F′)-convergent in F and this means that Tu is a Dieudonné operator. Then arguing as in the proof of Theorem 17, we obtain that Tu-m¯(A)≤ɛ and since the class of all Dieudonné operators from E to F is a closed linear subspace of (L(E,F),·) (see [17, Theorem 3.5]), we derive that m¯(A) is a Dieudonné operator.
6. Completely Continuous Operators on Cb(X,E)
Recall that a bounded linear operator T from a Banach space G to a Banach space F is said to be a Dunford-Pettis operator if zn→0 in G for σ(G,G′) implies T(zn)F→0 (see [43, Section 19]).
Definition 24.
A bounded linear operator T:Cb(X,E)→F is said to be completely continuous if T(fn)F→0 whenever (fn) is a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 in σ(E,E′) for each t∈X.
Using Theorem 5 one can get the following result.
Proposition 25.
Let T:Cb(X,E)→F be a bounded linear operator. Then the following statements are equivalent.
T is completely continuous.
T(fn)F→0 whenever fn→0 in σ(Cb(X,E),Cb(X,E)βσ′).
Theorem 26.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous and completely continuous operator and m∈Mσ(X,L(E,F′′)) its representing measure. Then the following statements hold.
T is strongly bounded.
For each A∈Ba, m¯(A):E→F is a Dunford-Pettis operator.
Proof.
(i) In view of [43, Theorem 19.1] and Proposition 25T maps σ(Cb(X,E), Cb(X,E)′) Cauchy sequences in Cb(X,E) onto norm convergent sequences in F. It follows that T is a Dieudonné operator and hence T is unconditionally converging. Thus T is strongly bounded (see Theorem 17).
(ii) Let A∈Ba and assume that xn→0 in E for σ(E,E′). Then supxnE<∞. Since T is strongly bounded, arguing as in the proof of Theorem 17 for a given ɛ>0 there exist Z∈Z with Z⊂A and U∈P with A⊂U such that
(30)supmy′U∖Z:y′∈BF′≤ɛ.
Then we can choose u∈Cb(X) with 0≤u≤1X, u|Z≡1, and u|X∖U≡0. Define Tu(x):=T(u⊗x) for x∈E. We shall show that Tu:E→F is a Dunford-Pettis operator. Let hn=u⊗xn for n∈N. Then hn(t)→0 in σ(E,E′) for each t∈X and supnhn<∞. It follows that Tu(xn)F=T(hn)F→0 and this means that Tu:E→F is a Dunford-Pettis operator (see [43, Theorem 19.1]). Then arguing as in the proof of (ii) of Theorem 17, we obtain that ‖Tu-m¯(A)‖≤ɛ. Since the class of all Dunford-Pettis operators from E to F is a closed linear subspace of (L(E,F),·) (see [28, page 27]), we derive that m¯(A) is a Dunford-Pettis operator.
Corollary 27.
Assume that E is a Schur space. Let T:Cb(X,E)→F be a (βσ,·)-continuous linear operator. The the following statements are equivalent.
T is strongly bounded.
T is completely continuous.
T is weakly completely continuous.
T is unconditionally converging.
∑n=1∞T(fn) converges unconditionally whenever (fn) is a uniformly bounded sequence in Cb(X,E) such that suppfn∩suppfm=∅ for n≠m.
Proof.
(i)⇒(ii) Assume that T is strongly bounded and (fn) is a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 in σ(E,E′) for each t∈X. It follows that fn(t)E→0 because E is supposed to be a Schur space. Hence by Theorem 11T(fn)F→0, as desired.
(ii)⇒(iii) It is obvious.
(iii)⇒(iv) See Proposition 21.
(iv)⇒(v) Assume that (iv) hold and let (fn) be a uniformly bounded sequence in Cb(X,E) such that suppfn∩suppfm=∅ for n≠m. Let C=supnfn and (an)∈l∞. Then
(31)supn∑i=1naifi≤Csupnan
and it follows that ∑n=1∞fn is wuc in Cb(X,E) (see [44]). Hence ∑n=1∞T(fn) converges unconditionally in F.
(v)⇒(i) It follows from Theorem 11.
Theorem 28.
Assume that E is separable. Let T:Cb(X,E)→F be a (βσ,·F)-continuous and strongly bounded operator and let m∈Mσ(X,L(E,F′′)) be its representing measure. Then the following statements are equivalent.
T is completely continuous.
limn∫Xfn,gyn′dλ=0 whenever (fn) is a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 in σ(E,E′) for t∈X and (yn′) is a sequence in BF′.
Here λ∈ca+(Ba) is a control measure for m¯y′:y′∈BF′ and for n∈N, gyn′ is an element of Lw*1(λ,E′) corresponding to m¯yn′ (see Proposition 16).
Proof.
(i)⇒(ii) Assume that T is completely continuous and let (fn) be a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 in σ(E,E′) for each t∈X and (yn′) is a sequence in BF′. Then, by Proposition 16,
(32)∫Xfn,gyn′dλ=yn′Tfn≤T(fn)F⟶0.
(ii)⇒(i) Assume that (ii) holds. Let (fn) be a uniformly bounded sequence in Cb(X,E) such that fn(t)→0 in σ(E,E′) for each t∈X. Choose a sequence (yn′) in BF′ such that yn′Tfn≥(1/2)‖T(fn)‖F. Hence, by Proposition 16,
(33)T(fn)F≤2yn′Tfn=2∫Xfn,gyn′dλ⟶0,
so T is completely continuous.
7. Weakly Compact Operators on Cb(X,E)
If X is a compact Hausdorff space (resp., X is a locally compact Hausdorff space), weakly compact operators T:C(X,E)→F (resp., T:Co(X,E)→F) have been studied intensively by Batt and Berg [19, 20], Brooks and Lewis [27], Bombal [24], and Saab [29]. The aim of this section is to extend a characterization of weakly compact operators T:Co(X,E)→F of [27, Theorem 4.1] to (βσ,·F)-continuous and weakly compact operators T:Cb(X,E)→F.
Theorem 29.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous linear operator and let m∈Mσ(X,L(E,F′′)) be its representing measure. Then the following statements hold.
Assume that T is weakly compact. Then T is strongly bounded and for each A∈Ba, m¯(A):E→F is a weakly compact operator.
Assume that E′ and E′′ have the RNP and T is strongly bounded and for each A∈Ba, m¯(A):E→F is a weakly compact operator. Then T is weakly compact.
Proof.
(i) In view of [45, Corollary 9.3.2.] the conjugate operator T′:F′→Cb(X,E)βσ′ maps BF′ onto a relatively weakly compact subset of (Cb(X,E)βσ′,·), where (y′∘T)(f)=∫Xfdm¯y′ for f∈Cb(X,E). Hence {m¯y′:y′∈BF′} is a relatively weakly compact subset of the Banach space cabv(Ba,E′), equipped with the total variation norm. Making use of the Bartle-Dunford-Schwartz theorem [13, Theorem 5, pages 105-106], we obtain that the set {m¯y′:y′∈BF′} is uniformly countably additive and, for each A∈Ba, the set {m¯y′(A):y′∈BF′} is relatively weakly compact in E′. Thus by Theorem 11T is strongly bounded and, since m¯(A)′(y′)=m¯y′(A), we derive that m¯(A):E→F is weakly compact.
(ii) By Theorem 11{m¯y′:y′∈BF′} is uniformly countably additive. Moreover, for each A∈Ba, {m¯y′:y′∈BF′} is relatively weakly compact in E′. This means that {m¯y′:y′∈BF′} is relatively weakly compact subset of Mσ(Ba,E′) (see [13, Theorem 5, pages 105-106]). Since Cb(X,E)βσ′={Φμ:μ∈Mσ(Ba,E′)}, {Φm¯y′:y′∈BF′} is a relatively weakly compact subset of Cb(X,E)βσ′. Hence according to [45, Corollary 9.3.2] T is weakly compact.
Corollary 30.
Assume that E is reflexive. Then for a (βσ,·F)-continuous linear operator T:Cb(X,E)→F the following statements are equivalent.
T is weakly compact.
T is strongly bounded.
As a consequence of Corollaries 13 and 30 we can state a generalization of the well known theorem due to Batt and Berg telling us that if X is a compact Hausdorff space, E is reflexive, and F contains no isomorphic copy of co, then every bounded linear operator T:C(X,E)→F is weakly compact (see [20, Theorem 9]).
Corollary 31.
Assume that E is reflexive and F contains no isomorphic copy of co. Then every (βσ,·F)-continuous linear operator T:Cb(X,E)→F is weakly compact.
8. Nuclear Operators on Cb(X,E)
Following [46, Ch. 3, §7] we have the following definition.
Definition 32.
A (βσ,·F)-continuous linear operator T:Cb(X,E)→F is said to be nuclear if it can be represented as
(34)Tf=∑n=1∞λnΦnfynforeachf∈Cb(X,E),
where (Φn) is a βσ-equicontinuous sequence in Cb(X,E)βσ′, (yn) is a bounded sequence in F, and (λn) is a sequence in R such that ∑n=1∞|λn|<∞.
In particular, an operator L∈L(E,F) is said to be nuclear if there exist sequences (xn′) in E′ and (yn) in F such that L is of the form
(35)Lx=∑n=1∞xn′xynforeachx∈E,
and ∑n=1∞xn′E′·ynF<∞. Then we say that ∑n=1∞(xn′⊗yn) represents a nuclear operator L. The nuclear norm of a nuclear operator L:E→F is defined by
(36)Lnuc:=inf∑n=1∞xn′E′·ynF,
where the infimum is taken over all sequences (xn′) and (yn) such that L(x)=∑n=1∞xn′(x)yn holds for each x∈E. The nuclear operators L:E→F form a normed space under the nuclear norm ·nuc, which we shall denote by N(E,F) (see [13, Proposition 2, page 170]).
If X is a compact Hausdorff space, then nuclear operators from the Banach space C(X,E) to F have been studied by Saab and Smith [31]. In this section we extend Proposition 1 of [31] to the completely regular setting.
Let Cb(X,E)βσ′′ stand for bidual of (Cb(X,E),βσ). Note that Cb(X,E)βσ′′=(Cb(X,E)βσ′,·)′. Then one can embed B(Ba,E) into Cb(X,E)βσ′′ by the mapping π¯:B(Ba,E)→Cb(X,E)βσ′′, where, for g∈B(Ba,E),
(37)π¯gΦμ:=∫Xgdμforμ∈Mσ(Ba,E′).
(Here Φμ(f)=∫Xfdμ for f∈Cb(X,E)).
Proposition 33.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous linear operator such that Tx:Cb(X)→F is weakly compact for each x∈E, and let m∈Mσ(X,L(E,F′′)) be its representing measure. Then the following statements hold.
(T′′∘π¯)(g)=∫Xgdm¯y′ for g∈B(Ba,E).
iF(m¯(A)(x))=(T′′∘π¯)(1A⊗x) for A∈Ba, x∈E.
Proof.
(i) Let T′:F′→Cb(X,E)βσ′ and T′′:Cb(X,E)βσ′′→F′′ stand for the conjugate and the biconjugate operators of T, respectively. Then for each y′∈F′, y′(T(f))=∫Xfdmy′=∫Xfdm¯y′ for f∈Cb(X,E), and hence, for g∈B(Ba,E),
(38)T′′∘π¯gy′=π¯(g)(T′(y′))=π¯(g)(y′∘T)=∫Xgdm¯y′.
(ii) Let A∈Ba. Then by (i) for each x∈E and y′∈F′ we get
(39)((T′′∘π¯)(1A⊗x))(y′)=∫X(1A⊗x)dm¯y′=m¯y′(A)(x)=y′(m¯(A)(x))=iF(m¯(A)(x))(y′).
Hence iF(m¯(A)(x))=(T′′∘π¯)(1A⊗x) for A∈Ba, x∈E.
For m¯:Ba→L(E,F) by m¯nuc(A) we denote the variation of m¯ on A∈Ba; that is,
(40)m¯nuc(A):=sup∑i=1nm¯(Ai)nuc,
where the supremum is taken over all finite Ba-partitions (Ai)i=1n of A.
Theorem 34.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous and nuclear operator and let m∈Mσ(X,L(E,F′′)) be its representing measure. Then the following statements hold.
T is strongly bounded.
For each A∈Ba, m¯(A)∈N(E,F).
|m¯|nuc(X)<∞ and m¯:Ba→N(E,F) is ·nuc-countably additive.
Proof.
(i) In view of [46, Ch. 3, §7, Corollary 1] T is (βσ,·F)-compact. Hence by Theorem 29T is strongly bounded.
(ii) Assume that T is of the form
(41)Tf=∑n=1∞λnΦnfynforeachf∈Cb(X,E),
where (Φn) is a βσ-equicontinuous sequence in Cb(X,E)βσ′,(yn) is a bounded sequence in F, and (λn) is a sequence in R such that ∑n=1∞λn<∞. Then for n∈N, Φn(f)=Φμn(f)=∫Xfdμn, where μn∈Mσ(Ba,E′) and μn(X)=Φμn (see Remark 9). It follows that supnμn(X)=supΦμn<∞. Assume that A∈Ba. Then for x∈E, y′∈F′, using Proposition 33 we get
(42)y′m¯Ax=T′′∘π¯1A⊗xy′=π¯(1A⊗x)(y′∘T)=π¯1A⊗x∑n=1∞λny′ynΦμn=∑n=1∞λny′ynπ¯1A⊗xΦμn=∑n=1∞λny′yn∫X1A⊗xdμn=∑n=1∞λny′(yn)μn(A)(x)=y′∑n=1∞λnμnAxyn.
Hence
(43)m¯(A)(x)=∑n=1∞λnμn(A)(x)ynforx∈E.
Note that μn(A)E′≤μn(A)≤μn(X), and hence
(44)∑n=1∞μn(A)E′λnynF≤∑n=1∞μnXλnynF≤supnμnXsupnynF∑n=1∞λn<∞.
This means that m¯(A):E→F is a nuclear operators, as desired.
(iii) To show that m¯nuc(X)<∞, assume (Ai)i=1k is a Ba-partition of X. Then using (43)(45)∑i=1km¯(Ai)nuc≤∑i=1k∑n=1∞λnμnAiE′ynF≤∑i=1k∑n=1∞λnμnAiynF≤∑i=1k∑n=1∞λnμnAiynF≤∑n=1∞λnμnXynF≤supnμnXsupnynF∑n=1∞λn<∞.
Hence m¯nuc(X)<∞, as desired. Now we will show that m¯:Ba→N(E,F) is ·nuc-countably additive. Let ɛ>0 be given. Since ∑n=1∞λnμn(X)≤supnμnX∑n=1∞λn<∞, one can choose nɛ∈N such that
(46)∑n=nɛ+1∞λnμn(X)≤ɛ2a,wherea=supnynF.
Since μn∈Mσ(Ba,E′), for n∈N, there exists k∈N such that
(47)λjμj⋃i=k∞Ai≤ɛ2nɛaforj=1,…,nɛ.
Hence
(48)m¯⋃i=k∞Ai-∑i=1k-1m¯Ainuc=m¯⋃i=k∞Ainuc≤∑n=1∞λnμn⋃i=k∞AiE′ynF≤a∑n=1nɛλnμn⋃i=k∞Ai+a∑n=nɛ+1∞λnμn⋃i=k∞Ai≤aɛ2a+aɛ2a=ɛ.
This means that m¯:Ba→N(E,F) is ·nuc-countably additive.
9. Strictly Singular Operators on Cb(X,E)Definition 35.
A bounded linear operator T:Cb(X,E)→F is said to be strictly singular if it does not have a bounded inverse on any infinite-dimensional subspace contained in Cb(X,E).
Bilyeu and Lewis [21, Theorem 4.1] showed that if X is compact, then every strictly singular operator T:C(X,E)→F is strongly bounded and, for each Borel set in X, m(A):E→F is strictly singular. Strictly singular operators T:C(X,E)→F have been studied by Bessaga and Pełczyński [44] and Abbott et al. [18].
Now we show an analogue of Theorem 4.1 of [21] for (βσ,·)-continuous and strictly singular operators T:Cb(X,E)→F, where X is a completely regular Hausdorff space.
Theorem 36.
Let T:Cb(X,E)→F be a (βσ,·F)-continuous and strictly singular linear operator and let m∈Mσ(X,L(E,F′′)) be its representing measure. Then the following statements hold.
T is strongly bounded.
For each A∈Ba, m¯(A):→F is strictly singular.
Proof.
Since T is strictly singular, T is unconditionally converging (see [47, Proposition 1.5]) and hence by Theorem 17T is strongly bounded. Suppose that there is A∈Ba such that m¯(A):E→F is not strictly singular. Then there is an infinite-dimensional subspace M of E so that m¯(A)M has a bounded inverse. Therefore, there is a>0 so that m¯(A)(x)F≥axE for each x∈M.
Let ɛ>0 be given such that 2ɛ<a. Hence by Corollary 12, there exist Z∈Z, Z⊂A and U∈P, U⊃A such that m~(U∖Z)≤ɛ. Choose a function uo∈Cb(X) with 0≤uo≤1X such that uo|Z≡1 and uo|X∖U≡0. For x∈E let hx=uo⊗x. Then, by Theorem 8,
(49)T(hx)F=∫Uhxdm¯F≥∫Zhxdm¯F-∫U∖Zhxdm¯F≥∫Zhxdm¯F-∫U∖Zhxdm¯F≥m¯(Z)(x)F-m~(U∖Z)xE≥m¯(A)(x)F-m¯(A∖Z)(x)F-ɛxE≥m¯(A)(x)F-m¯~(A∖Z)xE-ɛxE≥axE-ɛxE-ɛxE≥(a-2ɛ)xE.
Let EM={uo⊗x:x∈M}. Then EM is an infinite-dimensional subspace of Cb(X,E), and this means that T is not strictly singular, a contradiction.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
DinculeanuN.1967New York, NY, USAPergamon PressMR0206190DinculeanuN.2000John Wiley & SonsMR178243210.1002/9781118033012SentillesF. D.Bounded continuous functions on a completely regular space197216831133610.1090/s0002-9947-1972-0295065-1MR0295065WheelerR. F.A survey of Baire measures and strict topologies19831297190MR710569FontenotR. A.Strict topologies for vector-valued functions1974264841853MR0348463KhuranaS. S.Topologies on spaces of vector-valued continuous functions197824119521110.2307/1998840MR492297KhuranaS. S.OthmanS. I.Convex compactness property in certain spaces of measures1987279234534810.1007/BF01461727MR9195102-s2.0-34250105093KhuranaS. S.OthmanS. I.Completeness and sequential completeness in certain spaces of measures1995452163170MR1357072KhuranaS. S.VielmaJ.Weak sequential convergence and weak compactness in spaces of vector-valued continuous functions1995195125126010.1006/jmaa.1995.1353MR13528212-s2.0-27844486311GranirerE. E.On Baire measures on D-topological spaces196760122KatsarasA.Continuous linear functionals on spaces of vector-valued functions1974151319MR0629982NowakM.Operators on spaces of bounded vector-valued continuous functions with strict topologies2014201412407521DiestelJ.UhlJ. J.197715Providence, RI, USAAmerican Mathematical SocietyMathematical SurveysMR0453964LewisP. W.Strongly bounded operators197453120720910.2140/pjm.1974.53.207MR0367724LewisP. W.Variational semi-regularity and norm convergence19732602130MR0322508BrooksJ. K.LewisP. W.Operators on continuous function spaces and convergence in the spaces of operators1978292157177MR5068892-s2.0-004053002810.1016/0001-8708(78)90009-9AbbottC. A.BatorE. M.BilyeuR. G.LewisP. W.Weak precompactness, strong boundedness, and weak complete continuity1990108232533510.1017/s030500410006919xMR1074720AbbottC.BatorE.LewisP.Strictly singular and strictly cosingular operators on spaces of continuous functions1991110350552110.1017/S0305004100070584MR1120485BattJ.Applications of the Orlicz-Pettis theorem to operator-valued measures and compact and weakly compact transformations on the spaces of continuous functions196914907935BattJ.BergE. J.Linear bounded transformations on the space of continuous functions196942215239MR0248546BilyeuR.LewisP.Some mapping properties of representing measures19761092732872-s2.0-3425039132810.1007/bf02416964MR0425609BombalF.CembranosP.Characterization of some classes of operators on spaces of vector-valued continuous functions1985971137146MR76450210.1017/s0305004100062678BombalF.CembranosP.Dieudonné operators on C(K,E)198634301305BombalF.On weakly compact operators on spaces of vector valued continuous functions1986971939610.2307/2046087MR831394BombalF.PorrasB.Strictly singular and strictly cosingular operators on C(K, E)198914335536410.1002/mana.19891430125MR1018252BombalF.Rodriguez-SalinasB.Some classes of operators on C(K,E). Extension and applications1986471556510.1007/bf012025002-s2.0-34250123959MR855138BrooksJ. K.LewisP. W.Linear operators and vector measures197419213916210.1090/s0002-9947-1974-0338821-5MR0338821DobrakovI.On representation of linear operators on Co(T, X)1971211330MR0276804SaabP.Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions1984951101108MR72708410.1017/S030500410006134XSaabE.SaabP.On unconditionally converging and weakly precompact operators1991353522531MR1103684SaabP.SmithB.Nuclear operators on spaces of continuous vector-valued functions199133222323010.1017/S0017089500008259MR1108746SaabP.SmithB.Spaces on which unconditionally converging operators are weakly completely continuous19922231001100910.1216/rmjm/1181072710MR11837022-s2.0-84881140487SwartzC.Unconditionally converging operators on the space of continuous functions19721716951702MR0333815KatsarasA.Spaces of vector measures197520631332810.1090/S0002-9947-1975-0365111-8MR0365111TopsoeF.Compactness in spaces of measures197036195212GrothendieckA.Sur les applications lineaires faiblement compactnes d’espaces de type C(K)19535129173MR0058866DiestelJ.198492Berlin, GermanySpringerGraduate Texts in Mathematics10.1007/978-1-4612-5200-9MR737004GravesW. H.RuessW.Compactness in spaces of vector-valued measures and a natural Mackey topology for spaces of bounded maesurable functions19802180203DobrakovI.On integration in Banach spaces, I1970203511536MR0365138NowakM.Operators on the space of bounded strongly measurable functions2012388139340310.1016/j.jmaa.2011.10.033MR28697542-s2.0-84455205653CembranosP.MendozaJ.19971676Berlin, GermanySpringerLecture Notes in MathematicsBourgainJ.An averaging result for l1-sequences and applications to weakly conditionally compact sets in LX1197932428929810.1007/bf02760458MR5710832-s2.0-32044457165AliprantisC. D.BurkinshawO.1985119New York, NY, USAAcademic PressPure and Applied MathematicsMR809372BessagaC.PełczyńskiA.On bases and unconditional convergence of series in Banach spaces1958172151164MR0115069EdwardsR. E.1965New York, NY, USAHolt, Rinehart and WinstonMR0221256SchaeferH. H.1971New York, NY, USASpringerMR0342978HowardJ.The comparison of an unconditionally converging operator196933295298MR0247520