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p r =: p , with p the conjugate exponent of p, is a reflexive Fr´echet space, equipped with the seminorms
1/β(n) ∞ + β(n) qp ,n (x) := |xi | , x ∈ p , i=1
where β(n) := p + for n ∈ N. This family of Fr´echet spaces was studied in [18]. + By [1, Proposition 2.15] the Fr´echet space p , for 1 < p < ∞, is not uniformly − mean ergodic. So, again by Corollary 2.9, the (LB)-space p , for 1 < p < ∞, is not uniformly mean ergodic. 1 n
3. Mean ergodic results A sequence (Pn )∞ n=1 ⊆ L(X) is a Schauder decomposition of X if it satisfies: (S1) Pn Pm = Pmin{m,n} for all m, n ∈ N, (S2) Pn → I in Ls (X) as n → ∞, and (S3) Pn = Pm whenever n = m. By setting Q1 := P1 and Qn := Pn − Pn−1 for n ≥ 2 we arrive at a sequence of ∞ pairwise orthogonal projections (i.e., Qn Qm = 0 if n = m) satisfying n=1 Qn = I, with the series converging in Ls (X). If the series is unconditionally convergent in Ls (X), then {Pn }∞ n=1 is called an unconditional Schauder decomposition, [19]. Such decompositions are intimately associated with (non-trivial) spectral measures; see (the proof of) [4, Proposition 4.3] and [19, Lemma 5 and Theorem 6]. If X is
8
A.A. Albanese, J. Bonet and W.J. Ricker
barrelled, then (S2) implies that {Pn }∞ n=1 is an equicontinuous subset of L(X). According to (S1) each Pn and Qn , for n ∈ N, is a projection and Qn → 0 in Ls (X) as n → ∞. Condition (S3) ensures that Qn = 0 for each n ∈ N. Let {Pn }∞ n=1 ⊆ L(X) be a Schauder decomposition of X. Then the dual projections {Pnt }∞ n=1 ⊆ L(Xσ ) always form a Schauder decomposition of Xσ , [11, t ∞ p. 378]. If, in addition, {Pn }n=1 ⊆ L(Xβ ) is a Schauder decomposition of Xβ , then the original sequence {Pn }∞ n=1 is called shrinking, [11, p. 379]. Since (S1) and (S3) t clearly hold for {Pnt }∞ n=1 , this means precisely that Pn → I in Ls (Xβ ); see (S2). In dealing with the uniform mean ergodicity of operators the following notion, due to J.C. D´ıaz and M.A. Mi˜ narro, [7, p. 194], is rather useful. A Schauder decomposition {Pn }∞ in a lcHs X is said to have property (M ) if Pn → I in n=1 Lb (X) as n → ∞. Since every non-zero projection P in a Banach space satisfies P ≥ 1, it is clear that no Schauder decomposition in any Banach space can have property (M ). For non-normable spaces the situation is quite different. For instance, if X is a Fr´echet Montel space (resp. Fr´echet GDP-space, which is a larger class of spaces; see [4]), then every Schauder decomposition in X has property (M ); see [7] (resp. [4, Proposition 4.2]). The following two technical results will be needed latter. Lemma 3.1. Let X be a barrelled lcHs which admits a non-shrinking Schauder decomposition. Then there exists a Schauder decomposition {Pj }∞ j=1 ⊂ L(X) of X, a functional ξ ∈ X and a bounded sequence {zj }∞ j=1 ⊂ X with zj ∈ (Pj+1 −Pj )(X) such that | zj , ξ | > 12 for all j ∈ N.
Proof. Adapt the proof of Lemma 4.4 in [1].
Lemma 3.2. Let X be a barrelled lcHs which admits a Schauder decomposition without property (M ). Then there exists a Schauder decomposition {Pj }∞ j=1 ⊆ L((X) of X, a seminorm q ∈ ΓX and a bounded sequence {zj }∞ j=1 ⊂ X with zj ∈ (Pj+1 − Pj )(X) such that q(zj ) > 12 for all j ∈ N.
Proof. The proof of Lemma 4.5 in [1] also applies here.
Remark 3.3. Let {Pj }∞ j=1 be any Schauder decomposition in the complete barrelled lcHs X with ΓX a system of continuous seminorms generating the topology of X. Then {Pj }∞ j=1 is an equicontinuous sequence. Hence, for every p ∈ ΓX there exist q ∈ ΓX and Mp > 0 such that p(Pj x) ≤ Mp q(x),
x ∈ X,
for all j ∈ N. By setting r˜(x) := supj∈N r(Pj x), for every r ∈ ΓX , we obtain p(x) ≤ p˜(x) ≤ Mp q(x) ≤ Mp q˜(x),
x ∈ X.
˜ X := {˜ p : p ∈ ΓX } is also a system of continuous seminorms Accordingly, Γ generating the topology of X and satisfies p˜(Pj x) ≤ p˜(x),
x ∈ X, j ∈ N.
(3.1)
On Mean Ergodic Operators
9
The proof of the next result and Theorems 3.6 and 3.8 below follow those given in [1] for the corresponding result in Fr´echet spaces. We include the essential parts of these proofs to illustrate certain differences in the current setting and for the sake of self containment. Theorem 3.4. Let X be a complete barrelled lcHs which admits a non-shrinking Schauder decomposition. Then there exists a power bounded operator on X which is not mean ergodic. Proof. Let (Pj )j ⊂ L(X) denote a Schauder decomposition as given by Lemma 3.1 and define projections Qj := Pj − Pj−1 (P0 := 0) and closed subspaces Xj := Qj (X), j ∈ N. By Lemma 3.1 there exist a bounded sequence {zj }∞ j=1 ⊂ X with zj ∈ Xj+1 , and ξ ∈ X such that | zj , ξ | > 12 for all j ∈ N. Set ej := zj / zj , ξ ∈ Xj+1 . Then {ej }∞ j=1 is a bounded sequence of X and ej , ξ = 1 for all j ∈ N. By Remark 3.3 there exists a system ΓX of continuous seminorms generating the topology of X such that p(Pj x) ≤ p(x),
x ∈ X,
(3.2)
for all p ∈ ΓX and j ∈ N. Moreover, since ξ ∈ X , there exists p0 ∈ ΓX such that | x, ξ | ≤ p0 (x) for all x ∈ X. ∞ As in [10, p. 150], take a sequence a = {aj }∞ ⊆ R with j=1 j=1 aj = 1, n aj > 0, and set An := j=1 aj . For x ∈ X and integers m > n ≥ 2 we have m
⎛ Ak Qk x = ⎝
n−1
⎞ aj ⎠
j=1
k=n
m k=n
Qk x
+
m
⎛ aj ⎝
j=n
m
⎞ Qk x⎠ .
k=j
m ∞ ∞ Since k=1 Qk x sums to x in X, we see that { k=1 Ak Qk x}m=1 is a Cauchy sequence and hence, converges in X. Moreover, for each p ∈ ΓX , by (3.2) we have p
m
Ak Qk x
=
⎛ ⎞ m p⎝ aj (Pm − Pj−1 )x⎠ j=1
k=1
≤
m
aj (p(Pm x) + p(Pj−1 x)) ≤ 2p(x),
(3.3)
j=1
for each m ∈ N. Define a linear map Ta : X → X by Ta x :=
∞ k=1
Ak Qk x +
∞ j=2
Pj−1 x, ξ aj ej ,
x ∈ X.
(3.4)
10
A.A. Albanese, J. Bonet and W.J. Ricker
From (3.3) we obtain, for each p ∈ ΓX with p ≥ p0 , that p(Ta x)
≤
∞ ∞ p( Ak Qk x) + | Pj−1 x, ξ |aj p(ej ) j=2
k=1
≤
2p(x) +
∞
aj p0 (Pj−1 x)p(ej ).
j=2
Note that Mp := supj∈N p(ej ) < ∞, because {ej }∞ j=1 is bounded in X. Moreover, by (3.2) we have p0 (Pj−1 x) ≤ p0 (x) ≤ p(x) for all x ∈ X. Hence, p(Ta x) ≤ (2 + Mp )p(x) for all x ∈ X, where 2 + Mp depends only on p. arbitrary To show that Ta is power bounded, it suffices to show that for ∞ sequences a = {aj }∞ and b = {bj }∞ of positive numbers with j=1 aj = 1 = j=1 j=1 ∞ j=1 bj we have Ta Tb = Tc , with c a sequence of the same type. This is the claim in p. 150 of [10] which is purely algebraic and is proved on p. 151 of [10]. Finally, proceeding as in the final part of the proof of Theorem 1.5 of [1] one shows that Ker(I − Ta ) = {0} and ξ ∈ Ker(I − Tat ), i.e., Ker(I − Tat ) = {0}. Thus, we can apply Theorem 2.12 of [1] to conclude that Ta is not mean ergodic. Recall that a sequence {xn }∞ n=1 in a lcHs X is a basisif, for every x ∈ X, ∞ there is a unique sequence {αn }∞ n=1 ⊆ C such that the series n=1 αn xn converges to x in X. By setting fn (x) := αn we obtain a linear form fn : X → C which is called the nth coefficient functional associated to {xn }∞ n=1 . The functionals fn , ∞ n ∈ N, are uniquely determined by {xn }∞ and {(x , f n n )}n=1 is a biorthogonal n=1 sequence (i.e., xn , fm = δmn for m, n ∈ N). For each n ∈ N, the map Pn : X → X defined by n n Pn : x → fi (x)xi =
x, fi xi , x ∈ X, (3.5) i=1
i=1
is a linear projection with range equal to the finite-dimensional space span(xi )ni=1 . ∞ If, in addition, {fn }∞ n=1 ⊆ X , then the basis {xn }n=1 is called a Schauder basis ∞ for X. In this case, {Pn }n=1 ⊆ L(X) is clearly a Schauder decomposition of X and each dual operator Pnt : x →
n
xi , x fi ,
x ∈ X ,
(3.6)
i=1
for n ∈ N, is a projection with range equal to span(fi )ni=1 . Moreover, for every ∞ x ∈ X the series i=1 xi , x fi converges to f in Xσ . For this reason, {fn }∞ n=1 is also referred to as the dual basis of the Schauder basis {xn }∞ n=1 . The terminology “X has a Schauder basis” will also be abbreviated simply to “X has a basis”. Theorem 3.5. Let X be a complete barrelled lcHs with a Schauder basis and in which every relatively σ(X, X )-compact subset of X is relatively sequentially
On Mean Ergodic Operators
11
σ(X, X )-compact. Then X is reflexive if and only if every power bounded operator on X is mean ergodic. Proof. If X is reflexive, then X is mean ergodic by Proposition 2.3. Conversely, if X is not reflexive, then Theorem 1.2 of [1] shows that X admits a non-shrinking Schauder basis. By Theorem 3.4, X is not mean ergodic. Theorem 3.6. Let X be a complete barrelled lcHs which admits a Schauder decomposition without property (M ). Then there exists a power bounded, mean ergodic operator T ∈ L(X) which is not uniformly mean ergodic. Proof. Let {Pj }∞ j=1 ⊆ L(X) denote a Schauder decomposition as given by Lemma 3.2 and define projections Qj := Pj − Pj−1 (P0 := 0) and closed subspaces Xj := Qj (X) for all j ∈ N. By Lemma 3.2 there exist a bounded sequence {zj }∞ j=1 ⊆ X and a continuous seminorm q on X with zj ∈ Xj+1 and q(zj ) > 1/2 for all j ∈ N. Since {Pj }∞ j=1 is an equicontinuous sequence (because X is barrelled), we can apply Remark 3.3 to choose a system ΓX of continuous seminorms generating the topology of X such that p(Pj x) ≤ p(x),
x ∈ X,
(3.7)
for all p ∈ ΓX and j ∈ N. Clearly, there also exists p0 ∈ ΓX such that p0 ≥ q on X. Hence, p0 (zj ) > 1/2 for all j ∈ N. ∞ For any sequence a = {aj }∞ j=1 of positive numbers with j=1 aj = 1 we set n An := j=1 aj and define a linear map Ta : X → X by Ta x :=
∞
Ak Qk x,
x ∈ X.
k=1
As in the proof of Theorem 3.4 one shows that Ta is well defined, satisfies p(Ta x) ≤ 2p(x),
x ∈ X,
(3.8)
for all p ∈ ΓX , and is power bounded. Proceeding as in the proof of Theorem 5.2 of [1], one shows that Ker(I −Ta ) = {0} and Ker(I − Tat ) = {0} and hence, by Theorem 2.12 of [1], Ta is mean ergodic. It remains to show that T := Ta is not uniformly mean ergodic for the choice aj := 2−j . In this case, Ak = 1 − 2−k for all k ∈ N. Moreover, from Qj Qk = 0 whenever j = k and Q2k = Qk it follows that T mx =
∞
Am k Qk x,
x ∈ X,
k=1
for all m ∈ N. Hence, ∞
1 Ak · (1 − Ank )Qk x, T[n] x = n 1 − Ak
x ∈ X, n ∈ N.
k=1
Since T is mean ergodic, there exists P ∈ L(X) with T[n] → P in Ls (X).
(3.9)
12
A.A. Albanese, J. Bonet and W.J. Ricker Next, if x ∈ Xj for a fixed j ∈ N, by (3.9) we have that T[n] x =
1 Aj · (1 − Anj )x, n 1 − Aj
for all n ∈ N, as Qj Qk = 0 whenever j = k and Q2j = Qj . Since 0 < (1 − Anj ) < 1, it follows that 1 Aj p(x) p(T[n] x) ≤ n 1 − Aj for all p ∈ ΓX and n ∈ N. Therefore, p(T[n] x) → 0 as n → ∞ for all p ∈ ΓX . Since T[n] x → P x as n → ∞, we see that P x = 0. That is, P y = 0 for all y ∈ ∪∞ j=1 Xj . Since ∪∞ X is dense in X and P ∈ L(X), we obtain that P = 0 on X, that is, j=1 j T[n] → 0 in Ls (X). Suppose that T is uniformly mean ergodic. Then T[n] → 0 in Lb (X). In particular, since {zj }∞ j=1 is a bounded sequence in X, we have lim sup p(T[n] zj ) = 0
n→∞ j∈N
(3.10)
for all p ∈ ΓX . But, for all j ∈ N, p0 (T[2j ] zj ) >
j 1 [1 − (1 − 2−j )2 ], 4
with limj→∞ (1 − 2−j )2 = e−1 . This is in contradiction with (3.10). j
For Fr´echet spaces, the following result occurs in [1, Theorem 1.3]. Theorem 3.7. Let X be a complete barrelled lcHs with a Schauder basis and in which every relatively σ(X, X )-compact subset of X is relatively sequentially σ(X, X )-compact. Then X is Montel if and only if every power bounded operator on X is uniformly mean ergodic, that is, if and only if X is uniformly mean ergodic. Proof. Suppose that X is Montel. Then Proposition 2.4 implies that X is uniformly mean ergodic. Conversely, suppose that X is not Montel. Observe that the Schauder decomposition {Pn }∞ n=1 ⊂ L(X) induced by the basis of X has the property that each space Qn (X) := (Pn − Pn−1 )(X), n ∈ N, is Montel because dim Qn (X) = 1 for all n ∈ N. By [2, Theorem 3.7(iii)], the Schauder decomposition {Pn }∞ n=1 does not satisfy property (M ) and hence, Theorem 3.6 guarantees the existence of a power bounded, mean ergodic operator in L(X) which fails to be uniformly mean ergodic. Theorem 3.8. Let X be a sequentially complete lcHs which contains an isomorphic copy of the Banach space c0 . Then there exists a power bounded operator on X which is not mean ergodic. Proof. Suppose that J is a topological isomorphism from c0 into X. Let {en }∞ n=1 be the canonical basis of c0 . Then the elements yn := Jen form a Schauder basis of Y := J(c0 ).
On Mean Ergodic Operators
13
Denote by || ||c0 the norm in c0 and by ΓX a system of continuous seminorms generating the topology of X. Then, for all p ∈ ΓX , there exists Mp > 0 such that p(Jx) ≤ Mp ||x||c0 ,
x ∈ c0 .
There also exist p0 ∈ ΓX and K > 0 such that ||x||c0 ≤ Kp0 (Jx),
x ∈ c0 .
Therefore, we have that ∞ ∞ and p xj yj xj yj ≤ Mp sup |xj | sup |xj | ≤ Kp0 j∈N
j=1
(3.11)
j∈N
j=1
for all x = (xj )∞ j=1 ∈ c0 and p ∈ ΓX . 1 ∞ Let {en }∞ n=1 ⊂ denote the dual basis of {en }n=1 . For each n ∈ N, define −1 ∞ yn ∈ Y by yn := en ◦ J , in which case {yn }n=1 is the dual basis of {yn }∞ n=1 and | y, yn | ≤ Kp0 (y),
y ∈ Y,
as y = Jx for some x ∈ c0 . By the Hahn–Banach theorem, for each n ∈ N we can find fn ∈ X such that fn |Y = yn and Define xn :=
n i=1
| x, fn | ≤ Kp0 (x),
x ∈ X.
(3.12)
yi and gn := fn − fn+1 , for each n ∈ N, and observe that
xk , gn = xk , fn − xk , fn+1 = δkn
for all k, n ∈ N. We can then define projections Pn : X → X via n
x, gk xk , x ∈ X, Pn x := k=1
so that Pn (X) = span {xj }nj=1 = span {yj }nj=1 and Pn Pm = Pmin{n,m} . Set h := f1 and observe that n yj , f1 = 1, n ∈ N.
xn , h = j=1
On the other hand, xn ∈ (Pn − Pn−1 )(X) (with P0 :=0) for all n ∈ N, and n {xn }∞ is a bounded sequence in X because xn = J( j=1 ej ), n ∈ N. Since nn=1 j=1 ej c0 = 1, we have p(xn ) ≤ Mp ,
n ∈ N,
(3.13)
for all p ∈ ΓX . In particular, n 1 1 ej c0 = , p0 (xn ) ≥ K j=1 K
Moreover, the identities n Pn x = ( x, fk − x, fn+1 )yk , k=1
n ∈ N.
x ∈ X, n ∈ N,
14
A.A. Albanese, J. Bonet and W.J. Ricker
together with (3.11) and (3.12) imply that p(Pn x) ≤ Mp sup | x, fk − x, fn+1 | ≤ 2Mp Kp0 (x)
(3.14)
1≤k≤n
for all p ∈ ΓX , n ∈ N and x ∈ X. Accordingly, {Pn }∞ equicontinuous. n=1 ⊂ L(X) is ∞ ∞ Let a = {aj }j=1 be any sequence of positive numbers with j=1 aj = 1 and n set An := j=1 aj for n ∈ N. As in the statement of Theorem 3 of [10], we define Sa x := x −
∞
an Pn−1 x +
n=2
∞
Pn−1 x, h xn ,
x ∈ X.
n=2
Then by (3.13), (3.12) and (3.11) we have, for each x ∈ X, that p(Sa x)
≤ p(x) + 2Mp Kp0 (x) + Mp sup | Pn−1 x, h | n≥2
≤ p(x) + 2Mp Kp0 (x) + Mp Kp0 (Pn−1 x) ≤ p(x) + 2Mp Kp0 (x) + Mp K 2 Mp0 p0 (x) = (1 + 2Mp K + Mp K 2 Mp0 )p(x) for all p ∈ ΓX with p ≥ p0 . So, Sa ∈ L(X). The fact that Sa is power bounded follows from the Claim on p. 156 of [10], stating that Sa Sb = Sc for an appropriate c. It remains to show that Sa is not mean ergodic. For this, we can now proceed exactly as in the final part of the proof of Theorem 1.6 of [1].
4. Mean ergodicity of co-echelon spaces We wish to give an application of the previous results to K¨ othe co-echelon spaces. Let I be a countable index set. A K¨ othe matrix A = (an )∞ n=1 is an increasing sequence of strictly positive functions on I. Let V = (vn )∞ n=1 denote the associated decreasing sequence of functions vn := 1/an , n ∈ N. Define the inductive limits kp (V ) = kp (I, V ) = ind p (vn ), 1 ≤ p ≤ ∞, and k0 (V ) = k0 (I, V ) = ind c0 (vn ) , n
n
generated by the (weighted) Banach spaces
1/p I p p (vn ) = {x = (xi )i∈I ∈ C : qp,n (x) = (vn (i)|xi |) < ∞}, if 1 ≤ p < ∞ , i∈I
and ∞ (vn ) = {x = (xi )i∈I ∈ CI : q∞,n (x) = sup vn (i)|xi | < ∞} , i∈I
c0 (vn ) = {x = (xi )i∈I ∈ C : (vn (i)|xi |)i∈I converges uniformly to 0 in I} . I
That is, kp (V ) is the increasing union of the Banach spaces p (vn ), respectively c0 (vn ), for n ∈ N, endowed with the strongest locally convex topology under which
On Mean Ergodic Operators
15
the inclusion of each of these Banach spaces is continuous, i.e., kp (V ) is an (LB)space and so a barrelled, ultrabornological (DF)-space. The spaces kp (V ) are called co-echelon spaces of order p. Given a decreasing sequence V = (vn )∞ n=1 of strictly positive functions on I, set v¯(i) I ¯ <∞ V = v¯ = (¯ v (i))i∈I ∈ R+ : ∀n ∈ N sup i∈I vn (i) and associate to V¯ the following projective limit spaces Kp (V¯ ) = Kp (I, V¯ ) = proj p (¯ v ), if 1 ≤ p ≤ ∞; K0 (V¯ ) = K0 (I, V¯ ) = proj c0 (¯ v) . ¯ v ¯∈V
¯ v ¯∈V
These spaces are equipped with the complete locally convex topology given by the seminorms (qp,¯v )v¯∈V¯ , where
1/p (¯ v (i)|xi |)p , 1 ≤ p < ∞, and q∞,¯v (x) = sup v¯(i)|xi | . qp,¯v (x) = i∈I
i∈I
Then kp (V ) is continuously embedded in Kp (V¯ ) for 1 ≤ p ≤ ∞ or p = 0, with kp (V ) = Kp (V¯ ) for 1 ≤ p ≤ ∞. More precisely, kp (V ) = Kp (V¯ ) algebraically and topologically for 1 ≤ p < ∞ and k∞ (V ) = K∞ (V¯ ) algebraically. Moreover, k0 (V ) is, in general, a proper topological subspace of the barrelled (DF)-space K0 (V¯ ) such that its completion is equal to K0 (V¯ ). The (LB)-space kp (V ) is complete for 1 ≤ p ≤ ∞, and reflexive for 1 < p < ∞. In particular, the vectors ej = (δij )i∈I form a Schauder basis for kp (V ) if 1 ≤ p < ∞ or p = 0. For all these facts we refer to [3]. Proposition 4.1. Let V = (vn )∞ n=1 be a decreasing sequence of strictly positive functions on I and 1 < p < ∞. Then the reflexive (LB)-space kp (V ) (= Kp (V¯ ) algebraically and topologically) is uniformly mean ergodic if and only if it is a Montel space (hence, a (DFM)-space). Proof. Since the complete barrelled space kp (V ) admits a Schauder basis and its bounded sets are relatively sequentially σ(kp (V ), (kp (V )) )-compact, [6, Theorem 11, Examples 1,2], the result follows from Theorem 3.7. Proposition 4.2. Let V = (vn )∞ n=1 be a decreasing sequence of strictly positive functions on I. Then the following assertions are equivalent. (i) k1 (V ) is mean ergodic. (ii) k1 (V ) is uniformly mean ergodic. (iii) k1 (V ) is a Montel space (hence, a (DFM)-space). (iv) k1 (V ) does not contain an isomorphic copy of 1 . Proof. The complete barrelled (LB)-space k1 (V ) admits a Schauder basis and every relatively σ(k1 (V ), (k1 (V )) )-compact subset of k1 (V ) is relatively sequentially σ(k1 (V ), (k1 (V )) )-compact, [6, Theorem 11, Examples 1, 2]. So, by Theorem 3.7 we have (ii) ⇔ (iii), and by [3, Theorem 4.7] we have (iii) ⇔ k1 (V ) is reflexive. On the other hand, k1 (V ) is reflexive ⇔ (i); see Theorem 3.5.
16
A.A. Albanese, J. Bonet and W.J. Ricker
Next, (iii) ⇒ (iv) is obvious, because a Montel space cannot contain an isomorphic copy of any infinite-dimensional Banach space. (iv) ⇒ (iii): Suppose that k1 (V ) is not a Montel space. Then there exist an infinite set I0 ⊂ I and n ∈ N such that vm (i) = cm > 0 , ∀m ≥ n, inf i∈I0 vn (i) [3, Theorem 4.7]. Then, in the sectional subspace E0 of k1 (V ) defined by E0 := {x ∈ k1 (V ) : xj = 0 for all j ∈ I \ I0 }, the topology of 1 (vm ) coincides with that of 1 (vn ) for all m ≥ n. Indeed, for every x ∈ E0 and m ≥ n we have q1,m (x) ≤ q1,n (x) = vn (i)|xi | ≤ c−1 vm (i)|xi | = c−1 m m q1,m (x) . i∈I0
i∈I0
Consequently, the topology of k1 (V ) also coincides with that of 1 (vn ) in E0 . Hence, k1 (V ) contains an isomorphic copy of 1 , which is a contradiction. Proposition 4.3. Let V = (vn )∞ n=1 be a decreasing sequence of strictly positive functions on I. Then the following assertions are equivalent. (i) k∞ (V ) is mean ergodic. (ii) k∞ (V ) is uniformly mean ergodic. (iii) k∞ (V ) is a Montel space (hence, a (DFM)-space). (iv) k∞ (V ) does not contain an isomorphic copy of ∞ . (v) K0 (V¯ ) = K∞ (V¯ ) = k∞ (V ) algebraically and topologically. Proof. By the discussion just prior to Proposition 2.3, together with Proposition 2.4, it is clear that (iii) ⇒ (ii). That (ii) ⇒ (i) is obvious. Since ∞ is an infinitedimensional Banach space (i.e., its closed unit ball is not compact), it is clear that (iii) ⇒ (iv). Moreover, (iii) ⇔ (v) by [3, Theorem 4.7]. (iv) ⇒ (iii): Suppose that k∞ (V ) is not a Montel space. Then there exist an infinite set I0 ⊂ I and n ∈ N such that vm (i) = cm > 0 , ∀m ≥ n, inf i∈I0 vn (i) [3, Theorem 4.7]. Then, in the sectional subspace E0 of k∞ (V ) defined by E0 := {x ∈ k∞ (V ) : xj = 0 for all j ∈ I \ I0 }, the topology of ∞ (vm ) coincides with that of ∞ (vn ) for all m ≥ n. Indeed, for every x ∈ E0 and m ≥ n we have −1 q∞,m (x) ≤ q∞,n (x) = sup vn (i)|xi | ≤ c−1 m sup vm (i)|xi | = cm q∞,m (x) . i∈I0
i∈I0
Consequently, the topology of k∞ (V ) also coincides with that of ∞ (vn ) in E0 . Hence, k∞ (V ) contains an isomorphic copy of ∞ . This is a contradiction. (i) ⇔ (iv): Suppose that k∞ (V ) contains an isomorphic copy of ∞ . This implies that k∞ (V ) is not mean ergodic by [1, Remark 2.14(i)].
On Mean Ergodic Operators
17
Proposition 4.4. Let V = (vn )∞ n=1 be a decreasing sequence of strictly positive functions on I. Suppose that the (LB)-space k0 (V ) is complete. Then the following assertions are equivalent. (i) k0 (V ) is mean ergodic. (ii) k0 (V ) is uniformly mean ergodic. (iii) k0 (V ) is a Schwartz space (hence, a (DFS)-space). (iv) k0 (V ) does not contain an isomorphic copy of c0 . (v) k0 (V ) = k∞ (V ) algebraically and topologically. Proof. The complete, barrelled (LB)-space k0 (V ) admits a Schauder basis and every relatively σ(k0 (V ), (k0 (V )) )-compact subset of k0 (V ) is relatively sequentially σ(k0 (V ), (k0 (V )) )-compact, [6, Theorem 11, Examples 1, 2]. So, by Theorem 3.7 above and [3, Theorem 4.9] we have (ii) ⇔ k0 (V ) is a Montel space ⇔ (iii) ⇔ (v). Next, (ii) ⇒ (i) is obvious. Also, (iii) ⇒ (iv) is obvious, because a Schwartz space cannot contain an isomorphic copy of any infinite-dimensional Banach space. (iv) ⇒ (iii): Suppose that k0 (V ) is not a Schwartz space. Since k0 (V ) is complete, there exist an infinite set I0 ⊂ I and n ∈ N such that vm (i) = cm > 0 , ∀m ≥ n, inf i∈I0 vn (i) [3, Theorems 3.7, 4.9]. Then, in the sectional subspace E0 of k0 (V ) defined by E0 := {x ∈ k0 (V ) : xj = 0 for all j ∈ I \ I0 }, the topology of c0 (vm ) coincides with that of c0 (vn ) for all m ≥ n. Indeed, for every x ∈ E0 and m ≥ n we have −1 q∞,m (x) ≤ q∞,n (x) = sup vn (i)|xi | ≤ c−1 m sup vm (i)|xi | = cm q∞,m (x) . i∈I0
i∈I0
Consequently, the topology of k0 (V ) also coincides with that of c0 (vn ) in E0 . Hence, k0 (V ) contains an isomorphic copy of c0 . This is a contradiction. (i) ⇒ (ii): By Theorem 3.5 we have (i) ⇔ k0 (V ) is reflexive. Since k0 (V ) is complete, it is then also Montel by [3, Theorems 3.7, 4.7], thereby implying that (ii) holds via Theorem 3.7. Example. Every (LF)-space X (hence, every (LB)-space) satisfies the hypothesis of Proposition 2.5, because every linear continuous surjective map between two (LF)-spaces is necessarily open. But, in this setting, condition (iii) of Proposition 2.5 does not imply the condition (ii). Hence, also (iii) does not imply condition (i) as the following example illustrates. Let (an )∞ n=1 be a sequence of real numbers satisfying 1 < an+1 < an < a for ∞ some a ∈ R and for all n ∈ N. For each n ∈ N set vn := (ain )∞ i=1 and V := (vn )n=1 , where i ∈ I := N. Consider the co-echelon space k1 (V ) which is a Montel space (i) (hence, a (DFM)-space) because, for all n, m ∈ N with m > n, we have vvm = n (i) i am → 0 as i → ∞, [3, Theorem 4.7]. In particular, its (strong) topological an
18
A.A. Albanese, J. Bonet and W.J. Ricker
dual is the K¨ othe echelon Fr´echet space λ∞ (A) = λ0 (A), with A := (vn−1 )∞ n=1 , [3, Theorem 4.7]. Define T ∈ L(k1 (V )) by T x := ((1 − a−i )xi )∞ i=1 ,
x ∈ k1 (V ) .
It is easy to verify that Ker(I − T ) = {0} and that y := (a−i )∞ i=1 ∈ k1 (V ) does not belong to Im(I − T ), i.e., I − T is not surjective. So, condition (ii) of Theorem 2.5 does not hold. Since T m x = ((1−a−i )m xi )∞ i=1 for x ∈ k1 (V ) and for all m ∈ N, the sequence is bounded for all x ∈ k1 (V ). Indeed, given any x ∈ k1 (V ) there is {T m x}∞ m=1 n ∈ N such that x ∈ 1 (vn ), thereby implying that q1,n (T m x) = |(1 − a−i )m | · |xi |vn (i) ≤ |xi |vn (i) = q1,n (x) i∈N
i∈N
for all m ∈ N. So, the barrelledness of k1 (V ) implies that the sequence {T m}∞ m=1 ⊂ L(k1 (V )) is equicontinuous, i.e., for every p ∈ Γk1 (V ) the exists q ∈ Γk1 (V ) for which p(T m x) ≤ q(x) for all m ∈ N and x ∈ k1 (V ). In particular, T is power bounded. Since k1 (V ) is a complete (LB)-space and hence, a regular (LB)-space, given any bounded set B ⊂ k1 (V ) there exist k, n ∈ N such that B ⊂ kBn (Bn denotes the unit ball of 1 (vn )). On the other hand, since the inclusion map 1 (vn ) → k1 (V ) is continuous, given any p ∈ Γk1 (V ) there exists c > 0 such that p(x) ≤ cq1,n (x),
x ∈ 1 (vn ).
Therefore, for every m ∈ N we have 1 m 1 1 1 T x ≤c sup q1,n (T m x) ≤ c sup q1,n (x) ≤ ck sup p m m x∈kBn m x∈kBn m x∈B 1 m 1 m T x) → 0 as m → ∞. This shows that m T → 0 in and hence, supx∈B p( m Lb (k1 (V )). It remains to establish condition (iii) of Proposition 2.5. For this, we observe that ξ ∈ λ∞ (A) , T t ξ = ((1 − a−i )ξi )∞ i=1 ,
so that Ker(I − T t ) = {0}. Since T is power bounded and both Ker(I − T ) = {0} and Ker(I − T t) = {0}, we can apply [1, Theorem 2.12] to conclude that T is mean ergodic. But, k1 (V ) is a Montel space whose relatively σ(k1 (V ), (k1 (V )) )-compact subsets are relatively sequentially σ(k1 (V ), (k1 (V )) )-compact (see the discussion prior to Proposition 2.3). So, by Proposition 2.4, T is also uniformly mean ergodic. Hence, there is P ∈ L(k1 (V )) such that T[n] → P in Lb (k1 (V )). For each r ∈ N, let er be the element of k1 (V ) with 1 in the rth coordinate and 0’s elsewhere (we point out that {er }∞ r=1 is a Schauder basis for k1 (V )). Then, for all r ∈ N, T m er = (1 − a−r )m er → 0 as m → ∞ ,
On Mean Ergodic Operators so that T[n] er =
μ(1 − μ) er → 0 n(1 − μ)
as
19
m → ∞,
with μ := (1−a−r ). This implies that P = 0 and hence, that T[n] → 0 in Lb (k1 (V )), i.e., condition (iii) is satisfied. ∞ We remark that the operator T x := ((1 − 2−i )xi )∞ i=1 , for x = (xi )i=1 ∈ s (here s is the strong dual of the Fr´echet space s of all rapidly decreasing sequences, so that s is an (LB)-space), also satisfies condition (iii) of Proposition 2.5, but, fails condition (ii); the proof is similar to the previous one for T in k1 (V ).
References [1] A.A. Albanese, J. Bonet, W.J. Ricker, Mean ergodic operators in Fr´ echet spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 1–37. [2] A.A. Albanese, J. Bonet, W.J. Ricker, Grothendieck spaces with the Dunford–Pettis property. Positivity, in press. [3] K.D. Bierstedt, R.G. Meise, W.H. Summers, K¨ othe sets and K¨ othe sequence spaces. In: “Functional Analysis, Holomorphy and Approximation Theory”, J.A. Barroso (Ed.), North-Holland, Amsterdam, 1982, pp. 27–91. [4] J. Bonet, W.J. Ricker, Schauder decompositions and the Grothendieck and Dunford– Pettis properties in K¨ othe echelon spaces of infinite order. Positivity 11 (2007), 77–93. [5] J.M.F. Castillo, J.C. D´ıaz, J. Motos, On the Fr´echet space Lp− . Manuscripta Math. 96 (1998), 219–230. [6] B. Cascales, J. Orihuela, On compactness in locally convex spaces. Math. Z. 195 (1987), 365–381. [7] J.C. D´ıaz, M.A. Mi˜ narro, Distinguished Fr´ echet spaces and projective tensor product. Doˇ ga-Tr. J. Math. 14 (1990), 191–208. [8] N. Dunford, J.T. Schwartz, Linear Operators I: General Theory. 2nd Edition, Wiley– Interscience, New York, 1964. [9] R.E. Edwards, Functional Analysis. Reinhart and Winston, New York, 1965. [10] V.P. Fonf, M. Lin, P. Wojtaszczyk, Ergodic characterizations of reflexivity in Banach spaces. J. Funct. Anal. 187 (2001), 146–162. [11] N.J. Kalton, Schauder decompositions in locally convex spaces. Proc. Camb. Phil. Soc. 68 (1970), 377–392. [12] G. K¨ othe, Topological Vector Spaces I. 2nd Rev. Edition, Springer Verlag, Berlin– Heidelberg–New York, 1983. [13] G. K¨ othe, Topological Vector Spaces II. Springer Verlag, Berlin–Heidelberg–New York, 1979. [14] U. Krengel, Ergodic Theorems. Walter de Gruyter, Berlin, 1985. [15] M. Lin, On the uniform ergodic theorem. Proc. Amer. Math. Soc. 43 (1974), 337–340. [16] H.P. Lotz, Uniform convergence of operators on L∞ and similar spaces. Math. Z. 190 (1985), 207–220.
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[17] R.G. Meise, D. Vogt, Introduction to Functional Analysis. Clarendon Press, Oxford, 1997. [18] G. Metafune, V.B. Moscatelli, On the space p+ = q>p q . Math. Nachr. 147 (1990), 7–12. [19] S. Okada, Spectrum of scalar–type spectral operators and Schauder decompositions. Math. Nachr. 139 (1988), 167–174. [20] P. P´erez Carreras, J. Bonet, Barrelled Locally Convex Spaces. North Holland Math. Studies 131, Amsterdam, 1987. [21] K. Yosida, Functional Analysis. Springer Verlag, Berlin–Heidelberg, 1965. Angela A. Albanese Dipartimento di Matematica “E. De Giorgi” Universit` a del Salento Via Provinciale per Arnesano P.O. Box 193 I-73100 Lecce, Italy e-mail: [email protected] Jos´e Bonet Instituto Universitaro de Matem´ atica Pura y Aplicada IUMPA Edificio ID15 (8E), Cubo F, Cuarta Planta Universidad Polit´ecnica de Valencia E-46071 Valencia, Spain e-mail: [email protected] Werner J. Ricker Math.-Geogr. Fakult¨ at Katholische Universit¨ at Eichst¨ att–Ingolstadt D-85072 Eichst¨ att, Germany e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 21–39 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Fourier Series in Banach spaces and Maximal Regularity Wolfgang Arendt and Shangquan Bu Abstract. We consider Fourier series of functions in Lp (0, 2π; X) where X is a Banach space. In particular, we show that the Fourier series of each function in Lp (0, 2π; X) converges unconditionally if and only if p = 2 and X is a Hilbert space. For operator-valued multipliers we present the Marcinkiewicz theorem and give applications to differential equations. In particular, we characterize maximal regularity (in a slightly different version than the usual one) by Rsectoriality. Applications to non-autonomous problems are indicated. Mathematics Subject Classification (2000). Primary 42B15; Secondary 34G10. Keywords. Operator-valued multiplier, maximal regularity, non-autonomous problems.
0. Introduction The study of vector-valued Fourier transforms is on one hand motivated by the structure theory of Banach spaces where the validity of certain classical properties reflects geometric properties of the Banach spaces; on the other hand, it has fundamental applications in PDE. Of particular importance is the subject of operator-valued Fourier multipliers which have immediate applications to properties of maximal regularity for evolution equations. The aim of this article is to introduce into this subject and to show how it can be applied. The approach we use here is based on Fourier series. Given a Banach space X, the Fourier series of each f ∈ Lp (0, 2π; X) converges in Lp (0, 2π; X) if and only if X is a UMD-space and 1 < p < ∞. We show here that the Fourier series converges unconditionally in Lp (0, 2π; X) if and only if p = 2 and X is a Hilbert space (Theorem 1.5). Even though this result is known to specialists of unconditional The second author is supported by the NSF of China (Grant no. 10731020), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 200800030059) and the Alexander von Humboldt Foundation.
22
W. Arendt and S. Bu
structures in Banach spaces, it seems not be contained in the literature. The result can be reformulated by saying that each bounded sequence of scalar operators (λk I)k∈Z is a multiplier for Lp (0, 2π; X) if and only if p = 2 and X is a Hilbert space. The phenomenon that operator-valued versions of certain classical multiplier theorems are only valid in Hilbert spaces was first observed by Pisier (unpublished) as a consequence of Kwapien’s deep characterization of Hilbert spaces. In recent years the subject saw a spectacular revival. A break-through was the operator-valued Michlin multiplier theorem proved by Weis [W01] in 2001. Here we will concentrate on periodic multipliers, i.e., operator-valued versions of the Marcinkiewicz multiplier Theorem. If 1 < p < ∞ and X is a Hilbert space, we present the Marcinkiewicz multiplier Theorem for operator-valued sequences (Mk )k∈Z ∈ L(X) (Theorem 1.7 and Corollary 1.8). This result has a version which also holds in UMD-spaces but the notion of R-boundedness is needed. This is the content of Section 2. Then we show how the operator-valued Marcinkiewicz multiplier Theorem (Theorem 2.3) can be applied to vector-valued differential equations. Most natural is the periodic case (Section 3), but our emphasis is on the classical maximal regularity problem: u(t) ˙ = Au(t) + f (t), t ∈ (0, τ ) a.e. P0 (τ, p) u(0) = 0 . Here we consider an operator A ∈ L(D, X), where D is a Banach space which is continuously embedded in X. If for 1 ≤ p < ∞, P0 (τ, p) is well posed (i.e., for all f ∈ Lp (0, τ ; X) there is a unique solution u ∈ W 1,p (0, τ ; X) ∩ Lp (0, τ ; D) of P0 (τ, p)), then we show that the operator A is closed and R-sectorial without any assumptions on the spaces. This defers somehow from the usual setting since a priori, we do not consider A as an unbounded operator on X. To do so is motivated by the recent interesting applications of maximal regularity to the non-autonomous problem which we explain briefly in Section 5. In Section 4, we also show that conversely, problem P0 (τ, p) is well posed whenever 1 < p < ∞, X is a UMD-space and A is R-sectorial. In the present paper, we explain and complement results of [AB02] and the approach to maximal regularity via periodic multipliers chosen there. There is an alternative way based on the Michlin multiplier theorem, and we refer to [W01] and [KW04] for further information.
1. Vector-valued Fourier series and operator-valued Fourier multipliers
Let X be a Banach space and let Lp2π (X) := f : R → X measurable, f (t + 2π) = 2π f (t) a.e. and 0 f (t)p dt < ∞ the space of all X-valued 2π-periodic locally pintegrable functions on R, 1 ≤ p < ∞. Then Lp2π (X) is a Banach space for the
Fourier Series in Banach Spaces and Maximal Regularity norm
⎛ 1 f p := ⎝ 2π
2π
23
⎞1/p f (t)p dt⎠
.
0
If f ∈ Lp2π (X), then we denote by 1 fˆ(k) := 2π
2π
f (t)e−ikt dt
0
the kth Fourier coefficient of f , where k ∈ Z. The Fourier series of f +∞ ek ⊗ fˆ(k) k=−∞
converges in the sense of Ces`aro to f in Lp2π (X); i.e., σn − f p → 0 (n → ∞), m 1 n ikt ˆ where σn = n+1 (t ∈ R) and for m=0 k=−m ek ⊗ f (k). Here we let ek (t) = e ikt x ∈ X we define ek ⊗ x by (ek ⊗ x)(t) = e x (t ∈ R), where k ∈ Z. In particular, the space of all X-valued trigonometric polynomials n T (X) := ek ⊗ xk : n ∈ N, x−n , . . . , xn ∈ X k=−n
is dense in Lp2π (X) for all 1 ≤ p < ∞. Moreover, the Uniqueness Theorem holds: If f ∈ Lp2π (X) is such that fˆ(k) = 0 for all k ∈ Z, then f = 0 a.e. This allows us to define operator-valued multipliers in the following way. If Y is another Banach space, we denote by L(X, Y ) the space of all bounded linear operators from X to Y . When X = Y , we will simply denote it by L(X). Definition 1.1. Let X, Y be Banach spaces and let 1 ≤ p < ∞. A sequence (Mk )k∈Z ⊂ L(X, Y ) is an Lp -multiplier if for each f ∈ Lp2π (X), there exists g ∈ Lp2π (Y ) such that gˆ(k) = Mk fˆ(k) for all k ∈ Z. Now the Uniqueness Theorem guarantees that g is unique. It follows that the mapping f → g is linear. Thus, by the Closed Graph Theorem there exists a unique linear operator M ∈ L(Lp2π (X), Lp2π (Y )) such that (M f )∧ (k) = Mk fˆ(k) (k ∈ Z) (1.1) for all f ∈ Lp2π (X). The operators obtained in this way are exactly the translation invariant operators. To make this precise we consider the C0 -group U on Lp2π (X) given by (U(τ )f )(t) = f (t + τ ) (t ∈ R). The same group is also considered on Lp2π (Y ) without changing the name. Then an operator T ∈ L Lp2π (X), Lp2π (Y ) is called translation invariant, if U(t)T = T U(t) for all t ∈ R.
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W. Arendt and S. Bu
Proposition 1.2. Let T ∈ L Lp2π (X), Lp2π (Y ) , where X, Y are Banach spaces and 1 ≤ p < ∞. The following assertions are equivalent (i) T is translation invariant; (ii) there exist Mk ∈ L(X, Y ), such that (T f )∧ (k) = Mk fˆ(k), (k ∈ Z) for all f ∈ Lp2π (X). The following lemma is needed for the proof of Proposition 1.2. Lemma 1.3. Let g ∈ Lp2π (Y ), k ∈ Z be such that g(t + τ ) = eikτ g(t)
t-a.e.
for all τ ∈ R. Then there exists a unique y ∈ Y such that g = ek ⊗ y. 1,p (Y ) := u ∈ C(R; Y ) : Proof. Let B be the generator of U. Then D(B) = W2π u(t + 2π) = u(t) for all t ∈ R, and u ∈ Lp2π (Y ) and Bu = u for all u ∈ D(B). Here u is understood in the sense of distributions. Now the assumption on g says that U(τ )g = eikτ g for all τ ∈ R. Thus g ∈ D(B) and g = ikg. It follows that g is continuous, hence g ∈ C 1 (R; Y ). Consequently, g(t) = eikt g(0) for all t ∈ R. Proof of Proposition 1.2. (i)⇒(ii). Assume that T is translation invariant. Let k ∈ Z. For x ∈ X consider g := T (ek ⊗ x). Then U(τ )g = T U(τ )(ek ⊗ x) = eikτ T (ek ⊗ x) = eikτ g for all τ ∈ R. By Lemma 1.3 there exists a unique y ∈ Y such that T (ek ⊗ x) = ek ⊗ y.
(1.2)
We let Mk x := y. Then Mk : X → Y is linear and continuous. Moreover, by (1.2) one has (T f )∧ (k) = Mk fˆ(k) (1.3) for all f ∈ T (X). Since T (X) is dense in Lp2π (X) and T is continuous from Lp2π (X) to Lp2π (Y ), the identity (1.3) remains true for all f ∈ Lp2π (X). (ii)⇒(i). Let τ ∈ R and f ∈ Lp2π (X). Then for k ∈ Z one has ∧ U(τ )T f (k) = eikτ (T f )∧ (k) = eikτ Mk fˆ(k) ∧ = Mk eikτ fˆ(k) = Mk U(τ )f (k) ∧ = T U(τ )f (k). The Uniqueness Theorem implies that U(τ )T f = T U(τ )f .
Next we want to describe criteria which insure that a given sequence (Mk )k∈Z in L(X, Y ) is an Lp -multiplier. We say that a Banach space X is a Hilbert space, if there exists a scalar product · , · on X such that x, x 1/2 defines an equivalent norm on X. If p = 2, then even in the scalar case there are bounded sequences which are not Lp -multipliers. If both X, Y are Hilbert spaces, then each bounded
Fourier Series in Banach Spaces and Maximal Regularity
25
sequence (Mk )k∈Z is an L2 -multiplier. The following result shows that the converse remains true. Theorem 1.4. Let X, Y be Banach spaces. Assume that each bounded sequence (Mk )k∈Z ⊂ L(X, Y ) is an L2 -multiplier. Then both spaces X, Y are Hilbert spaces. Proof. It follows from the assumption that there exists C > 0 such that for every finite sequence (xk )k∈Z in X and Mk ∈ L(X, Y ) satisfying Mk ≤ 1, we have ek ⊗ Mk xk ≤ C ek ⊗ xk . (1.4) 2 2 L (0,2π;Y )
k∈Z
k∈Z
L (0,2π;X)
Let (xk )k∈Z be a finite sequence in X. There exist fk ∈ X such that fk (xk ) = xk and fk = 1. Let u ∈ X, u = 1 be fixed. Consider the linear operators Mk ∈ L(X, Y ) given by Mk (x) := fk (x)u (x ∈ X). Then Mk L(X,Y ) = 1. Moreover, Mk xk = fk (xk )u = xk u. It follows from (1.4) that 1/2 xk 2 ≤ C ek ⊗ xk . k∈Z
L2 (0,2π;X)
k∈Z
Thus X is of Fourier type 2. Hence X is a Hilbert space [Pie07, page 317]. Now we are going to show that Y is also a Hilbert space. For this we let (yk )k∈Z be a finite sequence in Y and let u ∈ X and f ∈ X be such that u = f = f (u) = 1. Consider the linear operators Nk ∈ L(X, Y ) given by Nk (x) := f (x)yk /yk (x ∈ X) when yk = 0, and Nk = 0 when yk = 0. Then Nk L(X,Y ) ≤ 1. It follows from (1.4) that ek ⊗ Nk xk ≤ C ek ⊗ xk 2 2 L (0,2π;Y )
k∈Z
k∈Z
L (0,2π;X)
for xk ∈ X. Taking xk = yk u. Then Nk xk = yk . It follows that 1/2 ek ⊗ yk ≤C yk 2 . 2 k∈Z
L (0,2π;Y )
k∈Z
We have shown that Y is of Fourier cotype 2. Thus Y is of Fourier type 2 and hence Y is a Hilbert space [Pie07, p. 316]. When X = Y , the preceding result can be improved. Instead of considering all bounded linear operator sequences, we only need to consider (λk I)k∈Z where (λk )k∈Z is a bounded scalar sequence. For the proof we need to introduce Rademacher functions. by rk the kth Rademacher function on [0, 1] We denote given by rk (t) = sgn sin(2k πt) , k = 1, 2, 3, . . . . Here we recall two fundamental properties of Rademacher functions which will be used in the proof of the next result. Let π : N → N be a bijection and let (xk )k≥1 be a finite sequence in X. Then it follows easily from the definition hat rk xk 2 = rπ(k) xk 2 . k
L (0,1;X)
k
L (0,1;X)
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W. Arendt and S. Bu
Kahane’s contraction principle states that for every finite sequence (xk )k≥1 in X and λk ∈ C with |λk | ≤ 1, λk rk xk 2 ≤ 2 rk xk 2 . L (0,1;X)
k
L (0,1;X)
k
(see, e.g., [LT79]). Let (Ω1 , Σ1 , μ1 ) and (Ω2 , Σ2 , μ2 ) be two measure spaces. We consider subsets J1 ⊂ L2 (Ω1 ), J2 ⊂ L2 (Ω2 ). Then by fk xk 2 gk xk 2 k
L (Ω1 ;X)
L (Ω2 ;X)
k
we mean that there exists a constant C > 0 depending only on X such that for every finite number of xk ∈ X, and every sequence (fk ) ⊂ J1 and (gk ) ⊂ J2 (satisfying possibly some further restriction to be made precise), one has 1 fk xk 2 ≤ gk xk 2 ≤ C fk xk 2 . C L (Ω1 ;X) L (Ω2 ;X) L (Ω1 ;X) k
k
k
Theorem 1.5. Let X be a Banach space. Assume that each bounded sequence (λk I)k∈Z defines an L2 -multiplier. Then X is a Hilbert space. Proof. We claim that for every finite number of trigonometric polynomials fk ∈ L2 (0, 2π) and every finite number of xk ∈ X, we have rk fk xk rmn,k fˆk (n)xk , (1.5) 2 2 L ([0,1]×[0,2π];X)
k
L ([0,1];X)
n,k
where mn1 ,k1 = mn2 ,k2 when (n1 , k1 ) = (n2 , k2 ). Indeed, it follows from the Kahane’s contraction principle that for every finite number of xk ∈ X, rk ek ⊗ xk rk xk . (1.6) 2 2 k
L ([0,1]×[0,2π];X)
k
L ([0,1];X)
On the other hand, since every bounded scalar sequence defines an L2 -multiplier by assumption, it follows that rk ek ⊗ xk 2 ek ⊗ xk 2 . (1.7) k
L ([0,1]×[0,2π];X)
k
L ([0,2π];X)
If f ∈ L2 (0, 2π), we let Δ(f ) := {n ∈ Z : fˆ(n) = 0} be the Fourier spectrum of f . Then there exist N1 , N2 , . . . ∈ N sufficiently large, so that {0} < Δ(eNk fk ) < Δ(eNl fl ) whenever k < l. Here by M1 < M2 we mean that k < l for every k ∈ M1 and l ∈ M2 , where M1 and M2 are subsets of N.
Fourier Series in Banach Spaces and Maximal Regularity
27
It follows from Kahane’s contraction principle, (1.6) and (1.7) that rk fk xk 2 L ([0,1]×[0,2π];X)
k
rk eNk en ⊗ fˆk (n) xk
L2 ([0,1]×[0,2π];X)
n
k
= rk eNk +n ⊗ fˆk (n)xk
L2 ([0,1]×[0,2π];X)
k,n
rmn,k rk eNk +n ⊗ fˆk (n)xk
L2 ([0,1]×[0,1]×[0,2π];X)
k,n
rmn,k eNk +n ⊗ fˆk (n)xk
L2 ([0,1]×[0,2π];X)
k,n
rmn,k fˆk (n)xk
L2 ([0,1];X)
k,n
.
We have shown that our claim (1.5) is true. Now under the assumption of the theorem, X is a UMD-space as the Hilbert transform is bounded on L2 (0, 2π; X). It follows that X has a non trivial cotype q < ∞. This implies that for every finite number of xk ∈ X, one has rk xk 2 gk xk 2 (1.8) L ([0,1];X)
k
L (Ω;X)
k
where the gk ’s are independent complex standard Gaussian variables on some probability space (Ω, Σ, p) [LeTa91, p. 253]. It follows from this and (1.5) that if fk ∈ L2 (0, 2π) are trigonometric polynomials, then gk fk xk 2 gn,k fˆk (n)xk 2 (1.9) L ([0,2π]×Ω;X)
k
L (Ω ;X)
n,k
where the gn,k ’s are independent complex standard Gaussian variables on some probability space (Ω , Σ , p ). We remark that (1.9) remains true for arbitrary fk ∈ L2 (0, 2π) by an approximation argument. Now assume that fk ∈ L2 (0, 2π) satisfy fk L2 (0,2π) = 1 and let hk := n gn,k fˆk (n). Then the hk ’s are also independent 2 1/2 ˆ = fk 2 = complex standard Gaussian variables on (Ω , Σ , p ) as n |fk (n)| 1. Thus we have by (1.8) gn,k fˆk (n)xk 2 = hk xk 2 rk xk 2 . n,k
L (Ω ;X)
k
On the other hand, if we let fk := and gk fk xk 2 k
L (Ω ;X)
k
L ([0,1];X)
√ 2N πχ[ k−1 , k ] for 1 ≤ k ≤ N . Then fk 2 = 1 N
L ([0,2π]×Ω;X)
=
N
1/2 xk 2 . k
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It follows from (1.9) that
xk 2
1/2
rk xk
k
L2 ([0,1];X)
k
.
It follows that X is of cotype 2 and type 2. Thus X is a Hilbert space by a result of S. Kwapien [Kw72]. Recall that a series
+∞
xk in a Banach space X is called unconditionally
k=−∞
convergent if
n
lim
n→∞
xπ(k)
k=−n
exists for each bijection π : Z → Z. This is equivalent to the fact that n
lim
n→∞
λk xk
k=−n
exists for each (λk )k∈Z ∈ ∞ (Z). Thus if the Fourier series +∞
ek ⊗ fˆ(k)
k=−∞
Lp2π (X),
converges unconditionally in then each bounded sequence (λk I)k∈Z with I the identity operator on X, is an Lp -multiplier. This implies that p = 2 (as a consequence of the scalar result mentioned above) and X is a Hilbert space by Theorem 1.5. Next we discuss convergence of the Fourier series. For f ∈ Lp2π (X) and n ∈ N, we let Sn (f ) := |k|≤n ek ⊗ fˆ(k) be the partial sum of the Fourier series; hence Sn ∈ L(Lp2π (X)) for n ∈ N. Theorem 1.6. Let X be a Banach space. The following conditions are equivalent. (i) X is a UMD-space; (ii) supn∈N Sn < ∞ for some 1 < p < ∞ (equivalently for all 1 < p < ∞); +∞ (iii) for each 1 < p < ∞ and each f ∈ Lp2π (X), the Fourier series ek ⊗ fˆ(k) converges to f in Lp2π (X);
k=−∞
(iv) there exists 1 < p < ∞ and for each f ∈ Lp2π (X), the Fourier series
+∞
ek ⊗
k=−∞
fˆ(k) converges to f in Lp2π (X); (v) the sequence (Mk )k∈Z given by Mk = I for k ≥ 0, Mk = −I for k < 0, is an Lp -multiplier for some 1 < p < ∞ (equivalently for all 1 < p < ∞). We refer to the literature (e.g., [Bur01]) for definition of UMD-spaces. Here we just mention that each Lp -space is a UMD-space whenever 1 < p < ∞. Moreover,
Fourier Series in Banach Spaces and Maximal Regularity
29
each UMD-space is reflexive and closed subspaces and quotient spaces of UMDspaces are UMD-spaces. The operator associated with the multiplier Mk = sgn(k)I on Lp2π (X) is called the Hilbert transform. Thus the Hilbert transform is bounded on Lp2π (X) if and only if 1 < p < ∞ and X is a UMD-space. For the operator-valued multiplier theorems we want to present here we need the notion of Rademacher type (or briefly type) and Rademacher cotype (or briefly cotype) of a Banach space. We refer to [LT79] [Pie07, p. 308] for the definitions and further properties of these notions. We just recall that every Lq -space with 1 ≤ q ≤ 2 is of type q and cotype 2. Every Lq -space with 2 ≤ q < ∞ is of cotype q and type 2. A Banach space X is of type 2 and of cotype 2 if and only if X is a Hilbert space [Kw72]. Next we formulate the variational form of the Marcinkiewicz multiplier theorem. Theorem 1.7. Let X, Y be UMD-spaces. Assume that X is of cotype 2 and Y is of type 2. Let (Mk )k∈Z ⊂ L(X, Y ) be a bounded sequence such that sup n∈N
n n−1 −1 −1 −2 2 M + − M k+1 Mk+1 − Mk < ∞. k
(1.10)
k=−2n
k=2n−1
Then (Mk )k∈Z is an Lp -multiplier for 1 < p < ∞. The hypothesis on X, Y are satisfied in particular when both spaces X and Y are Hilbert spaces. In that case Theorem 1.7 was proved by J. Schwartz [Sch61] in 1961. The scalar case is due to J. Marcinkiewicz and appeared in 1939 in Studia Mathematica. The more general case we present here can be obtained by an inspection of the proof of [AB02, Theorem 1.3] stopping on page 318 line 9 and [AB02, Proposition 1.13]. We mention a special case which turns out to the most suitable for generalizations Corollary 1.8 ([AB02, Theorem 1.3]). Let X, Y be UMD-spaces, X is of cotype 2 and Y is of type 2. Let (Mk )k∈Z ⊂ L(X, Y ) be a bounded sequence such that τ := sup |k|Mk+1 − Mk < ∞.
(1.11)
k∈Z
Then (Mk )k∈Z is an Lp -multiplier whenever 1 < p < ∞. Proof. For n ∈ N one has n n 2 −1 2 −1 1 2n − 2n−1 ≤τ = τ, Mk+1 − Mk ≤ τ k 2n−1 n−1 n−1 k=2
and similarly
k=2
n−1 −2 −1
Mk+1 − Mk ≤ τ.
k=−2n
Now the result follows from Theorem 1.7.
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Concerning operator-valued versions of the variational version of Marcinkiewicz Theorem (Corollary 1.8) on UMD-spaces instead of Hilbert spaces we refer to Z. Strkalj and L. Weis [SW07] (see also [CPSW00] for results on operator-valued Fourier multipliers with respect to more general Schauder decompositions).
2. The Marcinkiewicz multiplier theorem in the general case In order to describe the general periodic multiplier theorem we need the notion of R-boundedness. We recall that rk is the kth Rademacher function on [0, 1]. Definition 2.1. Let X, Y be Banach spaces. A set T ⊂ L(X, Y ) is called R-bounded if for all (equivalently for one) q ∈ [1, ∞) there exists a constant C > 0 such that n n rk Tk xk q ≤ C rk xk q k=1
L (0,1;Y )
k=1
L (0,1;X)
for all n ∈ N, x1 , . . . , xn ∈ X, T1 , . . . , Tn ∈ T . This notion of unconditional boundedness of a family of operators implies boundedness but is stronger in general. More precisely the following holds. Proposition 2.2 ([AB02, Proposition 1.13]). Let X, Y be Banach spaces. The following assertions are equivalent. (i) Each bounded T ⊂ L(X, Y ) is R-bounded; (ii) X is of cotype 2 and Y is of type 2. One immediate consequence of Proposition 2.2 is that each bounded subset of L(X) is R-bounded if and only if X is a Hilbert space. We note however that for an arbitrary Banach space a family T ⊂ L(X) of scalar operators is R-bounded if and only if it is bounded. Here we call T ∈ L(X) scalar, if it is of the form λI for some λ ∈ C. Now we can formulate the operator-valued Marcinkiewicz Theorem. Theorem 2.3 ([AB02, Theorem 1.3]). Let X, Y be UMD-spaces, 1 < p < ∞. Assume that (Mk )k∈Z ⊂ L(X, Y ) is such that both Mk : k ∈ Z , k(Mk+1 − Mk ) : k ∈ Z are R-bounded in L(X, Y ). Then (Mk )k∈Z is an Lp -multiplier. p It is known that when (Mk )k∈Z ⊂ L(X, Y ) is an L -multiplier, then the set Mk : k ∈ Z must be R-bounded [AB02, Proposition 1.11]. In general, it is not possible to replace the R-boundedness in Theorem 2.3 by norm boundedness. More precisely, the following is true.
Proposition 2.4 ([AB02, Proposition 1.17]). Let X, Y be Banach space and 1 ≤ p < Mk < ∞ ∞. Assume that every sequence (M ) ⊂ L(X, Y ) satisfying sup k k∈Z k∈Z p and supk∈Z k(Mk+1 − Mk ) < ∞, is an L -multiplier. Then X is of cotype 2 and Y is of type 2. In particular, when X = Y , then X is a Hilbert space.
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31
3. The periodic non-homogeneous problems We are now going to show how the multiplier theorem can be applied. Let X be a UMD-space and let D be a Banach space which is continuously embedded into X; we write D → X for short. Let A ∈ L(D, X), 1 < p < ∞. We consider the following problem. Given f ∈ Lp (0, 2π; X) we want to find a solution u of the problem ⎧ ⎨ u ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) u(t) ˙ = Au(t) + f (t) a.e. Pper ⎩ u(0) = u(2π). Here W 1,p (0, 2π; X) consists of those continuous functions u : [0, 2π] → X for which there exists u ∈ Lp (0, 2π; X) such that t u (s)ds (t ∈ [0, 2π]). u(t) = u(0) + 0
Equivalently, W (0, 2π; X) consists of those functions u ∈ Lp (0, 2π; X) for which u ∈ Lp (0, 2π; X), where u is defined in the sense of distributions. We say that problem Pper is well posed if for each f ∈ Lp (0, 2π; X), there exists a unique solution u of Pper . The following result characterizes well-posedness of the problem Pper . 1,p
Theorem 3.1. Let 1 < p < ∞. The following assertions are equivalent. (i) For each f ∈ Lp (0, 2π; X) there exists a unique solution of Pper ; (ii) for each k ∈ Z the operator ik − A ∈ L(D, X) is invertible and the family (ik − A)−1 : k ∈ Z is R-bounded in L(X, D). Theorem 3.1 is a consequence of the multiplier theorem. It is similar to [AB02, Theorem 2.3], where a stronger hypothesis on A is imposed, namely that A is closed as an unbounded operator on X. This means by definition that the graph of A G(A) := (x, Ax) : x ∈ D is closed in X × X. Condition (ii) does imply closedness of A. In fact, taking k = 0, (ii) implies that A is invertible. Now let (x, y) ∈ G(A). Then there exist xn ∈ D such that xn → x and Axn → y in X. It follows that xn = A−1 (Axn ) → A−1 y in D. Hence x = A−1 y ∈ D and Ax = y. We now give a proof of Theorem 3.1. Proof of Theorem 3.1. (ii)⇒(i). Assume condition (ii). Then A is invertible. It follows that the graph norm xA := Ax
(x ∈ D)
defines an equivalent norm on D. In fact, since A is an isometric isomorphism from (D, · A ) to X, it follows that (D, · A ) is complete. Since xA ≤ AxD , it follows from the Open Mapping Theorem that both norms on D are equivalent.
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Thus A is an isomorphism from D to X. It follows from (ii) that the family A(ik − A)−1 : k ∈ Z is R-bounded in L(X). Since ik(ik − A)−1 − A(ik − A)−1 = IX , we conclude that the family
(3.1)
k(ik − A)−1 : k ∈ Z
is R-bounded in L(X). Now [AB02, Theorem 2.3] implies that Pper is well posed. (i)⇒(ii). Assume that Pper is well posed. a) We claim that A is bijective. Let y ∈ X be given. Then for f (t) ≡ −y, there exists u ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) such that u(t) ˙ = Au(t) − y a.e. and 2π 1 u(0) = u(2π). Then x := 2π u(t)dt ∈ D and 0
2π 2π 1 1 Au(t)dt = u(t)dt ˙ − f (t)dt 2π 2π 0 0 0 1 u(2π) − u(0) + y = y. = 2π We have shown that A is surjective. Injectivity can be seen as follows. Let x ∈ D be such that Ax = 0. Then u(t) := x satisfies 0 = u(t) ˙ = Au(t) + 0. Hence u is a solution of Pper for f ≡ 0. Since also v ≡ 0 is a solution, it follows that x ≡ u ≡ 0. b) It follows from a) that A is invertible and hence that A is closed. Now [AB02, Theorem 2.3] implies that ik− A is bijective and the set k(ik − A)−1 : k ∈ Z is R-bounded in L(X), and so (ik − A)−1 : k ∈ Z is R-bounded in L(X, D) since A−1 : X → D is an isomorphism. Two consequences of Theorem 3.1 are remarkable. First of all it follows that well-posedness of Pper is independent of p ∈ (1, ∞). Secondly, D is dense in X. This follows from condition (ii) of Theorem 3.1, [ABHN01, Proposition 3.3.8] and the fact that X is reflexive. 1 Ax = 2π
2π
4. Maximal regularity Instead of Dirichlet boundary conditions we consider now the initial value problem. Again X is a UMD-space and D is a Banach space such that D → X. Let A ∈ L(D, X) be an operator, 1 < p < ∞ and τ > 0. Given f ∈ Lp (0, τ ; X) we want to find a solution of the problem ⎧ ⎨ u ∈ W 1,p (0, τ ; X) ∩ Lp (0, τ ; D) u(t) ˙ = Au(t) + f (t) a.e. P0 (τ, p) ⎩ u(0) = 0 . We say that problem P0 (τ, p) is well posed if for every f ∈ Lp (0, τ ; X), there exists a unique solution u of P0 (τ, p). This can be characterized as follows.
Fourier Series in Banach Spaces and Maximal Regularity
33
Theorem 4.1. Let 1 < p < ∞ and let τ > 0 be fixed. The following assertions concerning the operator A are equivalent. (i) Problem P0 (τ, p) is well posed; (ii) there exists ω ∈ R such that (λ − A) is invertible whenever Re λ > ω and the set {(λ − A)−1 : Re λ > ω} is R-bounded in L(X, D). If these equivalent conditions are satisfied, then D is dense in X and A (with domain D) generates a holomorphic C0 -semigroup on X. Implication (ii)⇒(i) is due to L. Weis [W01] who uses an operator-valued multiplier theorem on Lp (R, X). Here we will deduce this implication from the periodic multiplier theorem, Theorem 3.1. The proof of the implication (i)⇒(ii) is given in two steps. At first we show that A generates a holomorphic C0 -semigroup. This result is due to G. Dore [Do93, Theorem 2.2], who states it under the additional hypothesis that A is closed and D is dense in X. However, the proof sketched in [Do93] can be carried over to the more general situation. We give all the details to convince the reader. Once it is known that A generates a holomorphic C0 -semigroup one may again use Theorem 3.1 (or use [CP00]). Proof of Theorem 4.1. (i)⇒(ii). By the Closed Graph Theorem there exists a constant c1 ≥ 0 such that uW 1,p (0,τ ;X) ≤ c1 f Lp(0,τ ;X)
(4.1)
for every f ∈ Lp (0, τ ; X), where u denotes the unique solution of P0 (τ, p). a) First we show that there exists ω1 ∈ R such that λ − A is injective whenever Re λ ≥ ω1 . In fact, let λ ∈ C, x ∈ D, such that λx − Ax = 0. Then u(t) = 1 λt − 1)x is the unique solution of P0 (τ, p) for f ≡ x. Hence uW 1,p (0,τ ;X) ≤ λ (e c1 f Lp(0,τ ;X) = c1 τ 1/p x. Hence by (4.1) x
1 p 1 Re λpτ e −1 = u ˙ Lp (0,τ ;X) pRe λ ≤ c1 f Lp(0,τ ;X) = c1 τ 1/p x.
Thus if we choose ω1 > 0 so large that 1 Re λpτ e − 1 > cp1 τ, pRe λ whenever Re λ ≥ ω1 , then x = 0. b) Since W 1,p (0, τ ; X) → C([0, τ ]; X), there exists c2 ≥ 0 such that u(τ )X ≤ c2 uW 1,p (0,τ ;X) for all u ∈ W 1,p (0, τ ; X). Let fλ (t) :=
eλt 0
if t ≤ Re1 λ if Re1 λ < t ≤ τ
(4.2)
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W. Arendt and S. Bu
where Re λ > max
1
τ , ω1
=: ω2 . Then for x ∈ X one has
fλ ⊗ xLp (0,τ ;X) =
c3 x 1
,
(Re λ) p
p1 − 1) . Let u be the solution of P0 (τ, p) for the inhomogeneity τ fλ ⊗ x. Define R(λ)x := Re λ e−λt u(t) dt ∈ D. Then where c3 =
1 p p (e
0
τ (λ − A)R(λ)x
=
=
−Re λ
(e
−λt
τ
) u(t) dt − Re λ
0
0
τ
τ
−Re λ
(e−λt ) u(t) dt − Re λ
0
e−λt Au(t) dt
e−λt u (t) dt
0
1 Re λ
+Re λ =
−Re λe
e−λt fλ (t) dt x
0 −λτ
u(τ ) + x.
Define Sx := Re λe−λτ u(τ ). Then S : X → X is linear and by (4.1) and (4.2) Sx
≤ c2 Re λe−Re λτ uW 1,p (0,τ ;X) ≤ c1 c2 Re λe−Re λτ fλ Lp (0,τ ;X) x 1
= c1 c2 c3 (Re λ) p e−Re λτ x, where p1 + p1 = 1. Thus there exists ω ≥ ω2 such that SL(X) ≤ Re λ ≥ ω. Let Re λ ≥ ω. Since (λ − A)R(λ)x = x − Sx
1 2
whenever (4.3)
for all x ∈ X, and since I −S is surjective, it follows that (λ−A) is surjective. Hence by a) the operator (λ − A) : D → X is bijective. By using a similar argument used before the proof of Theorem 3.1, this implies already that the unbounded operator λ − A on X with domain D is closed. Thus the unbounded operator A on X with domain D is also closed. Moreover, by (4.3) (λ − A)−1 x = R(λ)x + (λ − A)−1 Sx . Hence 1 (λ − A)−1 L(X) ≤ R(λ)L(X) + (λ − A)−1 L(X) . 2 Consequently (λ − A)−1 L(X) ≤ 2R(λ)L(X) .
(4.4)
Fourier Series in Banach Spaces and Maximal Regularity
35
Let x ∈ X, then for u as above, R(λ)x
=
Re λ λ
τ 0
=
Re λ λ
−(e−λt ) u(t) dt
τ
e−λt u (t) dt − e−λτ u(τ ) .
0
Hence by (4.1) and (4.2) 1 τ p e−Re λp t dt u Lp (0,τ ;X) + c2 e−Re λτ uW 1,p (0,τ ;X) λR(λ)x ≤ Re λ 0
1/p 1 −Re λτ c1 xfλ Lp (0,τ ;X) + c2 e p Re λ 1/p 1 1/p −Re λτ x ≤ c1 c 3 + c2 (Re λ) e p ≤ c4 x
≤ Re λ
for some constant c4 > 0 whenever Re λ ≥ ω. Thus it follows from (4.4) that sup λ(λ − A)−1 L(X) < ∞ .
Re λ>ω
By [ABHN01, Proposition 3.3.8] this implies that D is dense in X (since X is reflexive) and that A generates a holomorphic C0 -semigroup (by [ABHN01, Corollary 3.7.17]). Now assertion (ii) can be deduced from Theorem 3.1 as in [AB02, Corollary 5.2 and the following lines]. Alternatively one can use [CP00]. (ii)⇒(i). Let 1 < p < ∞ be fixed and let τ > 0. We have to show that problem P0 (τ, p) is well posed. The assumption (ii) implies that A generates a holomorphic C0 -semigroup T (keep in mind that X is reflexive, so the domain D of A is dense in X by [ABHN01, Proposition 3.3.8]). This shows in particular that P0 (τ, p) has at most one solution. In fact, if u ∈ W 1,p (0, τ ; X)∩Lp (0, τ ; D) such that u(t) ˙ = Au(t) t and u(0) = 0, consider v(t) = u(s) ds. Then v ∈ C 1 ([0, τ ]; X) ∩ C([0, τ ]; D) is 0
a classical solution of v(t) ˙ = Av(t) and v(0) = 0. Hence v ≡ 0 by [ABHN01, Theorem 3.1.12] and so u ≡ 0. Thus it remains to show existence which we do now. 1st case: ω < 0, τ = 2π. If the constant ω of condition (ii) is negative, then we can apply Theorem 3.1. Thus, given f ∈ Lp (0, 2π; X) there exists w ∈ W 1,p (0, 2π; X)∩ Lp (0, 2π; D) satisfying w(t) ˙ = Aw(t) + f (t) a.e. Let v(t) := T (t)w(0). Then by [Lun95, 1.2.2 and 2.2.1], one has v ∈ W 1,p (0, 2π; X)∩Lp (0, 2π; D) and v(t) ˙ = Av(t) a.e., v(0) = w(0). Thus u := w − v is a solution of P0 (2π, p). 2nd case: ω < 0, τ > 0 arbitrary. Let f ∈ Lp (0, τ ; X). Define g(t) = rf (rt) where τ r := 2π . The operator rA satisfies condition (ii) as well. Then g ∈ Lp (0, 2π; X).
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W. Arendt and S. Bu
˙ = By the first case there exists v ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) such that v(t) 1,p rAv(t) + g(t) a.e. and v(0) = 0. Let u(t) := v(t/r). Then u ∈ W (0, τ ; X) ∩ Lp (0, τ ; D), u(0) = 0 and u(t) ˙ = 1r v˙ rt = Av rt + 1r g rt = Au(t) + f (t) a.e. Thus u is a solution of P0 (τ, p). 3rd case: The constant ω ∈ R and τ > 0 are arbitrary. Let ω1 > ω. Then the operator A−ω1 satisfies the assumptions of the 2nd case. Let f ∈ Lp (0, τ ; X). Then there exists v ∈ W 1,p (0, τ ; X)∩Lp (0, τ ; D) satisfying v(t) ˙ = Av(t)− w1 v(t)+ g(t), v(0) = 0 where g(t) = e−ω1 t f (t). Then u(t) = eω1 t v(t) is a solution of P0 (τ, p). An immediate consequence of Theorem 4.1 is the following. Corollary 4.2. If P0 (τ, p) is well posed for some τ > 0, 1 < p < ∞, then it is well posed for all τ > 0, 1 < p < ∞. Remark 4.3. It can be seen from the proof of Theorem 4.1 that the implication (i)⇒(ii) is always true, without any assumption on the Banach space X. We know that (i) does not imply that D is dense in X, in general. If X is reflexive, though, then (ii) implies density of D.
5. The non-autonomous equations Let X be a UMD-space and D a Banach space such that D → X. Given an operator A ∈ L(D, X) we say that A satisfies maximal regularity if condition (i) of Theorem 4.1 is satisfied. We know that this is independent of the choice of 1 < p < ∞ and τ > 0 (Corollary 4.2). Given 1 < p < ∞, τ > 0 we define the maximal regularity space M Rp (0, τ ) := W 1,p (0, τ ; X) ∩ Lp (0, τ ; D) which is a Banach space for the sum norm uMR := uW 1,p (0,τ ;X) + uLp(0,τ ;D) . By T rp := u(0) : u ∈ M Rp (0, τ ) we define the trace space. Then the following result on well-posedness can be obtained by a simple perturbation argument (see [Am04] or [ACFP07]). Theorem 5.1. Let A : [0, τ ] → L(D, X) be continuous and assume that A(t) satisfies maximal regularity for each t ∈ [0, τ ]. Then for each f ∈ Lp (0, τ ; X) and each initial value u0 ∈ T rp , there exists a unique u ∈ M Rp (0, τ ) satisfying u(t) ˙ = A(t)u(t) + f (t) a.e. u(0) = u0 . The non-autonomous problem with periodic boundary conditions is not as simple. Let A : [0, 2π] → L(D, X) be continuous such that A(0) = A(2π). Furthermore we want to assume that the injection D → X is compact. Then there are two type of results. In the first one we assume the same condition on A(t) as in Theorem 5.1 for Dirichlet boundary conditions. We define for 1 < p < ∞ the
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37
periodic maximal regularity space M Rper,p := u ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) : u(0) = u(2π) which is a Banach space for the natural norm uMR := uW 1,p (0,2π;X) + uLp(0,2π;D) . Theorem 5.2 ([AR09, Corollary 9.3]). Assume that (a) A(t) + is dissipative for all t ∈ [0, 2π] and some > 0 and that (b) A(t) satisfies maximal regularity (as defined above) for all t ∈ [0, 2π]. Then for 1 < p < ∞ and each f ∈ Lp (0, 2π; X), there exists a unique solution of u ∈ M Rper,p Pper u(t) ˙ = A(t)u(t) + f (t) a.e. If Pper is well posed for A then it is so for −A as well. But conditions (a) and (b) are not invariant by this inversion. This shows that conditions (a) and (b) are too strong in general. In the autonomous case, condition (ii) of Theorem 3.1 is equivalent to well-posedness of Pper . So also in the non-autonomous case, it is natural to assume that each A(t) satisfies this condition. However, this condition alone is too weak to deduce that Pper is well posed in this case, and in the finitedimensional case Floquet Theory is available. Still some significant results on the solutions of problem Pper are valid. In order to formulate them, we consider the operator DA : M Rper,p → Lp (0, 2π; X) given by DA u := u˙ − A(·)u(·). This is a bounded linear operator between the two Banach spaces M Rper,p and Lp (0, 2π; X). Note that M Rper,p is continuously (and compactly) embedded in Lp (0, 2π; X). So we may consider DA as an unbounded operator on the Banach space Lp (0, 2π; X). If each A(t) satisfies condition (ii) of Theorem 3.1, then this operator is indeed closed (as unbounded operator on Lp (0, 2π; X)). This is not obvious but a particular result of maximal regularity. Some more can be said. The operator DA is a Fredholm operator, that is, DA has finite-dimensional kernel and closed image R(DA ) with finite codimension in Lp (0, 2π; X) for each 1 < p < ∞. To say that Pper is well posed means that DA is invertible. This is stronger than Fredholm, but knowing the Fredholm property helps a lot to prove invertibility (under further assumptions as in Theorem 5.2 for example). We collect the information given here in the following concluding theorem. Theorem 5.3 ([ AR09, Theorem 4.1 and Corollary 3.7]). Assume that A(t) satisfies condition (ii) of Theorem 3.1 for each t ∈ [0, 2π]. Let 1 < p < ∞. Then the following holds. (a) DA is closed seen as unbounded operator on Lp (0, 2π; X); (b) DA is a Fredholm operator; (c) if σ(DA ) = C, then DA has compact resolvent and in particular Fredholm’s alternative holds: Pper is well posed if and only if DA is injective.
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W. Arendt and S. Bu
Acknowledgement We are most grateful to Quanhua Xu (Besan¸con) for some helpful comments in the context of Theorem 1.5.
References [Am04]
H. Amann: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4 (2004), 417–430. [ABHN01] W. Arendt, C. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Birkh¨ auser, Basel, 2001. [ACFP07] W. Arendt, R. Chill, S. Fornaro, C. Poupaud: Lp -maximal regularity for nonautonomous evolution equations. J. Diff. Equ. 237 (2007), 1–26. [AB02] W. Arendt, S. Bu: The operator-valued Marcinkiewicz multiplier theorems and maximal regularity. Math. Z. 240 (2002), 311–343. [AR09] W. Arendt, P. Rabier: Linear evolution operators on spaces of periodic functions. Comm. Pure and Applied Analysis 8 (2009), 5–36. [Bur01] D. Burkholder: Martingales and singular integrals in Banach spaces. In: Handbook of the Geometry of Banach Spaces, Vol. I (W.B. Johnson and J. Lindenstrauss Eds.), Elsevier, 2001, 233–269. [CPSW00] Ph. Cl´ement, B. de Pagter, F. A. Sukochev, H. Witvliet: Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), 135–163. [CP00] Ph. Cl´ement, J. Pr¨ uss: An operator-valued transference principle and maximal regularity on vector-valued Lp -spaces. In: Evolution Equations and Their Applications in Physical and Life Sciences (G. Lumer and L. Weis Eds.), Marcel Dekker, 2000, 67–78. [Do93] G. Dore: Lp -regularity of abstract differential equations. In: Functional Analysis and Related Topics (H. Komatsu Ed.), Springer LNM 1540 (1993), 25–38. [KW04] P. Kunstmann, L. Weis: Maximal Lp -regularity for parabolic equations, Fourier multipliers theorems and H ∞ -functional calculus. In: Functional Analytic Methods for Evolution Equations. Springer LNM 1855, 2004, 65–311. [Kw72] S. Kwapien: Isomorphic characterization of inner product spaces by orthogonal series with vector-valued coefficients. Studia Math. 44 (1972), 583–595 [LeTa91] M. Ledoux, M. Talagrand: Probability in Banach Spaces, Isoperimetry and Processes. Springer-Verlag, Berlin, 1991. [LT79] J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces II, Springer, Berlin, 1979. [Lun95] A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkh¨ auser, Basel, 1995. [Pie07] A. Pietsch: History of Banach Spaces and Linear Operators. Birkh¨ auser, Basel, 2007. [Sch61] J. Schwartz: A remark on inequalities of Calderon-Zygmund type for vectorvalued functions. Comm. Pure Appl. Math. 14 (1961), 785–799. [SW07] Z. Strkalj, L. Weis: On operator-valued Fourier multiplier theorem. Trans. Amer. Math. Soc. 359 (2007), 3529–3547.
Fourier Series in Banach Spaces and Maximal Regularity [W01]
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L. Weis: Operator-valued Fourier multiplier theorems and maximal Lp regularity. Math. Ann. 319 (2001), 735–758.
Wolfgang Arendt Institute of Applied Analysis University of Ulm D–89069 Ulm, Germany e-mail: [email protected] Shangquan Bu Department of Mathematical Science University of Tsinghua Beijing 100084, China e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 41–49 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Spectral Measures on Compacts of Characters of a Semigroup Dragu Atanasiu Abstract. In this note we give integral representations for some ∗-representations of the type U : S → B(H) where S is a commutative semigroup with involution and neutral element and B(H) are the bounded operators of the Hilbert space H. Mathematics Subject Classification (2000). Primary 47B15; Secondary 43A35. Keywords. Radon spectral measure, positive definite function.
1. Introduction In this paper we give, in Section 2, an integral representation for positive definite functions, defined on a commutative semigroup with involution and neutral element, which includes the Berg-Maserick theorem [3, p. 169, Theorem 2.1] (see also [2, p. 93, Theorem 2.5]) and the extension of this theorem from [5, p. 96, Corollary 1]. The integral representation proved in Section 2 is similar to the representation given in [1, p. 96, Theorem 2.5] but the proof is shorter. In [6, p. 2950, Theorem 2] it is shown, independent of the Banach algebra theory, that every ∗-representation U : S → B(H), where S is a commutative semigroup with involution and neutral element and B(H) are the bounded operators of the Hilbert space H has an integral representation with respect to a unique selfadjoint Radon spectral measure defined on the Borel sets of the space of characters of S with the pointwise convergence topology. This integral representation is used to give a new proof of the Gelfand-Naimark theorem for abelian C∗ -algebras. In Section 3 we obtain, using the result proved in Section 2, an integral representation which extends [6, p. 2950, Theorem 2]. In Section 4 are obtained conditions such that the support of the spectral measure considered in Section 3 is contained in sets which generalize the ball, the torus and the rectangle.
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In Section 5 we give a construction of the spectral measure from Section 3. This construction follows the way of the proof in [4, p. 435, Section 3]. In the last section we use the construction of the spectral measure from Section 5 to prove the Gelfand-Naimark theorem for abelian C∗ -algebras.
2. A Berg-Maserick type theorem 2.1. Definitions and notations In this paper (S, ·) is a commutative semigroup with involution and neutral element e (see [2, p. 86]). A function ϕ : S → C is positive definite if for every natural number n ≥ 1 and every choice of elements s1 , . . . , sn of S and complex numbers c1 , . . . , cn we have n cj ck ϕ(sj s∗k ) ≥ 0. j,k=1
A function v : S → [0, ∞) is an absolute value on S if v(e) = 1,v(s∗ ) = v(s) for all s ∈ S and v(st) ≤ v(s)v(t) for all s, t ∈ S. A function ϕ : S → C is v-bounded if for every s ∈ S we have |ϕ(s)| ≤ Cϕ v(s) where Cϕ is a positive real number. In [2, p. 90, Proposition 1.12] it is shown that we can always take Cϕ = ϕ(e) if the function ϕ is positive definite. Let Γ be a set. Let (aγ,t )γ∈Γ,t∈S be a family of complex numbers such that for every γ ∈ Γ we have aγ,t = 0 only for a finite number of t. For a function ϕ : S → C and γ ∈ Γ we denote by ϕγ : S → C the function defined by ϕγ (s) = aγ,t ϕ(ts), s ∈ S. t∈S
Let v be an absolute value on S. We denote by P the set {ϕ : S → C|ϕ and (ϕγ )γ∈Γ are positive definite ; ϕ v-bounded}. We also denote by C = {ρ : S → C|ρ(e) = 1, ρ(s∗ ) = ρ(s), ρ(st) = ρ(s)ρ(t), s, t ∈ S} the set of characters of S. We equip C with the topology of pointwise convergence. We consider the following subsets of C: V = {ρ ∈ C||ρ(s)| ≤ v(s), s ∈ S} and M = {ρ ∈ V|ργ (e) ≥ 0, ∀γ ∈ Γ}.
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2.2. The Berg-Maserick type theorem Theorem 2.1. For a function ϕ : S → C the following conditions are equivalent a) ϕ ∈ P; b) there is a unique positive Radon measure on M such that ρ(s)dμ(ρ). ϕ(s) = M
Proof. We only have to prove a ⇒ b. The uniqueness of the measure μ can be proved as in [2, p. 95]. We prove the existence of the measure μ. According to the Berg-Maserick theorem [2, p. 93, Theorem 2.5] there is a positive Radon measure ν on V such that ϕ(s) =
ρ(s)dν(ρ). V
For every γ ∈ Γ we obtain n
cj ck ϕγ (sj s∗k ) =
j,k=1
V t∈S
aγ,t ρ(t)|
n
cj ρ(sj )|2 ≥ 0.
j=1
Because for every continuous function f : V → C such that f ≥ 0 the function can be uniformly approximated on V by functions of the type n
√ f
cj ρ(sj )
j=1
this means that we have
V t∈S
aγ,t ρ(t)f (ρ)dν(ρ) ≥ 0
for every continuous function f : V → C such that f ≥ 0. Consequently for every γ ∈ Γ the measure aγ,t ρ(t)) · ν (ρ → t∈S
is positive which means that ν{ρ ∈ V|
aγ,t ρ(t) < 0} = 0.
t∈S
This finishes the proof.
Remark 2.2. If for some γ0 ∈ Γ we have ϕγ0 = 0 where ϕ ∈ P has the representing measure μ, then μ({ρ|ργ0 (e) = 0}) = 0.
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Corollary 2.3. For every ϕ ∈ P the following inequality holds |
n
cj ϕ(sj )| ≤ ϕ(e) sup |
n
cj ρ(sj )|
ρ∈M j=1
j=1
for every natural number n ≥ 1 and every choice of elements s1 , . . . , sn of S and of complex numbers c1 , . . . , cn . Proof. This corollary is an immediate consequence of Theorem 2.1.
3. An integral representation via spectral measures In this section H is a Hilbert space and B(H) is the space of bounded operators on H. A function U : S → B(H) is a ∗-representation if we have U(e) = I; U(s∗ ) = U(s)∗ , s ∈ S; U(st) = U(s)U(t), s, t ∈ S. With the theorem from the preceding section we can obtain as in [6, p. 2951] the following result Theorem 3.1. Let U : S → B(H) be a ∗-representation. We suppose that for every γ ∈ Γ the operator aγ,t U(t) t∈S
is positive. There exists a unique selfadjoint Radon spectral measure E : Bor(M) → B(H), where Bor(M) are the Borel sets of M, such that ρ(s)dE(ρ), s ∈ S, U(s) = M
where M = {ρ ∈ C|ργ (e) ≥ 0, ∀γ ∈ Γ; |ρ(s)| ≤ U(s) , s ∈ S}. Corollary 3.2. Suppose that (S, ·, +) is a commutative algebra equipped with a involution and neutral element e. Let U : S → B(H) be a linear ∗-representation. There exists a unique selfadjoint Radon spectral measure E : Bor(M) → B(H) such that U(s) = ρ(s)dE(ρ), s ∈ S M
where M is the set {ρ : S → C|ρ linear, ρ(e) = 1,ρ(s∗ ) = ρ(s),ρ(st) = ρ(s)ρ(t),|ρ(s)| ≤ U(s),s,t ∈ S}
Spectral Measures on Compacts of Characters of a Semigroup
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Proof. The equalities U((at1 + bt2 )s) − aU(t1 s) − bU(t2 s) = 0, a, b ∈ C; s, t1 , t2 ∈ S, and the way of proof in [6, p. 2951] imply that μ-almost every ρ ∈ M is linear (see Remark 2.2). Remark 3.3. We note that [6, p. 2949, Theorem 1] is a consequence of Corollary 3.2.
4. Examples of ∗-representations Definition 4.1. We say that a ∗-representation U : S → B(H) has an integral ∗-representation on a compact set M ⊂ C if there is a selfadjoint spectral Radon measure (necessarily unique) E : Bor(M) → B(H) such that U(s) = ρ(s)dμ(ρ), s ∈ S. M
A set G ⊂ S is a generator set if each element of S is a product of elements from G ∪ {g ∗ |g ∈ G}. Proposition 4.2. Let H be a Hilbert space. For every s ∈ S let U(s) : H → H be a linear application such that U(st) = U(s)U(t), U(s)x, y = x, U(s∗ )y , U(e) = I, s, t ∈ S, x, y ∈ H. If we suppose that G is a generator set of the semigroup S and that
(Mg2 I − U(gg ∗ ))x, x ≥ 0 , g ∈ G, x ∈ H, where Mg ≥ 0, the operator U(s) is continuous for every s ∈ S, and U has an integral ∗-representation on the compact M = {ρ ∈ C||ρ(g)| ≤ Mg , g ∈ G}. Proof. We have
U(g)x, U(g)x = U(gg ∗ )x, x ≤ Mg2 x, x g ∈ G, x ∈ H, and consequently the operator U(g) is continuous for every g ∈ G and we get U(g) ≤ Mg . From the fact that G is a generator set it results that for every s ∈ S the linear application U(s) is bounded. Now we can apply Theorem 3.1 to finish the proof. Corollary 4.3. Let H be a Hilbert space.For every s ∈ S let U(s) : H → H be a linear application such that U(st) = U(s)U(t), U(s)x, y = x, U(s∗ )y , U(e) = I, s, t ∈ S, x, y ∈ H. If we suppose that G is a generator set of the semigroup S and that
(Mg2 I − U(gg ∗ ))x, x = 0 , g ∈ G, x ∈ H,
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D. Atanasiu
the operator U(s) is continuous for every s ∈ S and U has an integral ∗-representation on the compact M = {ρ ∈ C||ρ(g)| = Mg , g ∈ G}. Proof. This result is a consequence of the preceding proposition and of Remark 2.2 Proposition 4.4. Let H be a Hilbert space. For every s ∈ S let U(s) : H → H be a linear application such that U(st) = U(s)U(t), U(s)x, y = x, U(s∗ )y , U(e) = I, s, t ∈ S, x, y ∈ H. If we suppose that G is a generator set of the semigroup S and that 1 1 Mg,τ I − U(g) − U(g ∗ ) x, x ≥ 0 , g ∈ G, τ ∈ {±1, ±i}, x ∈ H, 2τ 2τ where (Mg,τ )g∈G,τ ∈{±1,±i} is a family of real numbers such that Mg,τ > 0 τ ∈{±1,±i}
and −Mg,−1 ≤ Mg,1 , −Mg,−i ≤ Mg,i , g ∈ G, the operator U(s) is continuous for every s ∈ S and U has an integral ∗-representation on the compact M = {ρ ∈ C| − Mg,−1 ≤ Re ρ(g) ≤ Mg,1 , −Mg,−i ≤ Im ρ(g) ≤ Mg,i , g ∈ G}. Proof. From the equality 1 1 U(g) − U(g ∗ ) = Mg,τ I − 2τ 2τ τ ∈{±1,±i}
we obtain that
Mg,τ I
τ ∈{±1,±i}
1 1 ∗ U(g) − U(g ) x, x 0≤ Mg,τ I − 2τ 2τ " ! Mg,τ x, x , g ∈ G, x ∈ H, ≤ τ ∈{±1,±i}
which means that all the operators 1 1 Mg,τ I − U(g) − U(g ∗ ) , g ∈ G, 2τ 2τ are continuous. Now using the relations ⎛ ⎞ 1 1 U(g) − U(g ∗ ) , g ∈ G, 2U(g) = ⎝ τ Mg,τ ⎠ I − τ Mg,τ I − 2τ 2τ τ ∈{±1,±i}
τ ∈{±1,±i}
Spectral Measures on Compacts of Characters of a Semigroup
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and the fact that G is a generator set it results that for every s ∈ S the linear application U(s) is bounded. We finish the proof using Theorem 3.1.
5. A construction of the spectral measure From the following result we can obtain the selfadjoint Radon spectral measure from Theorem 3.1. Theorem 5.1. Let U : S → B(H) be a ∗-representation as in Theorem 3.1. Defining M as in Theorem 3.1 there is a ∗-representation μ : C(M) → B(H) where C(M) is the algebra {f : M → C|f
continuous}
with the involution f ∗ (ρ) = f (ρ), such that μ(f ) ≤ sup |f (ρ)|, μ(ρ → ρ(s)) = U(s)
and
ρ∈M
U(s) = sup |ρ(s)|. ρ∈M
Proof. Let U : S → B(H) be a ∗-representation as in Theorem 3.1. For every x ∈ H the function s → U(s)x, x is in P (see Section 2) if we take v(s) = U(s), s ∈ S. According to Corollary 2.3 we have # # # !⎛ ⎞ "## # # n # n # # # # # # # ⎝ # ⎠ c U(s ) x, x ≤
x, x sup c ρ(s ) j j j j # # # # ρ∈M # j=1 # # # j=1 It results because (U(s))s∈S are normal operators that # # # # n # # n # cj U(sj ) ≤ sup # cj ρ(sj )## j=1 ρ∈M # j=1 # and consequently the function μ(ρ →
n
cj ρ(sj )) =
j=1
n
cj U(sj )
j=1
is well defined. If we denote by A the algebra of functions M → C of the form ρ →
n j=1
cj ρ(sj )
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D. Atanasiu
it is immediate to verify that we have μ(f + g) = μ(f ) + μ(g),
μ(f ) = (μ(f ))∗ , f, g ∈ A.
μ(f g) = μ(f )μ(g) and μ(1) = I,
Now we construct our representation using the fact that according to the Stone-Weierstrass theorem the set A is dense in C(M) (see [2, p. 95]). Because from the definition of M in this theorem we have |ρ(s)| ≤ U(s), s ∈ S, for every ρ ∈ M and we also have | U(s)x, x | ≤ x, x sup |ρ(s)|, s ∈ S, x ∈ H, ρ∈M
it results that U(s) = sup |ρ(s)|, s ∈ S, ρ∈M
which finishes the proof.
6. The Gelfand-Naimark theorem for abelian C∗ -algebras Lemma 6.1. Let S ⊂ B(H) be a commutative algebra with involution and neutral element e. If M is the set {ρ : S → C|ρ linear, ρ(e) = 1, ρ(st) = ρ(s)ρ(t), ρ(s∗ ) = ρ(s), |ρ(s)| ≤ s, s, t ∈ S} there is a ∗-representation μ : C(M) → B(H) where C(M) is the algebra {f : M → C|f
continuous}
with the involution f ∗ (ρ) = f (ρ), such that μ(f ) ≤ sup |f (ρ)|, μ(ρ → ρ(s)) = s ρ∈M
and
s = sup |ρ(s)|. ρ∈M
Proof. Defining U : S → B(H) by U(s) = s this lemma is a consequence of Theorem 5.1. Theorem 6.2 (Gelfand-Naimark theorem). Let S ⊂ B(H) be an abelian C ∗ -algebra and M as in the preceding lemma. The function defined by s → G(s), where G(s) is the function M → C defined by G(s)(ρ) = ρ(s), is a ∗-isomorphism S → C(M) such that s = sup |ρ(s)|. ρ∈M
Proof. This result is a consequence of Lemma 6.1 if we notice that S is a complete space and that the linear subspace generated by {G(s)|s ∈ S} is dense in C(M).
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Acknowledgment I am indebted to the referee for valuable suggestions.
References [1] D. Atanasiu, Un th´eor`eme du type Bochner-Godement et le probl`eme des moments , J. Funct. Anal. 92(1990), 92–103. [2] C. Berg, J.P.R. Christensen and P. Ressel, Harmonic analysis on semigroups. Theory of positive definite and related functions, Springer-Verlag, 1984. [3] C. Berg and P.H. Maserick, Exponentially bounded positive definite functions, Illinois J. Math. 28(1984), 162–179. [4] G. Maltese, A representation theorem for positive functionals on involution algebras (Revised), Boll. U.M.I. (7)8-A(1994), 431–438. [5] P. Ressel, Integral representations on convex semigroups, Math. Scand. 61(1987), 93–101. [6] P. Ressel and W.J. Ricker, Semigroup representations, positive definite functions and abelian C∗ -algebras, Proc. Amer. Math. Soc. 126(1998), 2949–2955. Dragu Atanasiu Bor˚ as University S-501 90 Bor˚ as, Sweden e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 51–57 c 2009 Birkh¨ auser Verlag Basel/Switzerland
On Vector Measures, Uniform Integrability and Orlicz Spaces Diomedes Barcenas and Carlos E. Finol Abstract. Given a Banach space X and a probability space (Ω, Σ, μ), we characterize the countable additivity of the Dunford integral for Dunford integrable functions taking values in X as those weakly measurable functions f : Ω −→ X for which {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in some separable Orlicz space Lϕ (μ). We also provide a characterization of the Pettis integral of Dunford integrable functions by mean of weak compactness in separable Orlicz spaces and give a necessary and sufficient condition for the uniform integrability of {xf : x ∈ BX }, whenever f : Ω −→ X ∗ is Gel’fand integrable. Mathematics Subject Classification (2000). Primary 46G10; Secondary 28B05. Keywords. Vector measures, Pettis integral, Orlicz spaces.
1. Introduction and preliminaries We shall use the following notations: throughout, X will denote a Banach space with dual X ∗ and BX its closed unit ball; while the closed unit ball of X ∗ is denoted by BX ∗ . Regarding the integrals considered herein we shall follow ([3]), along with most definitions. Let us recall a result of De La Vall´ee Poussin’s ([9], Theorem 2, p. 3). Theorem 1.1. Let (Ω, Σ, μ) be a finite measure space. A subset A ⊂ L1 (μ) is uniformly integrable if and only if there is an N -function ϕ such that A is bounded in the Orlicz space Lϕ (μ). As to N -functions,their properties and various classes thereof, we refer to ([4] , [9]). An N -function ϕ is submultiplicative if, for all s, t > 0, we have that ϕ (st) ≤ ϕ (s) ϕ (t). Throughout, by (Ω, Σ, μ), we mean a finite measure space. Research supported partially by C.D.C.H. Universidad Central de Venezuela grant 03-00-60402005.
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Recall that a subset A of L1 (μ) is said to be uniformly integrable if it is bounded in L1 (μ) and for each ε > 0, there is δ > 0 such that μ(E) < δ ⇒ |f |dμ < ε E
uniformly in A. A weakly measurable function f : Ω → X is called Dunford integrable if for each x∗ ∈ X ∗ the measurable scalar function x∗ f is integrable. If f is Dunford integrable, then it can be proved ([8]) that for each measurable set E ∈ Σ, there ∗ is x∗∗ E ∈ X such that ∗∗ ∗ xE (x ) = x∗ f dμ, ∀x∗ ∈ X ∗ . E
The x∗∗ is called the Dunford integral of f over E and it is denoted functional ∗∗ by D E f dμ. If xE ∈ X ⊂ X ∗∗ ∀E ∈ Σ, we say function that the Dunford integral f is Pettis integrable. In this case we write P E f dμ instead of D E f dμ. Another integral we deal with in this paper is the so-called the Gel’fand integral. A function f : Ω → X ∗ is called Gel’fand integrable if for each x ∈ X, xf is integrable. Proceeding as in the case of Dunford integrable functions ([8]), ∗ if f is Gel’fand integrable, then for each measurable set E ∈ Σ, there is x∗∗ E ∈X ∗ ∗ such that x (x) = E xf dμ. The functional xE is called the Gel’fand integral of f over E. This fact is denoted by G E f dμ. Another characterization of uniformly integrability, obtained by Dunford and Pettis ([6], Theorem II, p. 39), reads as follows: Theorem 1.2. A subset A ⊂ L1 (μ) is uniformly integrable if and only if it is relatively weakly compact in L1 (μ). Since a strongly measurable function f : Ω → X is Pettis integrable if and only if it is Dunford integrable and the set {x∗ f : x∗ ∈ BX ∗ } is uniformly integrable in L1 (μ). J.Uhl([12]) has obtained the following consequence of De La Vall´ee Poussin’s theorem. Theorem 1.3. Let f : Ω → X be a strongly measurable function. Then, f is Pettis integrable with respect to μ, if and only if there is an N -function φ such that {x∗ f : x∗ ∈ BX ∗ } is bounded in Lφ (μ). Yet another improvement of the De La Vall´ee Poussin’s theorem is the following: Theorem 1.4 (([2], Theorem 2.5)). A subset A of L1 (μ) is uniformly integrable,if and and only if, there is a submultiplicative N -function ϕ such that A is relatively weakly compact in Lϕ (μ). The starting point for preparing this research has been the above Uhl’s and Alexopoulos’ theorems, together with a better understanding of vector integration ([3], [10], [11]).
On Vector Measures, Uniform Integrability and Orlicz Spaces
53
A weakly measurable function f : Ω → X is said to be determined by a weakly compact generated (W CG) subspace of X, if there is a weakly compact generated subspace D of X such that one of the following conditions is met. • If x∗ |D = 0, then x∗ f = 0 μ a.e. • For each x∗ ∈ X ∗ , there exists a sequence {ϕn }∞ n=1 of D-valued simple functions such that x∗ f = lim x∗ ϕn μ a.e. (see [11, Definition 2.1]). Regarding to the Pettis integrability of Dunford integrable functions, the following holds: Theorem 1.5. ([11]) A Dunford integrable function f : Ω → X is Pettis integrable if and only if D f dμ is a countably additive vector measure and f is weakly compact Ω
generated determined.
2. The results In this section we put together the main results of our paper. We start with a characterization of the countable additivity of the Dunford integral. Our characterization is similar to that of Pettis integral for strongly measurable functions, obtained by Uhl (Theorem 1.3). Theorem 2.1. Let (Ω, Σ, μ) be a finite measure space and f : Ω → X be a Dunford integrable function. Then the following statements are equivalent. (i) The Dunford integral of f is countably additive; that is, the set function D f dμ : Σ → X ∗∗ , defined by D f dμ (E) = D f dμ, E
is countable additive. (ii) There is a submultiplicative N -function ϕ such that the set {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in the Orlicz space Lϕ (μ). (iii) There is a submultiplicative N -function ϕ such that {x∗ f ∈ BX ∗ } is bounded in Lϕ (μ) and {x∗ f : x∗ ∈ BX ∗ } does not contain any basic sequence equivalent to the unit vector basis of l1 . Proof. Let f : Ω → X be a Dunford integrable function; put ν(E) = D f dμ (E ∈ Σ). E
It is easy to see that ν is a vector measure and according to ([11]), ν is a countably additive if and only if the operator: T : X ∗ → L1 (μ) defined by T (x∗ ) = x∗ f is weakly compact. This is equivalent to say that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in L1 (μ). By the Dunford Pettis characterization of relative weak compactness in L1 (μ), it is equivalent to the uniform integrability of {x∗ f : x∗ ∈ BX ∗ } in L1 (μ) and by Theorem 1.4, it is equivalent to the existence
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D. Barcenas and C.E. Finol
of a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in the Orlicz space Lϕ (μ). Therefore (i) and (ii) are equivalent statements. (iii)⇒(ii). Since submultiplicative functions obviously satisfy the Δ2 condition, for all t ≥ 0, then the corresponding Orlicz space is weakly sequentially complete. Therefore, if {x∗ f : x∗ ∈ BX ∗ } is bounded in Lϕ (μ), then, according to Rosenthal l1 theorem ([5], Theorem 2.e.5, p. 99), either {x∗ f : x∗ ∈ BX ∗ } contains a weakly Cauchy sequence or {x∗ f : x∗ ∈ BX ∗ } contains a subsequence equivalent to the unit vector basis of l1 . Thus, if (iii) holds, then every subset of the bounded set {x∗ f : x∗ ∈ BX ∗ } contains a weakly Cauchy sequence and because, Lϕ (μ) is weakly sequentially complete, then every subset of {x∗ f : x∗ ∈ BX ∗ } contains a weakly convergent sequence. By Eberlein Smulyan’s theorem, {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact and consequently (iii)⇒(ii). (ii)⇒(iii). If (ii) holds, then {x∗ f : x∗ ∈ BX ∗ } is bounded in Lϕ (μ) because it is relatively weakly compact in Lϕ (μ). Hence, applying again the relative weak compactness of {x∗ f : x∗ ∈ BX ∗ }, by Eberlein Smulyan’s theorem, each sequence contains a weakly convergent subsequence and by Rosenthal’s l1 theorem, {x∗ f : x∗ ∈ BX ∗ } does not contain any isomorphic copy of l1 . An alternative proof can be obtained by using the Bessaga Pelczyinski principle ([1], p. 14). From the above theorem ad Theorem 1.5, we have the following characterization of Pettis integrability of Dunford integrable functions. Corollary 2.2. Let f : Ω → X be a Dunford integrable function. Then the following statements are equivalent: (i) f is Pettis integrable. (ii) f is weakly compact generated determined and there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ). (iii) f is weakly compact generated determined and there is a submultiplicative N -function such that {x∗ f : x∗ ∈ BX ∗ } is bounded in Lϕ (μ) and does not contain any subsequence isomorphic to the unit vector basis of l1 . We restate Uhl’s theorem as follows: Theorem 2.3. A strongly measurable function f : Ω → X is Pettis integrable if and only if there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ). Proof. If f : Ω → X is strongly measurable, then its range is essentially separable and, consequently, weakly compact generated determined. On the other hand, being f Pettis integrable, then it is Dunford integrable with ν(E) = D E f dμ, a countably additive vector measure. Hence {x∗ f : x∗ ∈ BX ∗ } is uniformly integrable and consequently there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ).
On Vector Measures, Uniform Integrability and Orlicz Spaces
55
For the converse, we suppose that f is strongly measurable and that there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ). Since f is strongly measurable it has range weakly compactly generated determined. Since Lϕ (μ) ⊂ L1 (μ) ([4], [9]), we have that f is Dunford integrable with the Dunford integral of f countably additive. Therefore, f is Pettis integrable. We have the following version of Corollary 5.2 from ([7]). Corollary 2.4. Let f : Ω −→ X be a Dunford integrable function. If there is p > 1 such that {x∗ f : x∗ ∈ BX ∗ } is bounded in Lp (μ) then the Dunford integral of f is countably additive. Proof. Bounded subset of Lp (μ) (p > 1) are uniformly integrable in L1 (μ) and, consequently, the Dunford integral of f is countably additive. Regarding to the Gel’fand integral we have the following result. Theorem 2.5. Let X be a Banach space. The following statements are equivalent. (a) X does not contain a complemented copy of l1 . (b) A function f : Ω → X ∗ is Gel’fand integrable if and only if there is a submultiplicative N -function ϕ such that {xf : x ∈ BX } is relatively weakly compact in Lϕ (μ). Proof. If X does not contain a complemented copy of l1 , then X ∗ does not contain any copy of l∞ and by a theorem of Diestel and Faires, ([3], Theorem 2, p. 20), the Gel’fand integrable function f defines a countably additive vector measure. Define T : X → L1 (μ) by T (x) = xf ; T is a bounded linear operator and we want to prove that T is weakly compact. If ν(E) = G f dμ, then ν is a countably E
additive vector measure for which ν(E)(x) = 0 for every x ∈ X whenever μ(E) = 0. By ([3], Theorem I.2.1), given ε > 0, there is δ > 0 such that μ(E) < δ ⇒ ν(E) < ε, which implies
μ(E) < δ ⇒ sup
x∈BX
|xf | dμ < 4ε.
(2.1)
E
On the other hand, by the Bartle, Dunford and Schwartz theorem ([3], Corollary 6, p. 14), ν(Σ) is relatively weakly compact, which implies that xf dμ : E ∈ Σ, x ∈ BX E
is bounded in C. Consequently,
|xf |dμ < ∞.
sup x∈BX
Ω
(2.2)
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D. Barcenas and C.E. Finol
From (2.1) and (2.2) we conclude that {xf : x ∈ BX } is uniformly integrable and {xf : x ∈ BX } is relatively weakly compact in Lϕ (μ) for some submultiplicative N -function ϕ. Conversely, if X contains a complemented copy of l1 , then X ∗ contains a copy of l∞ ; proceeding as in ([3], Example II.3.3, p. 53), we may construct a c0 -valued Dunford integrable function whose Dunford integral is not countably additive. More precisely, if the Dunford integral of the function f : [0, 1] → c0 is not countably additive, Y is a closed subspace of X ∗ and T : c0 → Y is an isomorphism, then, the function g := T f is defined on [0, 1], with values in X ∗ , and it is easy to see that its Gel’fand integral is not countably additive. This function is also Gel’fand integrable when it is considered as taking values in l∞ , and the values of both integrals coincide. Since the Dunford integral of f is not countably additive, then {xf : x ∈ Bl1 } is not uniformly integrable in L1 [0, 1] and, according to Theorem 1.4, there is not any submultiplicative function ϕ with {xf : x ∈ Bl1 } relatively weakly compact in Lϕ (μ). It is natural to ask whether a Gel’fand integrable function f whose Gel’fand integral is countably additive is Pettis integrable. The answer to this question is negative as the following example shows: Example. Take X = C[0, 1]. Then X contains a copy of every separable Banach space and, consequently, X contains a copy of l1 which is uncomplemented in C [0, 1] , since ,otherwise, B [0, 1] = C ∗ [0, 1] would contain a copy of l∞ . Since C [0, 1] contains a copy of l1 , then its dual, B[0, 1], does not have the weak Radon-Nikodym property ([8] Theorem 12.1) and, therefore, there is a countably additive vector measure ν : Σ −→ B[0, 1] (Σ the Borel sets in [0, 1]) such that ν does not have a weak Radon-Nikodym derivative; on the other hand, by ([8]) Theorem 11.1 there is g : [0, 1] −→ B[0, 1], Gel’fand integrable, such that ν(E) = G gdμ. E
Plainly, g is not weakly equivalent to any Pettis integrable function. In fact; if there is f such that f is Pettis integrable and xf = xg for all x ∈ X, then x∗ (P gdu) = x∗ ( f du) ∀x∗ ∈ X ∗∗ and E ∈ Σ E
E
and so x∗ ν(E) = x∗ (P
f du) =
E
x∗ f dμ.
E
This implies, by Hahn-Banach’s theorem, that ν has a weak Radon-Nikodym derivative; a contradiction.
On Vector Measures, Uniform Integrability and Orlicz Spaces
57
Acknowledgment We wish to thank the referee for his suggestions which considerably improved the paper.
References [1] F. Albiac and N.J. Kalton,Topics in Banach Spaces Theory, Springer, 2006. [2] J. Alexopoulos, De La Vall´ee Poussin’s theorem and weakly compact sets in Orlicz spaces, QM 17 (1994), 231–238. [3] J. Diestel and J.J. Uhl, Vector Measures, Math. Survey 15, AMS. sre. Providence, RI (1977). [4] M.A. Krasnoselskii and V.B. Rutickii, Convex functions and Orlicz spaces, Noordhoff, Groningen 1961. [5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, 1977. [6] P. Meyer, Probabilit´es et Potentiels, Publications de L’Institut de Math. de L’Universit´e de Strasburg. Act. Sci. et Industrielles, Hermann, Paris 1966. [7] K. Musial, Topics in the theory of Pettis integration, Rend. Istit. mat. Univ. Trieste, 23 (1992), 177–262. [8] K. Musial, Pettis integral, Handbook of Measure Theory 531–568 (edited by E. Pap) North Holland, Amsterdam 2002. [9] M.M. Rao and Z. Ren, The Theory of Orlicz spaces, Marcel Dekker, New York, 1991. [10] S. Schwabik and Y. Guoju, Topics in Banach spaces integration, series in Real Analysis, vol. 10, World Scientific, Singapore (2005). [11] G. Stefansson, Pettis Integrability, Trans. A.M.S. 330 1 (1992), 401–418. [12] J.J. Uhl, A characterization of strongly measurable Pettis integrable functions, Proc. A.M.S., 34 2 (1972), 425–42. Diomedes Barcenas Departamento de Matem´ atica Universidad de Los Andes M´erida, 5101, Venezuela e-mail: [email protected] Carlos E. Finol Escuela de Matem´ atica Universidad Central de Venezuela Caracas, Venezuela e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 59–64 c 2009 Birkh¨ auser Verlag Basel/Switzerland
The Bohr Radius of a Banach Space Oscar Blasco Abstract. Let 1 ≤ p, q < ∞ and let X be a complex Banach space. For each ∞ n ∞ f (z) = n=0 xn z with f H (D,X) ≤ 1 we define Rp,q (f, X) = sup{r ≥ 0 : ∞ p x0 + ( n=1 xn r n )q ≤ 1} and denote the Bohr radius of X by Rp,q (X) = inf{Rp,q (f, X) : f H ∞ (D,X) ≤ 1}. The aim of this note is to study for which spaces X = Ls (μ) or X = s one has Rp,q (X) > 0. Mathematics Subject Classification (2000). Primary 46E40; Secondary 30B10. Keywords. Bohr radius, vector-valued analytic functions.
1. Introduction and preliminaries In 1914 H. Bohr [3] showed that ∞
1 |an |( )n ≤ f ∞ , 3 n=0
(1.1)
∞ where f (z) = n=0 an z n a bounded analytic function on the open unit disc. The value 1/3 is sharp. A bit later some other proofs of such inequality were given (see [9, 10]). Also several authors have found some extensions (see [4, 5, 8, 11]). Another basic inequality was discovered in [9, Corollary 2.7] much later and will play a special role for us, namely ∞ 1 2 |a0 | + |an |( )n ≤ 1, (1.2) 2 n=1 whenever f H ∞ ≤ 1 and the value 1/2 is sharp in this case. Later on some multidimensional analogues of Bohr’s inequality in which the disc D is replaced by a domain Ω ⊂ Cm were considered (see [1, 2]) and several applications of this multidimensional Bohr radius and connections concerning local Banach space theory have been recently achieved (see [7, 6]). Partially supported by Proyecto MTM2008-04594/MTM.
60
O. Blasco
Our point of view will be to keep D as domain for the functions but allow them to take values in a complex (possibly infinite-dimensional) Banach space. Throughout this paper X stands for a complex Banach space and H p (D, X), usual, for 1 ≤ p ≤ ∞, denotes the Hardy spaces of X-valued holomorphic functions from the unit disc. n Definition 1.1. Given f (z) = ∞ n=0 xn z with f H ∞ (D,X) ≤ 1 we denote $ % ∞ n R(f, X) = sup r ≥ 0 : xn r ≤ 1 . (1.3) n=0
Let us now define the Bohr’s radius of X by R(X) = inf{R(f, X) : f H ∞ (D,X) ≤ 1}. That is to say
$
R(X) = sup r ≥ 0 :
∞
(1.4) %
xn r ≤ f H ∞ (D,X) n
.
n=0
Since C is embedded into any complex Banach space we have, due to (1.1) that R(X) ≤ 13 for any Banach space. However the notion is not very useful even in the finite-dimensional case for dimension greater than one. Let us denote Cm p m the space Cm endowed with the norm the wp = ( i=1 |wi |p )1/p , or w∞ = supm i=1 |wi |. Theorem 1.2. Let m ≥ 2 and 1 ≤ p ≤ ∞. Then R(Cm p ) = 0. Proof. It suffices to do the case m = 2. In the case p = ∞ one can easily find f with f ∞ = 1 and R(f, Cm ∞ ) = 0, (take for instance f (z) = e1 + e2 z). This shows that R(C2∞ ) = 0. Assume 1 < p < ∞. Let us now use that limy→∞ y 1/p − (y − 1)1/p = 0 to get, for each ε > 0, a value 0 < γ < 1 such that 1 − (1 − γ)1/p < εγ 1/p .
(1.5)
Now define f (z) = ((1 − γ)1/p , γ 1/p z) = (1 − γ)1/p e1 + γ 1/p e2 z. Clearly sup|z|<1 f (z)p = 1. On the other hand using (1.5) one has x0 p + εx1 p = (1 − γ)1/p + εγ 1/p > 1. m This shows that R(f, Cm p ) ≤ ε. Hence R(Cp ) = 0. Assume now p = 1. As above for each ε > 0 we can find 0 < γ < 1 satisfying & √ 1 − 1 − γ < ε γ. (1.6)
and define
√ √ & γ 1−γ 1 & √ √ (1, 1) + (1, −1)z = ( 1 − γ + γz, 1 − γ − γz). f (z) = 2 2 2
The Bohr Radius of a Banach Space Observe that f (z)1
= ≤
61
& 1 & √ √ | 1 − γ + γz| + | 1 − γ − γz| 2 1/2 & √ √ 1 & √ | 1 − γ + γz|2 + | 1 − γ − γz|2 = 1. 2
On the other hand, from(1.6), x0 1 + εx1 1 =
& √ 1 − γ + ε γ > 1.
m This shows that R(f, Cm 1 ) ≤ ε. Hence R(C1 ) = 0.
However, following the observation in [9] and extending inequality (1.2), we are going to define another a modified Bohr radius which needs not be zero even for infinite-dimensional Banach spaces. Definition ∞1.3. Let 1 ≤ p, q < ∞ and let X be a complex Banach space. Given f (z) = n=0 xn z n with f H ∞ (D,X) ≤ 1 we denote
q $ ∞ % p n xn r ≤1 . (1.7) Rp,q (f, X) = sup r ≥ 0 : x0 + n=1
We now define Rp,q (X) = inf{Rp,q (f, X) : f H ∞ (D,X) ≤ 1}.
(1.8)
Of course R1,1 (X) = R(X) and we have the following chain of inclusions: Rp1 ,q1 (X) ≤ Rp2 ,q2 (X),
p1 ≤ p2 ,
q1 ≤ q2 .
(1.9)
To compute the precise value of Rp,q (Cm 2 ) is difficult in general, even for m = 1. In [9, Cor. 2.7] it was shown that R2,1 (C) = 12 . Let us adapt the same argument to cover the cases 1 ≤ p ≤ 2. p Proposition 1.4. If 1 ≤ p ≤ 2 then Rp,1 (C) = 2+p . ∞ Proof. Let f (z) = n=0 an z n belong to the unit ball of H ∞ (D, C). We recall the estimate, first observed by Wiener [3],
|an | ≤ 1 − |a0 |2
(1.10)
(see also [9] for a proof). From (1.10) one concludes that |a0 |p +
∞ n=1
|an |rn ≤ |a0 |p + (1 − |a0 |2 )
r 1−r
(1.11)
r Since 1 ≤ p ≤ 2 we estimate (1.11) by |a0 |p + 2p (1 − |a0 |p ) 1−r . Now |a0 |p + p p 2 r p p (1 − |a0 | ) 1−r ≤ 1 if and only if r ≤ 2+p . This gives that Rp,1 (f, C) ≥ 2+p . p Hence Rp,1 (C) ≥ 2+p .
62
O. Blasco z−a 1−az
For the converse we use Moebius transformations φa (z) = 2 ∞ n n Since φa (z) = −a + 1−a n=1 a z one obtains a ap +
for 0 < a < 1.
∞ 1 − a2 n n 1 − a2 ra . a r = ap + a n=1 a 1 − ra
This shows that Rp,1 (φa , C) =
1−ap 1−a p
(1 + a) + a( 1−a 1−a )
Taking limits as a → 1 one gets Rp,1 (C) ≤
.
p 2+p .
2. The Bohr radius Rp,q (X) for Lp -spaces Using the same example as in Theorem 1.2 one gets the following: Proposition 2.1. Rp,q (Cm ∞ ) = 0 for any m ≥ 2 and 1 ≤ p, q < ∞. Theorem 2.2. Let m ≥ 2. Then Rp,p (Cm 2 ) > 0 if and only if p ≥ 2. Proof. Assume p ≥ 2. From (1.9) it suffices to see that R1,2 (Cm 2 ) > 0. Now given f in the unit ball of H ∞ (D, Cm ) one has, in particular, that f 2H 2 (D,Cm ) = 2 2 ∞ 2 n=0 xn 2 ≤ 1. Therefore
2
∞
∞ ∞ n 2 2n x0 + xn r ≤ x0 + xn r n=1
n=1
n=1
r2 ≤ x0 + 2(1 − x0 ) 1 − r2 2 r . ≤ max 1, 2 1 − r2 2
r √1 , one obtains that for R1,2 (Cm ) ≥ √1 . Now, since 2 1−r 2 = 1 for r = 2 3 3 Conversely, assume now that 1 ≤ p < 2. Arguing as in Theorem 1.2, one has that for each ε > 0 we can find 0 < γ < 1 such that
(1 − γ)p/2 + εp γ p/2 > 1. (2.1) √ √ Now selecting f (z) = 1 − γe1 + γe2 z and using (2.1) we get Rp,p (f, Cm 2 ) ≤ ε. This implies that Rp,p (Cm 2 ) = 0. Let us now study the situation for Lp -spaces in the infinite-dimensional case. Theorem 2.3. Let 1 ≤ p, q, s < ∞ and let (Ω, Σ, μ) be a measure space such that there exists a sequence of pairwise disjoint sets with 0 < μ(An ) < ∞. Then Rp,q (Ls (μ)) = 0 whenever 1 ≤ q < s.
The Bohr Radius of a Banach Space Proof. Let 0 < β < 1 and a =
1−β 2−β .
63
Set
a0 = β 1/s μ(A0 )−1/s
and an = an/s μ(An )−1/s
for n ≥ 1. Now define φn = an χAn and Fβ (z) =
∞
φn z n .
n=0
Clearly Fβ belongs to the unit ball of H ∞ (D, H), because ∞ ∞ Fβ (z)sLs (μ) = |z|ns |an |s dμ ≤ |an |s μ(An ) = 1. n=0
An
On the other hand
q ∞ p φn Ls (μ) rn φ0 Ls (μ) +
n=0
=
ap0 μ(A0 )p/s
n=1
= β
p/s
+
= β p/s +
+
q
∞
an μ(An )1/s rn
n=1 ∞
(a
1/s
q n
r)
n=1
a1/s r 1 − a1/s r
q .
1/s
a r q Now β p/s + ( 1−a 1/s r ) ≤ 1 if and only if
r≤
a−1/s (1 − β p/s )1/q (1 − β p/s )1/q (2 − β)1/s = . 1 + (1 − β p/s )1/q (1 − β)1/s 1 + (1 − β p/s )1/q
Since 1/q > 1/s, taking limits as β goes to 1 one gets that Rp,q (Ls (μ)) = 0.
Corollary 2.4. Let 1 ≤ p, s < ∞ and 1 ≤ q < s. Then Rp,q (s ) = Rp,q (Ls (R)) = 0. A look to the proof in Theorem 2.2 shows that actually for X = L2 (μ) one gets R1,2 (L2 (μ)) ≥ √13 . Let us give some lower estimates of Rp,q (Ls (μ)) for some values of q ≥ s. As usual p stands for the conjugate exponent satisfying 1/p + 1/p = 1. Theorem 2.5. Let 1 < s < ∞, q = max{s, s } and 1 ≤ p ≤ q. Then Rp,q (Ls (μ)) ≥
(q q /q
p1/q . + pq /q )1/q
(2.2)
∞ n belong to the unit ball of Proof. Let X = Ls (μ) and let f (z) = n=0 xn z ∞ H (D, X). It follows easily from complex interpolation (considering X1 to be ∞ L1 (μ) or L∞ (μ) and X2 to be L2 (μ)) that ( n=0 xn q )1/q ≤ f H q (D,Ls (μ)) . In ∞ q q particular we have n=1 xn ≤ 1 − x0 , n ≥ 1.
64
O. Blasco Therefore
x0 + p
∞
q xn r
n
≤ x0 + (1 − x0 ) p
n=1
q
rq 1 − rq
q/q .
Hence, for p ≤ q, we estimate
q ∞ q rq p n xn r ≤ x0 p + (1 − x0 p ) x0 + p (1 − rq )q/q n=1 rq q . ≤ max 1, p (1 − rq )q/q Note that
q rq p (1−r q )q/q
= 1 gives the value r =
1/q (p q)
q /q )1/q (1+( p q)
.
References [1] L. Aizenberg Multidimensional analogues of Bohr’s theorem on power series Proc. Amer. Math. Soc. 128 (1999), 1147–1155. [2] L. Aizenberg, A. Aytuna and P. Djakov Generalization of a theorem of Bohr for basis in spaces of holomorphic functions in several variables J. Math. Anal. Appl. 258 (2001), 429–447. [3] H. Bohr A theorem concerning power series Proc. London Math. Soc. (2)13 (1914), 1–15. [4] E. Bombieri Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze Bull. Un. Mat. Ital. (3)17 (1962), 276–282. [5] E. Bombieri and J. Bourgain A remark on Bohr’s inequality Inter. Math. Res. Notices 80 (2004), 4307–4329. [6] A. Defant, D. Garc´ıa and M. Maestre Bohr’s power series theorem and local Banach space theory J. reine angew. Math. 557 (2003), 173–197. [7] A. Defant, C. Prengel, Christopher Harald Bohr meets Stefan Banach. Methods in Banach space theory, London Math. Soc. Lecture Note Ser., 337, Cambridge Univ. Press, Cambridge, (2006), 317–339, [8] P.B. Djakov and M.S. Ramanujan A remark on Bohr’s theorem and its generalizations J. Anal. 8 (2000), 65–77. [9] V. Paulsen, G. Popescu, D. Singh On Bohr’s inequality Proc. London Math. Soc. 85 (2002), 493–512. ¨ [10] S. Sidon Uber einen Satz von Herrn Bohr Math. Z. 26 (1927), 731–732. [11] M. Tomic Sur un th´eor`eme de H. Bohr Math. Scand. 11 (1962), 103–106. Oscar Blasco Departamento de An´ alisis Matem´ atico Universidad de Valencia E-46100 Burjassot Valencia, Spain e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 65–78 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology Oscar Blasco and Jan van Neerven Abstract. Let X and Y be Banach spaces and (Ω, Σ, μ) a finite measure space. In this note we introduce the space Lp [μ; L (X, Y )] consisting of all (equivalence classes of) functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)x is strongly μ-measurable for all x ∈ X and ω → Φ(ω)f (ω) belongs to L1 (μ; Y ) for all f ∈ Lp (μ; X), 1/p + 1/p = 1. We show that functions in Lp [μ; L (X, Y )] define operator-valued measures with bounded p-variation and use these spaces to obtain an isometric characterization of the space of all L (X, Y )-valued multipliers acting boundedly from Lp (μ; X) into Lq (μ; Y ), 1 q < p < ∞. Mathematics Subject Classification (2000). 28B05, 46G10. Keywords. Operator-valued functions, operator-valued multipliers, vector measures.
1. Introduction Let (Ω, Σ, μ) be a finite measure space and let X and Y be Banach spaces over K = R or C. In his talk at the 3rd meeting on Vector Measures, Integration and Applications (Eichst¨ att, 2008), Jan Fourie presented some applications of the following extension of an elementary observation due to Bu and Lin [2, Lemma 1.1]. Proposition 1.1. Let Φ : Ω → L (X, Y ) be a strongly μ-measurable function. For all ε > 0 there exists strongly μ-measurable function f ε : Ω → X such that for μ-almost all ω ∈ Ω one has f ε (ω) 1 and Φ(ω) Φ(ω)f ε (ω) + ε. The first named author is partially supported by the Spanish project MTM2008-04594/MTM. The second named author is supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).
66
O. Blasco and J. van Neerven
Recall that a function φ : Ω → Z, where Z is a Banach space, is said to be strongly μ-measurable if there exists a sequence of Σ-measurable simple functions φn : Ω → Z such that for μ-almost all ω ∈ Ω one has limn→∞ φn (ω) = φ(ω) in Z. In Proposition 1.1, the strong μ-measurability assumption on Φ refers to the norm of L (X, Y ) as a Banach space. The next two examples show that the conclusion of Proposition 1.1 often holds if we impose merely strong μ-measurability of the orbits of Φ. Example 1. Consider X = ∞ (Z), let T be the unit circle, and define Φ : T → ∞ (Z) = L (1 (Z), K)
by Φ(t) := (eint )n∈Z . For all x ∈ 1 (Z) the function t → Φ(t)x = n∈Z xn eint is continuous, but the function t → Φ(t) fails to be strongly measurable. Taking for f the constant function with value u0 ∈ 1 (Z), defined by u0 (0) = 1 and u0 (n) = 0 for n = 0, we have Φ(t) = |Φ(t)f (t)| = | u0 , Φ(t) | = 1 ∀t ∈ T. Example 2. Consider X = C([0, 1]) and define Φ : [0, 1] → M ([0, 1]) = L (C([0, 1]), K) by Φ(t) := δt . For all x ∈ X the function t → Φ(t)x = x(t) is continuous, but the function t → Φ(t) fails to be strongly measurable. If f : [0, 1] → X is a strongly measurable function such that (f (t))(t) = 1 for all t ∈ [0, 1] (e.g., take f (t) ≡ 1), we have Φ(t) = | f (t), Φ(t) | = 1
∀t ∈ [0, 1].
Thus it is natural to ask whether strong μ-measurability of Φ can be weakened to strong μ-measurability of the orbits ω → Φ(ω)x for all x ∈ X, or even to μmeasurability of the functions ω → Φ(ω)x. Although in general the answer is negative even when dim Y = 1 (Example 5), various positive results can be formulated under additional assumptions on X or Φ (Propositions 2.2, 2.4, and their corollaries). One of the applications of Proposition 1.1 was the study of multipliers between spaces of vector-valued integrable functions. In [5], for 1 p, q < ∞, Mult(Lp (μ; X), Lq (μ; Y )) is defined to be the space of all strongly μ-measurable functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)f (ω) belongs to Lq (μ; Y ) for all f ∈ Lp (μ; X). It is shown (see [5, Proposition 3.4]) that for 1 q < p < ∞ and 1/r = 1/q − 1/p one has a natural isometric isomorphism Mult(Lp (μ; X), Lq (μ; Y )) Lr (μ; L (X, Y )). We observe (Proposition 3.1) that the strong μ-measurability of Φ as function with values in L (X, Y ) is not really needed to define bounded operators from Lp (μ; X) into Lq (μ; Y ); it is possible to weaken the measurability assumptions on the multiplier functions by only requiring strong μ-measurability of its orbits. This will motivate the introduction of an intermediate space between Lp (μ; L (X, Y )) and the space Lps (μ; L (X, Y )) of functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)x belongs to Lp (μ; Y ) for all x ∈ X. This is done by selecting the functions in
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Lps (μ; L (X, Y )) for which ω → Φ(ω)f (ω) belongs to L1 (μ; Y ) for all f ∈ Lp (μ; X), 1/p+1/p = 1. We shall denote this space by Lp [μ; L (X, Y )]. We shall see that, for 1 p < ∞, functions in this space define L (X, Y )-valued measures of bounded p-variation (Theorems 3.5 and 3.8), and prove that one has a natural isometric isomorphism Mult[Lp (μ; X), Lq (μ; Y )] Lr [μ; L (X, Y )], where 1/r = 1/q − 1/p and Mult[Lp (μ; X), Lq (μ; Y )] is defined to be the linear space of all functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)x is strongly μmeasurable for all x ∈ X and ω → Φ(ω)f (ω) belongs to Lq (μ; Y ) for all f ∈ Lp (μ; X) (Theorem 3.6).
2. Strong μ-normability of operator-valued functions Let (Ω, Σ, μ) be a finite measure space and let X and Y be Banach spaces. Definition 2.1. Consider a function Φ : Ω → L (X, Y ). 1. Φ is called strongly μ-normable if for all ε > 0 there exists strongly μmeasurable function f ε : Ω → X such that for μ-almost all ω ∈ Ω one has f ε (ω) 1 and Φ(ω) Φ(ω)f ε (ω) + ε. 2. Φ is called weakly μ-normable if for all ε > 0 there exist strongly μ-measurable functions f ε : Ω → X and g ε : Ω → Y ∗ such that for μ-almost all ω ∈ Ω one has f ε (ω) 1, g ε (ω) 1, and Φ(ω) | Φ(ω)f ε (ω), g ε (ω) | + ε. Clearly, every weakly μ-normable function is strongly μ-normable. In the case Y = K the notions of weak and strong μ-normability coincide and we shall simply speak of normable functions. It will be convenient to formulate our results on μ-normability in the following more general setting. Let S an arbitrary nonempty set. A function f : Ω → S is called a Σ-measurable elementary function ' if for n 1 there exist disjoint sets An ∈ Σ and elements sn ∈ S such that n1 An = Ω and f = n1 1An ⊗ sn . Since no addition is defined in S, this sum should be interpreted as shorthand notation to express that f ≡ sn on An . A function g : S → R is called bounded from above if sups∈S g(s) < ∞. The set of all such functions is denoted by BA (S). Proposition 2.2. Let Φ : Ω → BA (S) be such that for all s ∈ S the function ω → (Φ(ω))(s) is μ-measurable. If there is a countable subset C of S such that for all φ ∈ Φ(Ω) we have sup φ(s) = sup φ(s), s∈S
s∈C
then for all ε > 0 there exists a Σ-measurable elementary function f ε : Ω → S such that for μ-almost all ω ∈ Ω one has sup (Φ(ω))(s) (Φ(ω))(f ε (ω)) + ε. s∈S
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Proof. The function ω → sups∈C (Φ(ω))(s) is μ-measurable, as it is the pointwise supremum of a countable family of μ-measurable functions. Let (s(n) )n1 be an enumeration of C. For n 1 put An := ω ∈ Ω : sup Φ(ω)(s) (Φ(ω))(s(n) ) + ε . s∈S
These sets are μ-measurable, and therefore there exist sets An ∈ Σ such ' 'n that μ(An ΔAn ) = 0. Also, n1 An = Ω. Put B1 := A1 and Bn+1 := An+1 \ m=1 Bn ' for n 1. The sets Bn are Σ-measurable, disjoint. Since B0 := Ω \ n1 Bn is a μ-null set in Σ, the function 1Bn ⊗ s(n) , f ε := n0
where s(0) ∈ S is chosen arbitrarily, has the desired properties.
From this general point of view one obtains the following corollary. Corollary 2.3. Let X and Y be Banach spaces and consider a function Φ : Ω → L (X, Y ). 1. If X is separable and ω → Φ(ω)x is μ-measurable for all x ∈ X, then Φ is strongly μ-normable; 2. If X and Y are separable and ω → | Φ(ω)x, y ∗ | is μ-measurable for all x ∈ X and y ∗ ∈ Y ∗ , then Φ is weakly μ-normable. Proof. To prove part 2 we apply Proposition 2.2 to the set S = BX×Y ∗ (the unit ball of X × Y ∗ with respect to the norm (x, y ∗ ) = max{x, y ∗}) and the functions ω → | Φ(ω)x, y ∗ |, and note that Σ-measurable elementary functions with values in a Banach space are strongly μ-measurable. Since X is separable, for C we may take a set of the form {(xj , yk∗ ) : j, k 1}, where (xj )j1 is a dense sequence in BX and (yk∗ )k1 is a sequence in BY ∗ which is norming for Y . The proof of part 1 is similar. Proof of Proposition 1.1. By assumption, Φ can be approximated μ-almost everywhere by a sequence of simple functions with values in L (X, Y ). Each one of the countably many operators in the ranges of these functions is normed by some ( of X such separable subspace of X. This produces a separable closed subspace X that for μ-almost all ω ∈ Ω, Φ(ω)L (X,Y ) = Φ(ω)L (X,Y ( ). Now we may apply part 1 of Corollary 2.3.
Instead of a countability assumption on the set S we may also impose regularity assumptions on μ and Φ: Proposition 2.4. Let μ be a finite Radon measure on a topological space Ω. Let Φ : Ω → BA (S) be such that for all s ∈ S the function ω → (Φ(ω))(s) is lower
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semicontinuous. Then for all ε > 0 there exists a Borel measurable elementary function f ε : Ω → S such that for μ-almost all ω ∈ Ω one has sup (Φ(ω))(s) (Φ(ω))(f ε (ω)) + ε. s∈S
Proof. Let us first note that the function m(ω) := sup(Φ(ω))(s) s∈S
is lower semicontinuous, since it is the pointwise supremum of a family of lower semicontinuous functions. In particular, m is Borel measurable. Fix ε > 0. Using Zorn’s lemma, let (Ωi )i∈I be a maximal collection of disjoint Borel sets such that the following two properties are satisfied for all i ∈ I: (a) μ(Ωi ) > 0; (b) there exists si ∈ S such that m(ω) (Φ(ω))(si ) + ε for all ω ∈ Ωi . Clearly, (a) implies that the index set I is countable. We claim that ) μ Ω\ Ωi = 0. i∈I
The proof is then finished by taking f ε := i∈I 1Ωi ⊗ si and extending this definition to the remaining Borel μ-null set by assigning an arbitrary constant value on it; by (b) and the claim, this function satisfies the required inequality μ-almost everywhere. ' To prove the claim let Ω := Ω\ i∈I Ωi and suppose, for a contradiction, that μ(Ω ) > 0. By passing to a Borel subset of Ω we may assume that supω ∈Ω m(ω ) < ∞. Let M := ess supω ∈Ω m(ω ). The set A := {ω ∈ Ω : m(ω ) M − 13 ε} is Borel and satisfies μ(A) > 0. Since μ is a Radon measure we may select a compact set K in Ω such that K ⊆ A and μ(K) > 0. For any ω ∈ K we can find s ∈ S such that m(ω ) (Φ(ω ))(s ) + 13 ε. By lower semicontinuity, the set O := ω ∈ Ω : (Φ(ω ))(s ) < (Φ(ω))(s ) + 13 ε is open and contains ω . Choosing such an open set for every ω ∈ K, we obtain an open cover of K, which therefore has a finite subcover. At least one of the finitely many open sets of this subcover intersects K in a set of positive measure. Hence, there exist ω0 ∈ K and s0 ∈ S, as well as an open set O0 ⊆ Ω such that ω0 ∈ O0 , μ(K ∩ O0 ) > 0, m(ω0 ) (Φ(ω0 ))(s0 ) + 13 ε, and (Φ(ω0 ))(s0 ) < (Φ(ω))(s0 ) + 13 ε
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for all ω ∈ O0 . Hence, for μ-almost all ω ∈ K ∩ O0 , m(ω) − 13 ε M − 13 ε m(ω0 ) (Φ(ω0 ))(s0 ) + 13 ε < (Φ(ω))(s0 ) + 23 ε. It follows that the Borel set (K ∩ O0 ) \ N , where N is some Borel set satisfying μ(N ) = 0, may be added to the collection (Ωi )i∈I . This contradicts the maximality of this family. Corollary 2.5. Let μ be a finite Radon measure on a topological space Ω and let X and Y be Banach spaces. Consider a function Φ : Ω → L (X, Y ). 1. If ω → Φ(ω)x is lower semicontinuous for all x ∈ X, then Φ is strongly μ-normable. 2. If ω → | Φ(ω)x, y ∗ | is lower semicontinuous for all x ∈ X and y ∗ ∈ Y ∗ , then Φ is weakly μ-normable. Here are two further examples. Example 3. Consider Ω = (0, 1), X = L1 (0, 1), Y = K, and let Φ : (0, 1) → L∞ (0, 1) = L (L1 (0, 1), K) be defined by Φ(t) := 1(0,t) . For all x ∈ L1 (0, 1) the t function t → Φ(t)x = 0 x(s) ds is continuous. Corollary 2.5 asserts that Φ is normable. In fact, for f (t) := 1t 1(0,t) one even has Φ(t) = |Φ(t)f (t)| = 1
∀t ∈ (0, 1).
Example 4. Let X1 , X2 be Banach spaces and let T : X1 → X2 be a bounded linear operator with T = 1. Consider Ω = [0, 1], X = C([0, 1], X1 ), Y = X2 and let Φ : Ω → L (X, Y ) be defined by Φ(t) := Tt , where Tt (x) = T (x(t)) for x ∈ X. For all x ∈ X the function t → Tt x is continuous. Corollary 2.5 asserts that Φ is weakly (and hence strongly) normable. In fact, for each ε > 0 and t ∈ [0, 1] we can select xε ∈ BX1 and y ∗ε ∈ BX2∗ such that | T xε , y ∗ε | > 1 − ε. Defining f ε := 1 ⊗ xε and g ε := 1 ⊗ y ∗ε one has Φ(t) | Φ(t)f ε (t), g ε (t) | + ε
∀t ∈ [0, 1].
In the Examples 1, 2 and 3 the norming was exact. The next proposition formulates a simple sufficient (but by no means necessary) condition for this to be possible: Proposition 2.6. Let X and Y be Banach spaces and consider a function Φ : Ω → L (X, Y ). 1. Suppose that Φ : Ω → L (X, Y ) is strongly μ-normable. If X is reflexive, there exists a strongly μ-measurable function f : Ω → X such that for μ-almost all ω ∈ Ω one has f (ω) 1 and Φ(ω) = Φ(ω)f (ω). 2. Suppose that Φ : Ω → L (X, Y ) is weakly μ-normable. If X and Y are reflexive, there exist strongly μ-measurable functions f : Ω → X and g : Ω → Y ∗ such that for μ-almost all ω ∈ Ω one has f (ω) 1, g(ω) 1, and Φ(ω) = | Φ(ω)f (ω), g(ω) |.
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Proof. We shall prove part 1, the proof of part 2 being similar. For every n 1 choose a strongly μ-measurable function fn : Ω → X such that for μ-almost all ω ∈ Ω one has fn (ω) 1 and Φ(ω) Φ(ω)fn (ω) + n1 . 2 Since μ is finite, the sequence (fn )∞ n=1 is bounded in the reflexive space L (μ; X) and therefore it has a weakly convergent subsequence (fnk )∞ . Let f be its weak k=1 limit. By Mazur’s theorem there exist convex combinations gj in the linear span of 1 (fnk )∞ k=j such that gj − f < j . By passing to a subsequence we may assume that limj→∞ gj = f μ-almost surely. Clearly, for μ-almost all ω ∈ Ω one has gj (ω) 1 and Φ(ω) Φ(ω)gj (ω) + n1j .
The result follows from this by passing to the limit j → ∞.
The following example shows that the separability condition of Proposition 2.2 and the lower semicontinuity assumption of Proposition 2.4 and its corollaries cannot be omitted, even when X is a Hilbert space and Y = K. Example 5. Let Ω = (0, 1), X = l2 (0, 1), and Y = K. Recall that l2 (0, 1) is the Banach space of all functions φ : (0, 1) → R such that φ2 := sup |φ(t)|2 < ∞, U ∈U
t∈U
where U denotes the set of all finite subsets of (0, 1). Note that for all φ ∈ l2 (0, 1) the set of all t ∈ (0, 1) for which φ(t) = 0 is at most countable; this set will be referred to as the support of φ. Define Φ : (0, 1) → L (l2 (0, 1), K) by Φ(t)φ := φ(t). Clearly, Φ(t) = 1 for all t ∈ (0, 1). Also, Φ(t)φ = 0 for all t outside the countable support of φ and therefore this function is always measurable. Suppose now that a strongly measurable function f : (0, 1) → l2 (0, 1) exists such that 1 |Φ(t)f (t)| + 12 for almost all t ∈ (0, 1). Let N be a null set such that this inequality holds for all t ∈ (0, 1) \ N . For t ∈ (0, 1) \ N it follows that |(f (t))(t)| 12 . Let fn : (0, 1) → l2 (0, 1) be simple functions such that limn→∞ fn = f pointwise almost everywhere, say on (0, 1) \ N for some null set N . The range of each fn consists of finitely many elements of l2 (0, 1), each of which has countable support. Therefore there exists a countable set B ⊆ (0, 1) such that the support of f (t) is contained in B for all t ∈ (0, 1) \ N . For t ∈ (0, 1) \ (N ∪ N ), the inequality |(f (t))(t)| 12 implies that t ∈ B. Hence, (0, 1) \ (N ∪ N ) ⊆ B, a contradiction.
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3. Spaces of operator-valued functions Throughout this section, (Ω, Σ, μ) is a finite measure space and X and Y are Banach spaces. We introduce the linear spaces M (μ; L (X, Y )) := {Φ : Ω → L (X, Y ) : Φ is strongly μ-measurable }, Ms (μ; L (X, Y )) := {Φ : Ω → L (X, Y ) : Φx is strongly μ-measurable ∀x ∈ X}, Mw (μ; L (X, Y )) := {Φ : Ω → L (X, Y ) : Φx is weakly μ-measurable ∀x ∈ X}. Two functions Φ1 and Φ2 in M (μ; L (X, Y )) are identified when Φ1 = Φ2 μ-almost everywhere, two functions Φ1 and Φ2 in Ms (μ; L (X, Y )) are identified when Φ1 x = Φ2 x μ-almost everywhere for all x ∈ X, and Φ1 and Φ2 in Mw (μ; L (X, Y )) are identified when Φ1 x, y ∗ = Φ2 x, y ∗ μ-almost everywhere for all x ∈ X and y∗ ∈ Y ∗. As special cases, for X = K we put M (μ; X) := M (μ; L (K, X)) (which coincides with Ms (μ; L (K, X))) and Mw (μ; X) := Mw (μ; L (K, X)). The following easy fact will be useful below. Proposition 3.1. For Φ ∈ Ms (μ; L (X, Y )) and f ∈ M (μ; X), g(ω) := Φ(ω)f (ω) defines a function g ∈ M (μ; Y ). Proof. For simple functions f this is clear. The general case follows from this, using that μ-almost everywhere limits of strongly μ-measurable functions are strongly μ-measurable. For 1 p ∞ we consider the normed linear spaces Lp (μ; L (X, Y )) := Φ ∈ M (μ; L (X, Y )) : ΦLp(μ;L (X,Y )) < ∞ , Lps (μ; L (X, Y )) := Φ ∈ Ms (μ; L (X, Y )) : ΦLps (μ;L (X,Y )) < ∞ , Lpw (μ; L (X, Y )) := Φ ∈ Mw (μ; L (X, Y )) : ΦLpw (μ;L (X,Y )) < ∞ , where
1/p Φ(ω)p dμ(ω) , Ω 1/p := sup Φ(ω)xp dμ(ω) ,
ΦLp(μ;L (X,Y )) := ΦLps (μ;L (X,Y ))
x 1
ΦLpw (μ;L (X,Y )) := sup
Ω
sup
x 1 y ∗ 1
1/p | Φ(ω)x, y ∗ |p dμ(ω) ,
Ω
with the obvious modifications for p = ∞. As special cases we write Lp (μ; X) := Lp (μ; L (K, X)) = Lps (μ; L (K, X)) and Lpw (μ; X) := Lpw (μ; L (K, X)). Note that all these definitions agree with the usual ones.
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Let us recall some spaces of vector measures that are used in the sequel. The reader is referred to [3] and [4] for the concepts needed in this paper. Fix 1 p ∞ and let E be a Banach space. We denote by V p (μ; E) the Banach space of all vector measures F : Σ → E for which 1 (1A ⊗ F (A)) < ∞, F V p (μ;E) := sup μ(A) Lp (μ;E) π∈P(Ω) A∈π
where P(Ω) stands for the collection of all finite partitions of Ω into disjoint sets of strictly positive μ-measure. Similarly we denote by Vwp (μ; E) the Banach spaces of all vector measures F : Σ → E for which 1 F Vwp (μ;E) := sup (1A ⊗ F (A)) p < ∞. μ(A) Lw (μ;E) π∈P(Ω) A∈π
In both definitions of the norm we make the obvious modification for p = ∞. Note that F V 1 (μ;E) and F Vw1 (μ;E) equal the variation and semivariation of F with respect to μ, respectively. It is well known that for 1 p < ∞ and 1/p + 1/p = 1 one has a natural isometric isomorphism
(Lp (μ; E))∗ V p (μ; E ∗ ). We now concentrate on the case E = L (X, Y ). For each Φ ∈ L1 (μ; L (X, Y )) one may define a vector measure F : Σ → L (X, Y ) by F (A) := Φ dμ A
which satisfies F V 1 (μ;L (X,Y )) = ΦL1 (μ;L (X,Y )) . In the next proposition we extend this definition to functions Φ ∈ Lps (μ; L (X, Y )), 1 < p < ∞. The case p = 1 will be addressed in Remark 3.3 and Theorem 3.8. Proposition 3.2. Assume that Φ ∈ Lps (μ; L (X, Y )) for some 1 < p < ∞. Define F : Σ → L (X, Y ) by F (A)x := Φ(ω)x dμ(ω), x ∈ X. A
Then F is an L (X, Y )-valued vector measure and, for any q ∈ [1, p], one has F Vwq (μ;L (X,Y )) ΦLqs (μ;L (X,Y )) . Proof. Let us first prove that F is countably additive. Let (An )n1 be a sequence of ' pairwise disjoint sets in Σ and let A = n1 An . Put T := F (A) and Tn := F (An ).
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Then, N N Tn = sup T x − Tn x T − n=1
x =1
n=1
= sup ' x =1
sup
x =1
nN +1
An
1/p Φ(ω)xp dμ(ω) μ
Ω
ΦLp(μ;L (X,Y )) Hence T =
Φ(ω)x dμ(ω) )
1/p An
nN +1
μ
)
1/p An
.
nN +1
n1
Tn in L (X, Y ). Next,
| F (A), e∗ |q 1/q q−1 π∈P(Ω) e∗ =1 A∈π (μ(A)) # * F (A) +# # ∗ # = sup sup sup αA , e # # (μ(A))1/q π∈P(Ω) e∗ =1 (αA ) q =1 A∈π F (A) sup αA = sup (μ(A))1/q L (X,Y ) π∈P(Ω) (αA ) q =1 A∈π F (A) = sup sup sup αA x (μ(A))1/q π∈P(Ω) (αA ) q =1 x =1 A∈π 1A = sup Φ(ω)x dμ(ω) sup sup αA 1/q (μ(A)) π∈P(Ω) (αA ) q =1 x =1 Ω A∈π 1/q sup Φ(ω)xq dμ(ω)
F Vwq (μ;L (X,Y )) = sup
x =1
sup
Ω
= ΦLqs (μ;L (X,Y )) .
Remark 3.3. The same results holds for functions Φ ∈ L1s (μ; L (X, Y )) provided the family {ω → Φ(ω)x : x ∈ BX } is equi-integrable in L1 (μ; X). The next definition introduces a new class of Banach spaces intermediate between Lp (μ; L (X, Y )) and Lps (μ; L (X, Y )). Definition 3.4. For 1 p ∞ we consider the Banach space Lp [μ; L (X, Y )] := {Φ ∈ Ms (μ; L (X, Y )) : ΦLp[μ;L (X,Y )] < ∞}, where
ΦLp[μ;L (X,Y )] :=
Φ(ω)f (ω) dμ(ω).
sup f Lp (μ;X) =1
Ω
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It is clear that Lp (μ; L (X, Y )) → Lp [μ; L (X, Y )] → Lps (μ; L (X, Y )) with contractive inclusion mappings. Using these spaces we can prove the following improvement of Proposition 3.2. Theorem 3.5. Let 1 < p < ∞. Then Lp [μ; L (X, Y )] → V p (μ; L (X, Y )) and the inclusion mapping is contractive. Proof. Using the inclusion into Lp [μ; L (X, Y )] → Lps (μ; L (X, Y )), from Proposition 3.2 we see that F (A)x := A Φ(ω)x dμ(ω) defines a vector measure F : Σ → L (X, Y ). Now, if π ∈ P(Ω), then for ε > 0 and each A ∈ π there exist xA ∈ BX and ∗ yA ∈ BY ∗ so that #, -#p ε # p ∗ # . Φ(ω)xA dμ(ω), yA F (A) < # # + card(π) A Hence, F (A)p (μ(A))p−1 A∈π #, -#p 1 # ∗ # Φ(ω)xA dμ(ω), yA # # +ε p−1 (μ(A)) A A∈π , -p 1 ∗ Φ(ω)x dμ(ω), β y +ε sup A A A 1/p (βA ) p =1 A∈π (μ(A)) A , p βA xA ∗ Φ(ω) dμ(ω) 1A ⊗ , 1 ⊗ y +ε sup A A (μ(A))1/p A∈π (βA ) p =1 Ω A∈π p βA xA dμ(ω) 1A ⊗ +ε sup Φ(ω) (μ(A))1/p (βA ) p =1 Ω A∈π p Φ(ω)f (ω) dμ(ω) + ε sup f Lp (μ;X) =1
Ω
ΦpLp [μ;L (X,Y )]
+ ε.
Since ε > 0 was arbitrary, this gives the result.
For 1 p, q < ∞ we define Mult[Lp (μ; X), Lq (μ; Y )] to be the linear space of all Φ ∈ Ms (μ; L (X, Y )) such that ω → Φ(ω)f (ω) belongs to Lq (μ; Y ) for all f ∈ Lp (μ; X). By a closed graph argument the linear operator
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MΦ : f → Φf is bounded, and the space Mult[Lp (μ; X), Lq (μ; Y )] is a Banach space with respect to the norm ΦMult[Lp (μ;X),Lq (μ;Y )] := MΦ L (Lp (μ;X),Lq (μ;Y )) . We refer to [5] for further details and and some results on spaces of multipliers between different spaces of vector-valued functions, extending those proved in [1] for sequence spaces. Theorem 3.6. Let X and Y be Banach spaces and let 1 q < p < ∞. We have a natural isometric isomorphism Mult[Lp (μ; X), Lq (μ; Y )] Lr [μ; L (X, Y )], where 1/r = 1/q − 1/p. Proof. The case q = 1 corresponds to r = p and the result is just the definition of the space Lp [μ; L (X, Y )]. Assume 1 < q < p and Φ ∈ Lr [μ; L (X, Y )]. Let f ∈ Lp (μ; X). Then for any φ ∈ Lq (μ) we have that ω → f (ω)φ(ω) belongs to Lr (μ; X). Hence Φ(ω)f (ω)|φ(ω)| dμ(ω) ΦLr [μ;L (X,Y )] φLq (μ) f Lp(μ;X) . Ω
Taking the supremum over the unit ball of Lq (μ) the first inclusion is achieved. Conversely, let Φ ∈ Mult[Lp (μ; X), Lq (μ; Y )]. Let g ∈ Lr (μ; X), and choose ψ ∈ Lq (μ) and f ∈ Lp (μ; X) in such a way that g = ψf and gLr (μ;X) = ψLq (μ) f Lp(μ;X) . Now observe that Φ(ω)g(ω) = ψ(ω)Φ(ω)f (ω) ∈ L1 (μ; Y ) and Φ(ω)g(ω) dμ(ω) ψLq (μ) ΦMult[Lp (μ;X),Lq (μ;Y )] f Lp(μ;X) .
Ω
The next result establishes a link with the notion of strong μ-measurability. Proposition 3.7. Let X be a Banach space, 1 p ∞, and let Φ ∈ Lp [μ; L (X, Y )] be strongly μ-normable. Then ω → Φ(ω) belongs to Lp (μ) and 1/p Φ(ω)p dμ(ω) ΦLp [μ;L (X,Y )] . Ω
Proof. By assumption, for any ε > 0 there exists f ε ∈ M (μ; X) such that for μ-almost all ω ∈ Ω one has f ε (ω) 1 and Φ(ω) Φ(ω)(f ε (ω)) + ε. If εn ↓ 0, then for μ-almost all ω ∈ Ω Φ(ω) = lim Φ(ω)f εn (ω). n→∞
The strong μ-measurability of ω → Φ(ω)x for all x ∈ X implies the strong μmeasurability of the functions ω → Φ(ω)f εn (ω). It follows that ω → Φ(ω) is μ-measurable.
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Let φ ∈ Lp (μ) and consider ω → φ(ω)f ε (ω) ∈ Lp (μ; X). Then Φ(ω)|φ(ω)| dμ(ω) Φ(ω)(φ(ω)f ε (ω)) dμ(ω) + εφL1 (μ) Ω
Ω
ΦLp[μ;L (X,Y )] φLp (μ) + εφL1 (μ) .
Since ε > 0 was arbitrary, this gives the result.
By invoking Proposition 2.2 we shall now deduce some further results under the assumption that the space X is separable. The first should be compared the remarks preceding Proposition 3.2 (where functions Φ ∈ L1 (μ; L (X, Y )) are considered) and Remark 3.3 (where functions Φ ∈ L1s (μ; L (X, Y )) are considered). Theorem 3.8. Let X be a separable Banach space and let Φ ∈ L1 [μ; L (X, Y )] be given. Define F : Σ → L (X, Y ) by F (A)x := Φ(ω)x dμ(ω), x ∈ X. A
Then F is an L (X, Y )-valued vector measure and F V 1 (μ;L (X,Y )) ΦL1 [μ;L (X,Y )] . Proof. First we prove that F is countably ' additive. Let (An )n1 be a sequence of pairwise disjoint sets in Σ and let A = n1 An . Put T := F (A) and Tn := F (An ). Combining Proposition 2.2 and Proposition 3.7 one obtains that Φ ∈ L1 (μ). Hence, N N Tn = sup T x − Tn x T − n=1
x =1
n=1
= sup ' x =1
nN +1
Φ(ω)x dμ(ω)
An
Φ(ω) dμ(ω).
'
An
Hence T = n1 Tn in L (X, Y ). Next, using that F (A) A Φ(ω) dμ(ω), from Proposition 3.7 we conclude that F (A) ΦL1 [μ;L (X,Y )] . F V 1 (μ;L (X,Y )) = sup nN +1
π∈P(Ω) A∈π
Our final result extends the factorization result that was used in the proof of Theorem 3.6. Theorem 3.9. Let 1 p1 , p2 , p3 < ∞ satisfy 1/p1 = 1/p2 +1/p3 and let X be a separable Banach space. A function Φ ∈ Ms (μ; L (X, Y )) belongs to Lp1 [μ; L (X, Y )] if and only if Φ = ψΨ for suitable functions ψ ∈ Lp2 (μ) and Ψ ∈ Lp3 [μ; L (X, Y )]. In this situation we may choose ψ and Ψ in such a way that ΦLp1 [μ;L (X,Y )] = ψLp2 (μ) ΨLp3 [μ;L (X,Y )] .
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Proof. To prove the ‘if’ part let Φ ∈ Lp1 [μ; L (X, Y )]. Using Proposition 3.7 together with Proposition 2.2 one has that Φ ∈ Lp1 (μ). Put $ Φ(t) if Φ(t) = 0, Φ(t)p1 /p3 Φ(t) p1 /p2 ψ(t) := Φ(t) , Ψ(t) := 0 if Φ(t) = 0.
Clearly ψ ∈ Lp2 (μ) and Ψ ∈ Lp3 [μ; L (X, Y )]. Now for each g ∈ Lp3 (μ; X), invoking Proposition 3.1, one has that Ψg ∈ M (μ, Y ) and Ψ(t)g(t) Φ(t)p1 /p3 g(t). Hence the right-hand side defines a function in L1 (μ) and therefore Ψg ∈ L1 (μ, Y ). The above decomposition satisfies the required identity for the norms. To prove the ‘only if’ part let ψ ∈ Lp2 (μ) and Ψ ∈ Lp3 [μ; L (X, Y )] be given. For each f ∈ Lp1 (μ; X) we have ψf ∈ Lp3 (μ; X). Hence Ψ(ψf ) ∈ L1 (μ; Y ).
References [1] J.L. Arregui, O. Blasco, (p, q)-Summing sequences of operators. Quaest. Math. 26 (2003), no. 4, 441–452. [2] Q. Bu and P.-K. Lin, Radon-Nikodym property for the projective tensor product on K¨ othe function spaces. J. Math. Anal. Appl. 293 (2004), no. 1, 149–159. [3] J. Diestel, J.J. Uhl, Vector measures. Mathematical Surveys 15, Amer. Math. Soc., Providence (1977). [4] N. Dinculeanu Vector measures. Pergamon Press, New York (1967). [5] J.H. Fourie and I.M. Schoeman, Operator-valued integral multiplier functions. Quaest. Math. 29 (2006), no. 4, 407–426. Oscar Blasco University of Valencia Departamento de An´ alisis Matem´ atico E-46100 Burjassot, Valencia, Spain e-mail: [email protected] Jan van Neerven Delft University of Technology Delft Institute of Applied Mathematics P.O. Box 5031 NL-2600 GA Delft, The Netherlands e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 79–87 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Defining Limits by Means of Integrals Antonio Boccuto and Domenico Candeloro Abstract. A particular notion of limit is introduced, for Riesz space-valued functions. The definition depends on certain ideals of subsets of the domain. It is shown that, according with our definition, every bounded function with values in a Dedekind complete Riesz space admits limit with respect to any maximal ideal. Mathematics Subject Classification (2000). Primary 28B15; Secondary 46G10. Keywords. Riesz space, ideal convergence, integral.
1. Introduction In this paper a definition of convergence with respect to ideals is introduced. To every ideal I of subsets of an abstract non-empty set T a dual filter is associated in a natural way: it is the family of the complements of all elements of I. If the dual filter is an ultrafilter, then the involved ideal is maximal, and vice versa. It is shown that every bounded map, taking values in any Dedekind complete Riesz space, admits a “limit” with respect to any maximal ideal. This result can be achieved in the real-valued case by simply integrating with respect to a suitable “ultrafilter measure”, since all bounded functions are integrable. In the general case, the goal is obtained by using the powerful tools of the Chojnacki integral (see [3]) and the representation of Riesz spaces as ideals of suitable spaces of continuous functions (Maeda-Ogasawara-Vulikh theorem, [7, 8, 9]). Applications are given in finding extensions of finitely additive measures.
2. Preliminaries We begin with introducing the following basic notions (see also [4, 5, 6]). Definition 2.1. Let I denote any fixed ideal of subsets of an abstract set T . We say that I is admissible if it contains all finite subsets of T . In particular, the ideal consisting precisely of the finite subsets of T will be denoted by If in , and of course
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is the minimal admissible ideal. So, any admissible ideal must contain If in . On the opposite side, I is called maximal if no proper ideal in T strictly contains I. Since the family of all complements of elements from an ideal I forms a filter F , called the dual filter of I, we can say that I is admissible if and only if, for every element t0 ∈ T , the set T \ {t0 } belongs to the dual filter of I: filters of this kind are also called free. On the other hand, I is maximal if and only if its dual filter U is an ultrafilter: any subset of T either belongs to I or to U. Thanks to the Axiom of Choice, it is well known that any filter is contained in some ultrafilter (and therefore any ideal is contained in a maximal ideal). We will deal with two-valued additive set functions P : P(T ) → {0, 1} with P (T ) = 1. Such maps are also called ultrafilter measures: indeed, the family of all sets U such that P (U ) = 1 is an ultrafilter in T and consequently the family of all P -null sets is a maximal ideal. Assume now that I is any admissible ideal in an abstract set T , and let f : T → R be any real-valued mapping. We say that f is I-convergent to an element x ∈ R, if for every ε > 0 the set {t ∈ T : |f (t) − x| > ε} belongs to I. When this is the case, we also write x = I − limt∈T f (t). It is well known that the I-limit is unique (if it exists), and enjoys all linearity and monotonicity properties of usual limits, see [5]. However we mainly will be concerned with ideal convergence in the case of maximal ideals. Indeed, we have (see also [1]): Proposition 2.2. Let P : P(T ) → {0, 1} be an ultrafilter measure, and f : T → R any bounded function. Then, if I denotes the ideal of P -null sets, the function f has limit with respect to I, and we have I − lim f (t) = f (t) dP. t∈T
T
Conversely, given any maximal ideal I of subsets of T , any bounded mapping f : T → R has limit w.r.t. I, and this limit is nothing but the integral of f with respect to a suitable ultrafilter measure. Proof. Let us denote by U the dual filter of I. Then it is easy to see that sup inf f (t) = inf sup f (t)
U ∈U t∈U
U ∈U t∈U
and that the common value is the integral T f (t) dP : we shall denote by J this quantity. Now, fix arbitrarily ε > 0, and set Aε := {t ∈ T : |f (t)−J| > ε}. We claim that Aε belongs to I: indeed, if Aε ∈ U we should have either inf U∈U supt∈U ≤ J −ε or supU ∈U inf t∈U ≥ J + ε, and both these cases are impossible. Hence the first part is proved. Conversely, given any maximal ideal I in T and any bounded function f : T → R, the I-limit of f is nothing but the integral w.r.t. the (unique) ultrafilter measure whose null sets are precisely the elements of I.
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In the sequel, we shall extend our notion of limit to the case of Riesz spacevalued mappings, and show that any bounded function taking values in every Dedekind complete Riesz space has limit (at least in a weak sense) when I is maximal.
3. I-convergence in Riesz spaces From now on we shall assume that R is a Dedekind complete Riesz space. Our purpose here is to show that, if f : T → R is any bounded map, then, for every maximal ideal I in T , f has I-limit in R, at least in a weak sense. We start with the following definition. Definition 3.1. Let R be any Dedekind complete Riesz space, and f : T → R be any map. Given an ideal I of subsets of T , we say that f has I-limit in R if there exist an element l in R and a decreasing net (rλ )λ in R, with inf λ rλ = 0 and such that for each λ the set {t ∈ T : |f (t) − l| ≤ rλ } is the complement of some element N ∈ I. In other terms, we have I − lim f (t) = l if and only if inf sup f (t) = l = sup inf f (t),
U ∈U t∈U
U ∈U t∈U
where U is the dual filter of I. We shall say that a Dedekind complete Riesz space R has the strong limit property if, for any abstract set T , any bounded mapping f : T → R, and any maximal ideal I in T , there exists in R the I-limit of f . We first observe that the space R has the strong limit property, thanks to 2.2. From this, it clearly follows that the space RD , for any discrete space D, has the strong limit property, when endowed with the natural order and algebraic structure. We shall now give a further example. Let us assume that an abstract discrete space D is fixed, and denote by C(βD) the space of all continuous mappings ψ : βD → R, where βD denotes the ˇ Stone-Cech compactification of D. The space C(βD) has a natural ordering, and is stable under arbitrary suprema and infima of bounded subsets. Given any bounded map ψ : D → R, we denote by ψ the unique continuous extension of ψ to βD. Moreover, given a bounded family (ψh )h∈H of mappings from D to R, the notation suph ψh always means the pointwise supremum; in case the. mappings ψh are elements from C(βD), the lattice supremum will be denoted by h∈H ψh . Similar notations will be adopted for infima.
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Now, it is not difficult to check that, given any uniformly bounded family (ψh )h∈H of functions defined on D and taking values in R, we always have
/ 0 ψ h (d) = sup ψh (d), ψ h (d) = inf ψh (d), h∈H
h∈H
h∈H
h∈H
for all d ∈ D. Fix any abstract set T , together with an ultrafilter U of subsets of T , and choose any bounded mapping f : T → C(βD). We shall state the following result, whose proof now is straightforward. Lemma 3.2. Let us set: J1 :=
0 /
f (t),
/ 0
J2 :=
U ∈U t∈U
f (t),
U∈U t∈U
where the involved suprema and infima are taken in C(βD). Then we have J1 = J2 . The assertion of Lemma 3.2 is precisely that C(βD) has the strong limit property. We now turn to the concept of weak limit. To this aim, we need some definitions, concerning embeddings of a Dedekind complete Riesz space R. Definition 3.3. We say that R is embedded in another Dedekind complete Riesz space R∗ if there exist a one-to-one Riesz homomorphism (called embedding) j : R → R∗ such that j(R) is a Dedekind complete subspace of R∗ , and an onto homomorphism π : R∗ → R (called projection) such that π ◦ j is the identity map on R. When this happens, we also say that R∗ is an extension of R. In general, embeddings reduce the gap existing between the two quantities 0 / / 0 f (t), and f (t), F ∈F t∈F
F ∈F t∈F
whenever f : T → R is a bounded map, and F any filter in T : indeed, we have Theorem 3.4. Assume that f : T → R is a bounded map, and F is any filter in T . If R∗ is any extension of R, with embedding j and projection π, then we always have
/ 0 / 0 f (t) ≤ π j(f (t)) F ∈F t∈F
≤π
0 / F ∈F t∈F
F ∈F t∈F
j(f (t))
≤
0 /
f (t).
F ∈F t∈F
The proof is straightforward. Let us turn now to the definition of maximal extension.
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Definition 3.5. Let R be any Dedekind complete Riesz space, and R∗ any extension of R. We say that the extension of R into R∗ is maximal if, for every abstract set T , every ultrafilter U in T , and any bounded mapping f : T → R, it holds:
0 / / 0 π j(f (t)) = π j(f (t)) . U ∈U t∈U
U∈U t∈U
If this happens, then the common value will be called weak limit of f with respect to the dual ideal I of U. In particular, if R has an extension R∗ with the strong limit property, then R is obviously a maximal extension. We shall see later that this is the case, for example, if R has a strong unit, i.e., a strictly positive element u such that for every x ∈ R there is a positive real number λ > 0 for which |x| ≤ λu. More generally, given a Riesz space R, we say that a bounded mapping f : T → R has weak limit with respect to some ideal I in T , if the range of f is contained in a Dedekind complete space R0 with a maximal extension R0∗ , and
/ 0 0 / j(f (t)) = π j(f (t)) , π ∗
F ∈F t∈F
F ∈F t∈F
where F is the dual filter of I. In this case, the weak limit, i.e., the common value, is an element l of R0 , and we write l := (w) − I − lim f (t). Thanks to 3.4, we can see that, in case f : T → R is a bounded map, and I is any ideal in T , the existence of the strong I-limit implies that any weak limit coincides with it. Moreover, it is not difficult to check that the weak limit, when existing, enjoys the usual linearity properties, provided it is performed always with respect to the same maximal extension. We now turn to show that a weak I-limit always exists, as soon as f : T → R is a bounded mapping and I is a maximal ideal in T . To this aim, we recall the following version of the Maeda-Ogasawara-Vulikh Theorem (see [7]; [9], Theorems V.3.1, p. 131 and V.4.2, p. 138). Theorem 3.6. Every Dedekind complete Riesz space R is algebraically and lattice ( Ω : f is continuous, and isomorphic to an order dense ideal of C∞ (Ω) = {f ∈ R {ω ∈ Ω : |f (ω)| = +∞} is nowhere dense in Ω}, where Ω is a suitable compact Hausdorff extremely disconnected topological space. Moreover, if R has a strong order unit, then it is algebraically and lattice isomorphic to C(Ω) = {f ∈ RΩ : f is continuous}. We also recall the following well-known result, which in turn is related to the Maeda-Ogasawara-Vulikh representation theorem (see [3, 8]).
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Theorem 3.7. Let Ω be any Hausdorff, compact, extremely disconnected space, and denote by D the support set of Ω, endowed with the discrete topology. Then, Ω ˇ (with its original topology) is a subspace of the Stone-Cech compactification βD and there exists a continuous onto mapping r : βD → Ω, (retract) such that r|Ω is the identity map. Thanks to the previous theorems, we can deduce that, if R is a Dedekind complete Riesz space endowed with a strong order unit, then R admits a maximal extension. Indeed, we can view R as the space C(Ω), for a suitable Stone space Ω, and C(Ω) is embedded in C(βD), according with Theorem 3.7: indeed, for any mapping φ ∈ C(Ω) we can define j(φ) : βD → R as follows: j(φ)(ξ) = φ(r(ξ)), for all ξ ∈ βD, and also we can define a projection π : C(βD) → C(Ω) as: π(ψ) := ψ|Ω for all ψ ∈ C(βD). As already observed in Lemma 3.2, the space C(βD) has the strong limit property, hence the claim is proved. We now turn to the general case of a bounded map f , taking values in a Dedekind complete Riesz space R. Let us fix a Dedekind complete Riesz space R, and fix any abstract set T , together with a maximal ideal I of subsets of T , and its dual ultrafilter U. Theorem 3.8. Let f : T → R be any bounded mapping. Then there exists a weak I-limit of f , in the sense of Definition 3.5. Proof. We note that, if f : T → R is bounded and e is the least upper bound of the range of |f |, then the range of f is contained in the vector lattice V [e] := {b ∈ R : there exists λ > 0 with |b| ≤ λ e}. Since R is Dedekind complete, V [e] is too, and hence V [e] is a Dedekind complete Riesz space with a strong order unit. So, for the previous remark, V [e] has a maximal extension, and this shows that f has weak limit in V [e], hence in R. More precisely, as soon as f : T → R is a bounded map, and e is any positive element of R dominating |f |, we can choose the space C(βD) as an extension of V [e], and define f0 (t)(ξ) = f (t)(r(ξ)) for all t ∈ T and ξ ∈ βD; then, denoting by l the strong limit of f0 , the weak limit of f is π(l), i.e., l|Ω , where Ω is the Stone space such that V [e] is isomorphic to C(Ω).
4. Applications In this section we shall prove that I-limits can be used to extend measures. Definition 4.1. Given an abstract non-empty set T , and fixed a non-trivial algebra A of subsets of T , let us denote by D the family of all finite tagged partitions of T into non-empty subsets from A: this means that every element Π ∈ D is a finite collection of pairwise disjoint elements from A, say A1 , . . . , Ak , (which we shall call intervals), whose union is T , with attached a corresponding finite number of elements t1 , . . . , tk from T , such that ti ∈ Ai for all i = 1, . . . , k.
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Fixed two such partitions, Π1 and Π2 , we say that Π2 refines Π1 if any interval from Π2 is a subset of some interval from Π1 (without any condition on the attached points). Fixed any partition Π ∈ D, we denote by D(Π) the family of all partitions which refine Π. Now, if we consider D as an abstract set, the families D(Π) form, as Π runs in D, a filter basis in D: this means that, if we collect all the families (of subsets of D), which contain some family of the type D(Π), they form a filter in D. This filter is called the refinement filter of D, based on A. Assume now that a non-empty abstract set T is fixed, with an algebra A of subsets of T . Let us suppose that a bounded finitely additive measure m : A → R is given. Then we have: Proposition 4.2. There exists at least one finitely additive measure m ( : P(T ) → R such that m ( |A = m. Proof. For any Π ∈ D, say Π = {(A1 , t1 ), (A2 , t2 ), . . . , (Ak , tk )}, define Σ(f, Π) =
k
f (tj )m(Aj ),
j=1
for all bounded maps f : T → R. Now, let U be any ultrafilter on D containing the refinement filter. Denoting by e any strong order unit in R such that the range of m is contained in V [e], it is clear that Σ(f, ·) is a bounded map from D to V [e], and hence has weak I-limit, with respect to the dual ideal I of U: let us put J (f ) := (w) − I − lim Σ(f, Π), Π∈D
for every bounded map f : T → R, and with respect to the same maximal extension of V [e]. We claim that the mapping m(A) ( := J (1A ), defined on all subsets A of T , is a finitely additive extension of m. First of all, it is easy to show that J is a linear mapping. So, it remains to prove that J (1A ) = m(A) whenever A ∈ A. Indeed, let us fix A ∈ A, and set f = 1A . Let now ΠA be the partition whose intervals are A and T \ A. For any Π ∈ D(ΠA ) we have clearly Σ(f, Π) = m(A), from which it is easy to deduce that J (f ) = m(A), i.e., the claim. This concludes the proof. A similar argument leads to prove the following result, whose meaning is that, when a finitely additive measure is defined in the whole of P(T ), then every bounded function on T admits integral. Corollary 4.3. Let P : P(T ) → R be any positive finitely additive measure, and let f : T → R be any bounded map. Now, consider an ultrafilter U containing the refinement filter like in the previous proof, with A = P(T ). Denoted by I the dual ideal of U, for each bounded map f : T → R let us define f dP := (w) − I − lim Σ(f, Π), T
where Σ(f, Π) has the usual meaning.
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Then, f → T f dP is a linear monotonic functional, which coincides with the elementary integral, whenever f has finite range. Further applications concern the existence of invariant measures: for example, we can see that, given any non-empty set T and any map τ : T → T , it is always possible to construct a finitely additive τ -invariant measure μ : P(T ) → R. More precisely, we have the following result. Theorem 4.4. Let T be any abstract non-empty set, and τ : T → T be a mapping. Assume that m : A → R is any τ -invariant bounded finitely additive measure, defined in some algebra A of subsets of T . (This means that, for all A ∈ A, we have τ −1 (A) ∈ A and m(A) = m(τ −1 (A)).) Then there exists a bounded finitely additive measure μ : P(T ) → R, extending m and still invariant with respect to τ , i.e., μ(A) = μ(τ −1 (A)), for all A ⊂ T . Proof. Thanks to 4.2, we can find a bounded finitely additive extension m0 of m to the whole of P(T ). Then define a sequence (mn )n of measures, by induction: mn (A) = mn−1 (τ −1 (A)), for all A ⊂ T, n ≥ 1. Clearly, all measures mn coincide with m in A. Then set: μn := n1 ni=1 mi , for all n ≥ 1. If e denotes any upper bound for the range of |m0 |, then it is clear that each measure μn has the same upper bound, and that |μn (A) − μn (τ −1 (A))| ≤ 2e n for all A ⊂ T . Now choose any maximal admissible ideal I in the set N, and any maximal extension of V [e], in order to construct a (pointwise) weak I-limit μ0 of the sequence (μn )n . Then μ0 is the required extension: indeed, by construction it is additive and extends m, so the only property we have to prove is invariance; but for all sets A ∈ A we have |μ(A) − μ(τ −1 (A))| = | (w) − I − lim μn (A) − (w) − I − lim μn (τ −1 (A)) | 2e = 0, ≤ lim n n which concludes the proof.
References [1] E. Barone, A. Giannone, R. Scozzafava, On some aspects of the theory and applications of finitely additive probability measures. Pubbl. Istit. Mat. Appl. Fac. Ingr. Univ. Stud. Roma, Quaderno No. 16 (1980), 43–53. [2] A. Boccuto, D. Candeloro, Sandwich theorems, extension principle and amenability. Atti Sem. Mat. Fis. Univ. Modena 42 (1994), 257–271. [3] W. Chojnacki, Sur un th´eor`eme de Day, un th´eor`eme de Mazur-Orlicz et une g´en´eralisation de quelques th´eor`emes de Silverman. Colloq. Math. 50 (1986), 257– 262.
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[4] J.S. Connor, Two-valued measures and summability. Analysis 10 (1990), 373–385. ˇ at, W. Wilczy´ [5] P. Kostyrko, T. Sal´ nski, I-convergence. Real Anal. Exch. 26 (2000/ 2001), 669–685. [6] B.K. Lahiri, P. Das, I and I ∗ -convergence in topological spaces. Math. Bohemica 130 (2005), 153–160. [7] F. Maeda, T. Ogasawara, Representation of Vector Lattices. J. Sci. Hiroshima Univ. Ser. A 12 (1942), 17–35 (Japanese). [8] J. Rainwater, A note on projective resolutions. Proc. Am. Math. Soc. 10 (1959), 734–735. [9] B.Z. Vulikh, Introduction to the theory of partially ordered spaces. Wolters-Noordhoff Sci. Publ., Groningen, 1967. Antonio Boccuto and Domenico Candeloro Dipartimento di Matematica e Informatica via Vanvitelli, 1 I-06123 Perugia, Italy e-mail: [email protected] [email protected]
Operator Theory: Advances and Applications, Vol. 201, 89–97 c 2009 Birkh¨ auser Verlag Basel/Switzerland
A First Return Examination of Vector-valued Integrals Donatella Bongiorno Abstract. We prove that for each Bochner integrable function f there exists a trajectory yielding the Bochner integral of f , and that on infinite-dimensional Banach spaces there exist Pettis integrable functions f such that no trajectory yields the Pettis integral of f . Mathematics Subject Classification (2000). Primary 28B05, 48G10; Secondary 26A42. Keywords. Trajectory, first return integral, Bochner integral, Pettis integral, McShane integral.
1. Introduction Let X be a Banach space. It is well known that the Bochner integral of a function f : [0, 1] → X can be obtained as a limit of suitable Riemann sums, in particular it is McShane integrable (see [7, Theorem 5.1.5]). Unfortunately the partitions used in the those theories are not completely free, as in the Riemann integral. In view to cover this gap, U.B. Darji and M.J. Evans [1] proved that if f : [0, 1]n → R is Lebesgue integrable then it is possible to find a dense sequence t (in the sequel called trajectory) such that for each ε > 0 there exists a positive constant δ with 1 f (r(t, J)) |J| − f < ε, (1) 0 x∈J∈P
for each partition P of [0, 1] with mesh(P) = sup{|J| : J ∈ P} < δ. Here r(t, J) denotes the first point of t belonging to J. In this paper we investigate the possible extension of Darji-Evans theorem to vector-valued functions. For simplicity we restrict our attention to the case of functions of one real variable.
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It is well known that there are three possible distinct extensions of the Lebesgue integral, in case the range space is an infinite-dimensional Banach space; namely the Bochner integral, the Pettis integral and the McShane integral. Definition. Let X be a Banach space and let f : [0, 1] → X be Bochner (resp. Pettis or McShane) integrable with integral A. We say that a trajectory t yields the integral A if for each ε > 0 there is a constant δ > 0 such that f (r(t, J))|J| − A < ε, (2) J∈P
for each partition P of [0, 1] with mesh(P) < δ. According with this definition the mentioned Darji-Evans theorem in [0, 1] can be reformulated as follows. Theorem. If f : [0, 1] → R is Lebesgue integrable, then there exists a trajectory t yielding the Lebesgue integral of f . In this paper we will prove that this theorem holds also for Bochner integrable functions (Theorem 1) and cannot holds for Pettis and McShane integrable functions (Theorems 2 and 3, respectively). We also prove that, in the case that the dual of the range space is w∗ -separable, a class of Pettis integrable functions (not Bochner integrable) for which the Darji-Evans theorem holds is the class of scalarly null functions (Theorem 4).
2. Preliminaries The set of all real (resp. natural) numbers is denoted by R (resp. N). The Lebesgue measure of a subset E of R is denoted by |E|. We write a.e. for almost everywhere. The word measurable is always refereed to the Lebesgue measure. A sequence t ≡ {tn } of distinct points of [0, 1], dense in [0, 1], is called trajectory. Given a trajectory t and an interval J ⊂ [0, 1] we denote by r(t, J) the first element of t that belongs to J. We call partition of [0, 1]' any finite collection of non-overlapping compact n intervals J1 , . . . , Jn such that i=1 Ji = [0, 1]. Given a partition P = {J1 , . . . , Jn } we set mesh(P)=supi |Ji |. Throughout this paper we denote by X a fixed Banach space, by · its norm, by B(x, r) the ball with center x and radius r, and by X ∗ the topological dual of X. We recall that a function f : [0, 1] → X is said to be strongly measurable if there exists a sequence of simple functions {fn } such that fn (x) − f (x) → 0, a.e. in [0, 1]. A strongly measurable function f : [0, 1] → X is said to be Bochner integrable if there exists a sequence of simple functions {fn } such that 1 lim fn (x) − f (x) dx = 0. n
0
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1 1 In this case limn 0 fn dx is called the Bochner integral of f on [0, 1], where 0 fn dx is the Bochner integral of the simple function fn defined in the obvious way. It is known that a strongly measurable function f : [0, 1] → X is Bochner 1 integrable if and only if 0 f dx < ∞ (see [2, Theorem 2, §2, Chapter II]). A function f : [0, 1] → X is said to be weakly measurable if for each x∗ ∈ X ∗ the scalar function x∗ f is measurable. A weakly measurable function f : [0, 1] → X is said to be Pettis integrable if for each measurable set E ⊂ [0, 1] there exists xE ∈ X such that for all x∗ ∈ X ∗ the scalar function x∗ f is Lebesgue integrable and ∗ x (xE ) = x∗ f dx. E
In this case xE is called the Pettis integral of f on E. Each Bochner integrable function is McShane integrable and each McShane integrable function is Pettis integrable with the same value of the integrals (see [7, Theorems 5.1.2 and 6.1.2]). Pettis and McShane integrals coincide if X is super reflexive and if X = c0 (Γ), where Γ is a generic set (see [3]). A function f : [0, 1] → X is said to be scalarly negligible if x∗ f = 0 a.e. in [0, 1]. Each scalarly negligible function is Pettis integrable with Pettis integral equal to zero.
3. Bochner integrable functions Theorem 1. If f : [0, 1] → X is Bochner integrable, then there exists a trajectory t in [0, 1] yielding the Bochner integral of f . Within the proof of this theorem we need the following variant of the Lusin theorem: (RLT ) If F ⊂ [0, 1] is measurable, f : [0, 1] → X is strongly measurable and ε > 0 then there is a closed nowhere dense set E ⊂ F such that |F \ E| < ε and f |E is continuous. Proof of (RLT). By the Lusin theorem for strongly measurable functions (see [4, Theorem 3, Chapter III, §15, No. 8]) there exists a closed set S ⊂ F such that |F \ S| < ε/2 and f |S is continuous. If the interior of S is empty we take E = S. Otherwise we define S1 ⊂ S, removing a countable dense subset in the interior of S, and we use the regularity of the Lebesgue measure to find a closed set E ⊂ S1 such that |S1 \ E| < ε/2. So E is a nowhere closed subset of F with |F \ E| ≤ |F \ S| + |S \ S1 | < ε and f |E is continuous. Proof of Theorem 1. In this proof we use some ideas from [5]. For each n ∈ N let Fn = {t ∈ [0, 1] : f (t) ≤ n}. We start by finding inductively a sequence {En } of closed nowhere dense sets such that En ⊂ En+1 , En ⊂ Fn , |Fn \ En | < 1/2n , and f |En is continuous, for each n ∈ N. To find a nowhere dense closed set E1 ⊂ F1 such that |F1 \ E1 | < 1/2, and f |E1 is continuous, we apply (RLT ) to the strongly measurable function f , to the
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set F1 , and to the constant 1/2. Then, assumed that E1 ⊂ E2 ⊂ · · · ⊂ En−1 have been defined, we proceed as follows. We apply (RLT ) to the strongly measurable function f , to the set Fn \ En−1 , and to the constant 1/2n . So we find a nowhere dense closed set A ⊂ (Fn \ En−1 ) such that |(Fn \ En−1 ) \ A| < 1/2n , and f |A is continuous. We define En = En−1 ∪ A. Then En is nowhere dense and closed, En−1 ⊂ En , En ⊂ Fn−1 ∪ A ⊂ Fn , |Fn \ En | = |(Fn \ En−1 ) \ A| < 1/2n, and f |En is continuous. ' Let E = n En . Since En is closed and nowhere dense, then [0, 1] \ En is open and dense. Therefore, by the Baire category theorem, [0, 1] \ E is dense in [0, 1]. Let {rn } ⊂ [0, 1] \ E be dense in [0, 1]. We define inductively a trajectory t = {tn } as follows. By the definition of {rn } it follows r1 ∈ [0, 1] \ E1 . Let I be the connected component of [0, 1] \ E1 containing r1 . We define t1 = inf I. Then, assumed that t2 , . . . , tn−1 have been defined, by the definition of {rn } it follows that rn ∈ [0, 1] \ Em , for m = n, n + 1, . . . . Since the set {t1 , t2 , . . . , tn−1 } is finite, there exists a first index kn such that rn belongs to some connected component J of [0, 1] \ Ekn and one of the extreme points of J is not in {t1 , t2 , . . . , tn−1 }. We define tn as this extreme point. Remark that, by the density of {rn }, t coincide with the class of all extreme points of the connected components of the open sets [0, 1] \ E1 , . . . , [0, 1] \ En , . . . . Remark also that, by the definition of Fn , limn |Fn | = 1. Then, since |En | = |FN | − |FN \ En | ≥ |Fn | −
1 , 2n
we have limn |En | = 1. Therefore, if T ⊂ [0, 1] is an interval, there exists nT ∈ N such that T ∩ EnT = ∅. Moreover, since EnT is nowhere dense, it is T ⊂ EnT . Let I be a connected component of [0, 1] \ EnT such that I ∩ T = ∅. Then T contains at least one extreme point of I. As remarked before, this extreme point belongs to t; consequently t is dense in [0, 1], therefore t is a trajectory. Now we remark that: p) r(t, J) ∈ En implies J ∩ En = ∅, for each interval J ⊂ [0, 1] and for each n ∈ N. In fact, by the definition of t, the condition r(t, J) ∈ En implies r(t, J) ∈ Em for some m > n; i.e., r(t, J) is one of the end points of a connected component I of [0, 1] \ Em . Moreover, since En ⊂ Em , there exists a connected component (a, b) of [0, 1] \ En such that I ⊂ (a, b). Using again the particular construction of t, it follows that the end points of (a, b) belong to t and both precede r(t, J). Then a, b ∈ J. So, by r(t, J) ∈ I ⊂ (a, b), we have J ⊂ (a, b). Consequently J ∩ En = ∅. Finally we prove that t yields the Bochner integral of f . Given ε > 0, let N be such that ∞ n+1 (f + 1) ≤ ε, and ≤ ε. (3) 2n [0,1]\EN N
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Moreover let g be the extension of f |EN to [0, 1] such that g is linear on each connected component of [0, 1] \ EN . It is easy to prove that g is continuous on [0, 1] and that g(x) ≤ N for each x ∈ [0, 1]. Then, g being Riemann integrable, there exists a positive constant δ such that 1 g(r(t, Ji ))|Ji | − g < ε, (4) 0 i
for each usual partition P = {Ji } of [0, 1] with mesh(P) < δ. We set fP (t) = f (r(t, Ji )), if t ∈ Ji , gP (t) = g(r(t, Ji )), if t ∈ Ji . Therefore
1
fP = 0
f (r(t, Ji ))|Ji |,
i 1
gP = 0
g(r(t, Ji ))|Ji |.
i
' By E \ EN = n≥N (En+1 \ En ) and by |[0, 1] \ E| = 0 it follows that, for almost all t ∈ [0, 1] \ EN there exists n ≥ N such that t ∈ En+1 \ En . Let J ∈ P such that t ∈ J. Then, by p) we infer r(t, J) ∈ En+1 . So fP (t) = f (r(t, J)) ≤ n + 1 a.e. in [0, 1]. Thus, since En+1 \ En ⊂ Fn+1 \ En = (Fn+1 \ Fn ) ∪ (Fn \ En ), by (3) we have fP ≤ [0,1]\EN
≤
∞ N ∞
(n + 1)|En+1 \ En | (n + 1)(|Fn+1 \ Fn | + |Fn \ En |)
(5)
N
(f + 1) +
≤ [0,1]\EN
≤
∞ n+1 N
2n
2ε.
Moreover, since f > N on [0, 1] \ FN , by (3) it follows gP ≤ N · |[0, 1] \ EN | [0,1]\EN
≤ ≤ ≤
N · (|[0, 1] \ FN | + |FN \ EN |) N f + N 2 [0,1]\FN 2ε.
(6)
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Therefore, if P = {Ji } is a partition of [0, 1] with mesh(P) < δ, by |FN \ EN | < 1/2N and by (3), (4), (5) and (6) we infer: 1 f (r(t, Ji )) |Ji | − f 0 i 1 ≤ (f (r(t, Ji )) − g(r(t, Ji ))) |Ji | + g(r(t, Ji )) |Ji | − g 0 i i 1 + f − g 0 1
fP − gP + ε + f − g [0,1]\EN 0 fP + gP + ε ≤ [0,1]\EN [0,1]\EN + f + g + g [0,1]\EN [0,1]\FN FN \EN N ≤ 5ε + f + f + N 2 [0,1]\EN [0,1]\FN ≤ 8ε.
≤
This completes the proof.
4. Pettis integrable functions Theorem 2. There exists a strongly measurable function f : [0, 1] → l2 (N), Pettis integrable, such that no trajectory in [0, 1] yields the Pettis integral of f . Proof. Let {en } be the canonical basis of l2 (N). For n ≥ 2 let {an } be a decreasing sequence of positive numbers such that an − an+1 = 1/n2 and limn an = 0. We define ∞ n en χEn (x), f (x) = n=2
where En = (an+1 , an ]. The function f is strongly measurable and Pettis integrable with integral ∞ f= n|E ∩ En | en , xE = E
n=2
for each measurable set E ⊂ [0, 1]. In fact, for each x∗ = (x1 , x2 , . . . , xn , . . . ) ∈ l2∗ (N) we have ∞ x∗ f = nxn |E ∩ En | = x∗ (xE ). E
n=2
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Remark that f is not Bochner integrable, since 1 ∞ ∞ 1 = +∞. f = n|En | = n 0 2 2 Now we show that no trajectory in [0, 1] yields the Pettis integral of f . Let t be an arbitrary trajectory in [0, 1], and let 0 < ε < 1/2. ∞ Given δ > 0, let n0 ∈ N be such that an0 < δ and n0 n1 en < 1 − 2ε. First of all we remark that an0 > 1/n0 .
(7)
In fact an0
=
(an0 − an0 +1 ) + (an0 +1 − an0 +2 ) + · · · + (an0 +k − an0 +k+1 ) + · · ·
=
|En0 | + |En0 +1 | + · · · + |En0 +k | + · · · 1 1 1 + + ···+ + ··· n20 (n0 + 1)2 (n0 + k)2 1 1 1 + + ···+ + ··· n0 (n0 + 1) (n0 + 1)(n0 + 2) (n0 + k)(n0 + k + 1) 1 1 1 1 + − + ··· − n0 n0 + 1 n0 + 1 n0 + 2 1 . n0
= > = =
Moreover remark that, by the definition of f , it follows f (r(t, [0, an0 ])) = n en , for some n ≥ n0 . Then, by (7), we have f (r(t, [0, an0 ]) an0 ≥ n0 an0 > 1.
(8)
Now let P be a partition with mesh(P) < δ such that i) [0, an0 ] ∈ P; ii) [an0 , an0 −1 ], . . . , and [a3 , a2 ] are union of elements of P; iii) if J ∈ P with inf J = an+1 , n = 2, . . . , n0 − 1, then |J| < ε/(2n + 1) n0 . For n = 2, . . . , n0 − 1 we denote by Jn ∈ P the interval with inf Jn = an+1 . Then by definition of f and by iii) it follows 1 f (r(t, J)) |J| − en n J⊂[an+1 ,an ] = f (r(t, Jn )) |Jn | + f (r(t, J)) |J| − n|En | en Jn =J⊂[an+1 ,an ]
= f (r(t, Jn )) |Jn | + n|En \ Jn | en − n|En |en = f (r(t, Jn )) |Jn | − n|Jn |en .
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Hence
1 en n
f (r(t, J)) |J| −
J⊂[an+1 ,an ]
= 0, if r(t, Jn ) = an+1 = ((n + 1)en+1 − nen ) |Jn |, if r(t, Jn ) = an+1 .
Consequently 1 ε < (2n + 1)|Jn | < f (r(t, J)) |J| − en . n n0 J⊂[an+1 ,an ]
(9)
Thus, by (7), (8) and (9) we infer ∞ 1 en f (r(t, J)) |J| − n n=1 J∈P ∞ 0 −1 1 n 1 ≥ f (r(t, [0, an0 ])) an0 − − en f (r(t, J)) |J| − en n n n=n0 n=2 J⊂[an+1 ,an ] ≥ 1 − ε − (1 − 2ε) = ε .
This completes the proof.
Theorem 3. There exists a strongly measurable function f : [0, 1] → l (N), McShane integrable, such that no trajectory in [0, 1] yields the McShane integral of f . 2
Proof. Let f : [0, 1] → l2 (N) be defined as in Theorem 2. It was proved that f is strongly measurable and Pettis integrable on [0, 1]. Then, by a theorem proved by R.A. Gordon in [6] (see also [7, Theorem 6.2.1]) f is McShane integrable on [0, 1] with the same value of the integrals. Therefore no trajectory yields the McShane integral of f , by Theorem 2. Theorem 4. If X ∗ is w∗ -separable and f : [0, 1] → X is scalarly null, then there exists a trajectory t in [0, 1] yielding the Pettis integral of f . ' Proof. Let {x∗n } be w∗ -dense in X ∗ , and let E = n En , where En = {t ∈ [0, 1] : x∗n f (t) = 0}. By hypothesis |E| = 0. Then x∗n f (t) = 0 for each n ∈ N and each t ∈ [0, 1]\E. Now, for x∗ ∈ X ∗ and t ∈ [0, 1]\E there exists a sequence {nk } of indices such that x∗nk f (t) → x∗ f (t). Consequently x∗ f (t) = 0 and f (t) = sup x∗ =1 |x∗ f (t)| = 0, for each t ∈ [0, 1] \ E. Let t = {tn } be a trajectory in [0, 1] such that tn ∈ E, for each n. Then, for each δ > 0 and each partition P with mesh(P) < δ, we have J∈P f (r(t, J))|J| = 0. This completes the proof, since the Pettis integral of f is zero. Acknowledgment The author is indebted to the referee for various helpful comments that helped improve the paper.
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References [1] U.B. Darji and M.J. Evans, A first return examination of the Lebesgue integral, Real Anal. Exchange, 27 (2001/2002), 573–581. [2] J. Diestel and J.J. Uhl,Vector measures, Math. Surveys, 15 (1977). [3] L. Di Piazza and D. Preiss, When do McShane and Pettis integrals coincide?, Illinois J. Math., 47 (2003), 1177–1187. [4] N. Dinculeanu, Vector measures, Pergamon Press, Oxford, 1967. [5] D. Fremlin, Notes on first-return integration, Preprint available at http://www.essex.ac.uk/maths/staff/fremlin/n07k04.ps [6] R.A. Gordon, The McShane integral of Banach-valued functions, Illinois J. of Math., 34 (1990), 557–567. [7] S. Schwabik and Y. Guoju, Topics in Banach space integration. Series in Real Analysis, 10. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Donatella Bongiorno Dipartimento di Metodi e Modelli Matematici Viale delle Scienze I-90100 Palermo, Italy e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 99–107 c 2009 Birkh¨ auser Verlag Basel/Switzerland
A Note on Bi-orthomorphisms Gerard Buskes, Robert Page, Jr. and Rusen Yilmaz Abstract. We show that the space of bi-orthomorphisms forms a vector lattice.The space of orthomorphisms on a semiprime f -algebra is a vector sublattice of the space of bi-orthomorphisms and an ideal in the case that the f -algebra is Dedekind complete. Mathematics Subject Classification (2000). 46A40, 46B42. Keywords. Separately order bounded, bilinear map of order bounded variation, disjointness preserving bilinear operator, bi-orthomorphism.
1. Introduction Let A be a vector lattice. The authors of [13] were the first to study what, in this paper, we will call bi-orthomorphisms, i.e., bilinear maps A × A → A that are orthomorphisms in each variable separately. Their paper is based on a certain calculus for order bounded bilinear maps with values in a Dedekind complete vector lattice (Theorem 2.3 of [13]). The bilinear map φ : R2 × R2 → R defined by φ((x1 , x2 ), (y1 , y2 )) = (x1 − x2 )(y1 − y2 ) provides a counterexample to the Kantorovich-like formulas for the positive part and absolute value of bilinear maps in Theorem 2.3 of [13], though a calculus of order bounded variation like in [5] does yield the right formulas. One of the goals of [13] is to understand the space of quasimultipliers in relation to the space of bi-orthomorphisms, in case that A is an f -algebra. In that direction the main results in [13] (e.g., Theorem 4.7) are correct, because the lattice calculations for bi-orthomorphisms that are needed for these results are, in fact, more elementary than the Kantorovich-like formulas (and do not need Dedekind completeness of the range space) as we will show in this paper. We prove in this note that the space Orth(A, A) of bi-orthomorphisms is a vector lattice, which contains Orth(A) as a vector sublattice in case that A is a semiprime f -algebra. We prove that Orth(A) is an ideal in the vector lattice Orth(A, A) in case A is a semiprime Dedekind complete f -algebra. Our approach to the vector ¨ I˙ TAK Grant CODE The third named author gratefully acknowledges support from the T UB 2221.
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lattice structure of Orth(A, A) starts with (the more general) bilinear maps that are disjointness preserving and order bounded in each variable separately. In that direction we slightly finesse Theorem 3.4 in [8], resulting in a more concise proof of a theorem that follows from combining Theorems 1.12 and 2.11 in the Ph.D. thesis by Page (see [9]). For general information about vector lattices we refer to [8]. We remark that bilinear maps that are separately band preserving have recently been studied in the paper [5]. For a general survey on bilinear maps on products of vector lattices we refer to [4].
2. Preliminaries Let A and B be vector lattices. An operator T : A → B is said to be disjointness preserving if a ⊥ b implies T a ⊥ T b for all a ∈ A and b ∈ B. An operator T : A → A is called band preserving if a ⊥ b in A implies T a ⊥ b, where a ⊥ b means |a| ∧ |b| = 0. Clearly, every band preserving operator is disjointness preserving. A band preserving operator which is also order bounded is said to be an orthomorphism. The set of all orthomorphisms on A is denoted by Orth(A). If A is an Archimedean vector lattice, then Orth(A) is an Archimedean f -algebra under multiplication by composition, with the identity operator I on A as a multiplicative identity. It is well known that every order bounded operator T : A → B between two Archimedean vector lattices which preserves disjointness has a modulus |T | and |T |(|x|) = |T (|x|)| = |T (x)| holds for all x ∈ A. From here on, let A, B, and C be Archimedean vector lattices. A bilinear operator T : A × B → C is said to be of order bounded variation if, for all (x, y) ∈ A+ × B + , the set N,M n,m
|T (an , bm )| : an ∈ A+ , bm ∈ B + and
N n=1
an = x,
M
bm = y (N, M ∈ N)
m=1
is order bounded in C. T is said to be order bounded if for all (x, y) ∈ A+ × B + we have that {T (a, b) : 0 ≤ a ≤ y, 0 ≤ b ≤ y} is order bounded. We denote by Lbv (A, B; C) the set of all bilinear operators of order bounded variation and by Lb (A, B; C) the vector space of all order bounded bilinear maps. T is positive if for all x ∈ A+ and y ∈ B + we have T (x, y) ∈ C + . We denote by Lr (A, B; C) the span of the positive bilinear maps in Lb (A, B; C). Each element of Lr (A, B; C) is called regular. The vector spaces Lbv (A, B; C) and Lb (A, B; C) are ordered vector spaces under the ordering defined by T1 ≤ T2 if T2 − T1 is positive. The inclusions Lr (A, B; C) ⊂ Lbv (A, B; C) ⊂ Lb (A, B; C) hold. The converse inclusions need not hold, even if C = R, as was shown in [10]. However, if C is Dedekind complete, then Lr (A, B; C) = Lbv (A, B; C) and this space is then a Dedekind complete vector lattice. In addition, if C is Dedekind
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101
complete then for any T ∈ Lbv (A, B; C), the modulus |T | is determined by the following formula N M N,M |T |(x, y) = sup |T (an , bm )| : an ∈ A+ , bm ∈ B + , an = x, bm = y n,m
n
m
and |T (x, y)| ≤ |T |(|x|, |y|) for each x ∈ A and each y ∈ B (see [5, 7]). A bilinear map T : A × B → C is said to be separately disjointness preserving if +
+
(a1 , b1 ) ⊥ (a2 , b2 ) in A × B implies T (a1 , b) ⊥ T (a2 , b) and T (a, b1) ⊥ T (a, b2 ) for all a ∈ A and b ∈ B. A bilinear operator T : A × A → A is called separately band preserving if x ⊥ y in A
implies
T (x, z) ⊥ y and T (z, x) ⊥ y
for all z ∈ A. A bilinear operator T : A × B → C is called separately order bounded (respectively a Riesz bimorphism) if a → T (a, y) (a ∈ A)
and
b → T (x, b) (b ∈ B)
are order bounded (respectively a Riesz homomorphism). T is a Riesz bimorphism if and only if |T (a, b)| = T (|a| , |b|) for all a ∈ A and b ∈ B. Clearly, every separately band preserving bilinear operator is disjointness preserving. A separately band preserving bilinear operator which is also separately order bounded is called a bi-orthomorphism and the set of all bi-orthomorphisms of A × A into A is denoted by Orth(A, A). Finally we will use Fremlin’s Archimedean tensor product of two Archimedean vector lattices. In [6], Fremlin introduced for every two Archimedean vector lattices ¯ which is called the Archimedean E and F a new Archimedean vector lattice E ⊗F vector lattice tensor product of E and F , defined by the following universal prop¯ such that whenever G is a erty: there exists a Riesz bimorphism E × F → E ⊗F vector lattice and T is a Riesz bimorphism E × F → G then there exists a unique ¯ → G for which Riesz homomorphism T ⊗ : E ⊗F T (x, y) = T ⊗ (x⊗ y) (x ∈ E, y ∈ F ). ¯ has He also showed that the Archimedean vector lattice tensor product E ⊗F the following additional universal property. For every positive map T of E × F into any uniformly complete (hence Archimedean) vector lattice G there exists a ¯ → G such that T (x, y) = T ⊗ (x ⊗ y) for all unique positive linear map T ⊗ : E ⊗F x ∈ E, y ∈ F . It follows immediately that the map S → S ◦ ⊗ defines a bijective ¯ G) onto Lr (E, F ; G), where Lr (E ⊗F, ¯ G) denotes the positive map from Lr (E ⊗F, ¯ to G. partially ordered vector space of all linear regular maps from E ⊗F A slight extension of Fremlin’s universal property is the following result in [5]. Theorem 2.1. For every bilinear map T of E × F into any uniformly complete (hence Archimedean) vector lattice G which is of order bounded variation there ¯ → G such that T (x, y) = exists a unique order bounded linear map T ⊗ : E ⊗F
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G. Buskes, R. Page, Jr. and R. Yilmaz
T ⊗ (x ⊗ y) (x ∈ E, y ∈ F ). The map T → T ⊗ is a linear order isomorphism ¯ G), where Lb (E ⊗F, ¯ G) denotes the partially between Lbv (E, F ; G) and Lb (E ⊗F, ¯ to G. ordered vector space of all order bounded linear maps from E ⊗F
3. Separately disjointness preserving operators The modulus of any separately disjointness preserving separately order bounded bilinear map exists as we will prove in this section. We need the following Lemma, the equivalent of which for linear maps is well known (see, e.g., Theorem 1.1 in [11]). Lemma 3.1. Let E, F and G be vector lattices. Let T : E × F → G be a bilinear map and assume that N M N,M sup |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = x, ym = y n,m
n
m
exists in G for all x ∈ E + and all y ∈ F + . Then the map N M N,M |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = x, ym = y (x, y) → sup n,m
n
m
(x ∈ E , y ∈ F ) extends to a linear map E × F → G which in the ordered vector space Lbv (E, F ; G) is the least upper bound of {T, −T }. +
+
Proof. Let Gδ be the Dedekind completion of G. We interpret the map T : E ×F → G as a map T : E × F → Gδ and accordingly we read the formula in the lemma in Gδ rather than in G. According to [5] there exists a linear map E × F → Gδ which extends N M N,M |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = x, ym = y (x, y) → sup n,m
n
m
and that linear map is the modulus |T | of T in Lbv (E, F ; G ). This |T | has all its values in G. Moreover, in the ordered vector space Lbv (E, F ; G) we have that |T | ≥ T and |T | ≥ −T . It is immediate that for any S ∈ Lbv (E, F ; G) which is an upper bound for {T, −T } we have S ≥ |T |. The lemma follows. δ
The next theorem generalizes Theorem 3.4 in [8] and combines in one proof (rather than two separate arguments) the information from Theorems 1.12 and 2.11 in Page’s Ph.D. thesis [9] (see our comments after the theorem). Since the Ph.D. thesis of Page is less easily accessible, and for completeness sake, we have included the result with the more concise proof. Theorem 3.2. Let E, F and G be vector lattices. Let T : E × F → G be a separately disjointness preserving bilinear map. Then the following are equivalent. (1) T is separately order bounded. (2) T is order bounded.
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(3) T is of order bounded variation. (4) T is regular. Moreover, if T satisfies any of the above, then |T | exists and ||T |(e, f )| = |T (e, f )| = |T |(|e|, |f |) for all (e, f ) ∈ E × F. In particular, |T | is a Riesz bimorphism. Proof. Obviously (4)⇒(3)⇒(2)⇒(1). Thus it suffices to show (1)⇒(4). Assume that T : E × F → G is separately order bounded. Let e ∈ E + and f ∈ F + . Define the maps Tf : E → G by Tf (x) = T (x, f ) (x ∈ E) e T : F → G by e T (y) = T (e, y) (y ∈ F ). Since e T and Tf are order bounded operators which preserve disjointness by the hypotheses, |e T | and |Tf | exist and
|e T |(f ) = |e T (f )| = |T (e, f )| = |Tf (e)| = |Tf |(e). Now let x = (x1 , . . . , xm ) and y = (y1 , . . . , yn ) be partitions of e ∈ E + and f ∈ F + respectively (as defined in [5]). Then we have n m m n m n |T (xi , yj )| = |T (xi , yj )| = |xi T (yj )| i=1 j=1
i=1
=
= =
j=1
m n i=1 m i=1 m i=1
i=1
j=1
m n |xi T |( |xi T |(yj ) = yj )
j=1
|xi T |(f ) = |Tf (xi )| =
i=1 m i=1 m
|xi T (f )| =
j=1 m
|T (xi , f )|
i=1
m |Tf |(xi ) = |Tf | xi
i=1
i=1
= |Tf (e)| = |T (e, f )|. Thus, by the previous lemma, |T | exists in Lbv (E, F ; G) and |T | (e, f ) is given by the formula N M N,M sup |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = e, ym = f . n,m
n
m
for e ∈ E and f ∈ F . Then T is regular in Lbv (E, F ; G), which proves (1)⇒(4). Moreover, |T |(e, f ) = |T (e, f )| for e ∈ E + and f ∈ F + . Using the latter and the fact that T is separately disjointness preserving, we obtain that for arbitrary e ∈ E and f ∈ G the elements in {|T | (e+ , f + ), |T | (e+ , f − ), |T | (e− , f + ), |T | (e− , f − )} are pairwise disjoint hence +
+
||T | (e, f )| = |T | (e+ , f + ) + |T | (e+ , f − ) + |T | (e− , f + ) + |T | (e− , f − ) = |T | (|e| , |f |). Then |T | is a Riesz bimorphism. This proves the theorem.
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Some remarks are in order. First of all we comment that, in general, separately order bounded bilinear maps are not necessarily order bounded, as was shown by Swartz in [12]. Secondly, in Theorem 3.4 of [8] Kusraev and Tabuev proved (2) ⇔ (4) (as well as the fact that |T | is a Riesz bimorphism). Page proved (3) ⇒ (4) in Theorem 2.11 of his Ph.D. thesis [9] (as well as the fact that |T | is a Riesz bimorphism) and he proves (1) ⇒ (3) in Theorem 1.12. The proof by Kusraev and Tabuev is similar to our proof above of (1) ⇒ (4) while our proof above combines the two proofs by Page. We also, as an example, make explicit a comment from the introduction, which is one of our reasons for writing this note. Example 3.3. For the bilinear map φ : R2 × R2 → R defined by φ((x1 , x2 ), (y1 , y2 )) = (x1 − x2 )(y1 − y2 ) we have that φ is of order bounded variation. Hence |φ| exists and its values are calculated via the formula in Section 1. Consequently, φ also is order bounded but |φ| cannot be calculated via a Kantorovic formula for the absolute value as formulated in Theorem 2.3 of [13]. We collect a number of corollaries to Theorem 3.2. As a first easy application, we add a statement to the equivalences in Theorem 3.2 which is the analogue for bilinear maps of the first part of Theorem 3.3 in [2]. Corollary 3.4. Let E, F and G be vector lattices. Let T : E × F → G be a separately disjointness preserving bilinear map. Then each of (1)–(4) in Theorem 3.2 is equivalent to | (x1 , x2 ) | ≤ | (y1 , y2 ) | ⇒ |T (x1 , x2 ) | ≤ |T (y1 , y2 ) | for all (x1 , x2 ) , (y1 , y2 ) ∈ E × F.
()
Proof. Notice that | (x1 , x2 ) | ≤ | (y1 , y2 ) | if and only if |x1 | ≤ |y1 | and |x2 | ≤ |y2 |. If T is separately order bounded then (using the notation of the proof of Theorem 3.2 in this paper) by Theorem 3.3 in [2] |T (x1 , x2 ) | = |x1 T (x2 )| ≤ |x1 T (y2 )|, while by the same reasoning |x1 T (y2 )| = |T (x1 , y2 ) | = |Ty2 (x1 ) | ≤ |Ty2 (y1 ) | = |T (y1 , y2 ) |. Hence (1) in 3.2 implies the statement () above. Conversely, the statement () obviously implies order boundedness of T . The proof of the next corollary is left to the reader. In case T is order bounded it is also contained in Theorem 3.4 of [8]. Corollary 3.5. If T is separately disjointness preserving and separately order bounded, then |T |, T + and T − are Riesz bimorphisms and |T | = T + − T − . Our next corollary deals with bi-orthomorphisms. Corollary 3.6. Let A be a vector lattice. If T ∈ Orth(A, A), then |T |, T + and T − are also in Orth(A, A).
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Proof. Assume that x ⊥ y in A and z ∈ A. If T ∈ Orth(A, A), then |T | exists by the above theorem and ||T |(x, z)| ∧ |y| = |T (x, z)| ∧ |y| = 0
and
||T |(z, x)| ∧ |y| = |T (z, x)| ∧ |y| = 0.
holds, and so |T | ∈ Orth(A, A). It follows that T + and T − are also in Orth(A, A). Corollary 3.7. If A is an Archimedean vector lattice, then Orth(A, A) is a vector lattice. The next lemma for order bounded separately disjointness preserving operators is Theorem 3.6 of [8]. Lemma 3.8. Let E, F and G be Archimedean vector lattices. If T : E × F → G is separately disjointness preserving and separately order bounded, then there exists ¯ → G such that an order bounded disjointness preserving operator T ⊗ : E ⊗F T (x, y) = T ⊗ (x ⊗ y) (x ∈ E, y ∈ F ) and |T |⊗ = |T ⊗ |. Proof. Let Gδ be the uniform completion of G. By Theorem 2.1, the map T → T ⊗ ¯ ; Gδ ) is a linear order isomorphism. Since T + ∧T − = from Lbv (E, F ; Gδ ) to Lb (E ⊗F + ⊗ 0 we have that (T ) ∧ (T − )⊗ = 0 and T ⊗ = (T + )⊗ − (T − )⊗ . Also (T + )⊗ and (T − )⊗ are Riesz homomorphisms by the definition of tensor product. But (T + )⊗ and (T − )⊗ take their values in G, and then so does T ⊗ . Hence T ⊗ is order bounded disjointness preserving and |T |⊗ = (T + + T − )⊗ = (T + )⊗ + (T − )⊗ = (T ⊗ )+ + (T ⊗ )− = |T ⊗ |.
An analogue of a known Radon-Nikodym Theorem for disjointness preserving linear maps is next. It generalizes Proposition 3.7 (3) of [8] (where G is Dedekind complete and T is order bounded). Theorem 3.9. Let E, F and G be Archimedean vector lattices. If G is uniformly complete and S : E × F → G is a positive bilinear map and T : E × F → G is a separately disjointness preserving map which is separately order bounded and S ≤ |T | then there exists π ∈ Orth(I(T (E × F ))) (where I(T (E × F )) is the ideal generated by T (E × F ) in G) such that S = π ◦ |T |. If G is Dedekind complete then π can be selected in Orth(G). The latter theorem follows immediately from the following polar decomposition theorem for separately order bounded separately disjointness preserving bilinear operators. Theorem 3.10. Let E, F and G be vector lattices. If G is uniformly complete and T : E ×F → G is a separately order bounded and separately disjointness preserving operator, then there exists a bijective π ∈ Orth(I(T (E × F ))) such that T = π ◦ |T | (where I(T (E × F )) is the ideal generated by T (E × F ) in G). If G is Dedekind complete then π can be selected in Orth(G).
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¯ → G is order bounded and disjointness preservProof. The map T ⊗ : E ⊗F ing. By the polar decomposition theorem ([3, Theorem 7]), there exists π ∈ ¯ ))) such that T ⊗ = π ◦ |T ⊗ |. First note that Orth(I(T ⊗ (E ⊗F T ⊗ ◦ ⊗ = (π ◦ |T ⊗ |) ◦ ⊗ = π ◦ |T |⊗ ◦ ⊗ = π ◦ |T |. Next we observe that (π ◦ |T |)⊗ (x ⊗ y) = (π ◦ |T |)(x, y) = (T ⊗ ◦ ⊗)(x, y) = T ⊗ (⊗(x, y)) = T ⊗ (x ⊗ y) for all x ∈ E and x ∈ F , and so T ⊗ = (π ◦ |T |)⊗ . Then T = π ◦ |T | because the map T → T ⊗ is injective. ¯ )) = I(T (E × F )). Indeed, certainly T (E × F ) ⊂ Note also that I(T ⊗ (E ⊗F ⊗ ¯ ¯ , and therefore T (E ⊗F ). On the other hand, ⊗(E × F ) is majorizing in E ⊗F ⊗ ⊗ ¯ ¯ I (T (E ⊗F )) ⊂ I (T (E × F )). Then I (T (E ⊗F )) = I (T (E × F )) and π ∈ Orth (I (T (E × F ))). As a consequence we are able to locate Orth(A) as an ideal in Orth(A, A)) when A is a Dedekind complete semiprime f -algebra. Proposition 3.11. (i) If A is a semiprime f -algebra then Orth(A) is a vector sublattice of Orth(A, A). (ii) If A is a semiprime Dedekind complete f -algebra then Orth(A) is an order ideal in Orth(A, A). Proof. We first prove (i). Let A be a semiprime f -algebra. The map ϕ : Orth(A) → Orth(A, A) defined by (ϕ(π))(f, g) = π(f )g (π ∈ Orth(A) and f, g ∈ A) is an injective Riesz homomorphism. This proves (i). Now let A be a semiprime Dedekind complete f -algebra. Let T : A × A → A be a bilinear map of order bounded variation and let π in Orth(A) be such that |T | ≤ |ϕ(π)|. By 3.9, there exists π ∈ Orth(A) such that T + = π ◦ |ϕ(π)|. Now π ◦ |ϕ(π)|(f, g) = π (|π| (f )g) = π (|π| (f ))g. Hence T + ∈ ϕ(Orth(A)). Similarly T − ∈ ϕ(Orth(A)) and consequently T ∈ ϕ(Orth(A)). This proves (ii). Example 3.12. In general, Orth(A) = Orth(A, A). Indeed, take A = C[0, 1] and define an f -algebra multiplication by (f ∗ g)(x) = x · (f · g)(x) (f, g ∈ A and x ∈ [0, 1]). Then the ordinary multiplication (f, g) → f ·g is in Orth(A, A) but not in Orth(A). We ask the following question. Question 3.13. When is Orth(A, A) an f -algebra? When exactly Orth(A, A) is an f -algebra is indeed unclear. However, the results above lead to the following improvement of Theorem 4.7 in [13], to which we refer for the definition of a minimal ultra-approximate identity. The proof follows the same lines as [13] with a small addition at the end of the proof of Theorem 4.6 in [13] where it needs to be observed that band preserving operators on a Banach lattice are indeed orthomorphisms. Proposition 3.14. Let A be an f -algebra which is also a Banach algebra with a minimal ultra-approximate identity. Then Orth(A, A) is a Banach f -algebra.
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References [1] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, 1985. [2] Y.A. Abramovich, E.L. Arensen, and A.K. Kitover, Banach C(K)-modules and Operators Preserving Disjointness, Pitman Research Notes in Mathematics Series, 277, Longman Scientific & Technical, Harlow, 1992. [3] K. Boulabiar and G. Buskes, Polar decomposition of order bounded disjoint preserving operators, Proc. Amer. Math. Soc. 132 (2005), no. 3, 799–806. [4] Q. Bu, G. Buskes, and A.G. Kusraev, Bilinear maps on products of vector lattices: A survey, Positivity, Trends in Mathematics, Positivity, 97–126, Trends Math., Birkh¨ auser, Basel, 2007. [5] G. Buskes and A. van Rooij, Bounded variation and tensor products of Banach lattices, Positivity 7 (2003), 47–59. [6] D.H. Fremlin, Tensor products of Archimedean vector lattices, Amer. J. Math. 94 (1972), 778–798. [7] A.G. Kusraev, When all separately band preserving bilinear operators are symmetric, Vladikavkaz Mat. Zh. 9, no. 2 (2007), 22–25. [8] A.G. Kusraev and S.N. Tabuev, On disjointness preserving bilinear operators, Vladikavkaz Math. Zh. 6, no. 1 (2004), 58–70. [9] R. Page, On bilinear maps of order bounded variation, Thesis University of Mississippi, 2005. [10] A.L. Peressini and D.R. Sherbert, Ordered topological tensor products, Proc. London Math. Soc. 19 (1969), 177–190. [11] A. van Rooij, When do the regular operators between two Riesz spaces form a Riesz space?, Report 8410, Catholic University Nijmegen, (1984). [12] C. Swartz, Bilinear mappings between lattices, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 33(81) (1989), no. 2, 147–152. [13] R. Yilmaz and K. Rowlands, On orthomorphisms, quasi-orthomorphisms and quasimultipliers, J. Math. Anal. Appl. 313 (2006), 120–131. [14] A.C. Zaanen, Riesz Spaces II, North-Holland (1983). Gerard Buskes Department of Mathematics, University of Mississippi University, MS 38677, USA e-mail: [email protected] Robert Page, Jr. Department of Mathematics, Framingham State College Framingham, MA 01701, USA Rusen Yilmaz Department of Mathematics, Faculty of Arts and Science Rize University 53100 Rize, Turkey e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 109–113 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Compactness of Multiplication Operators on Spaces of Integrable Functions with Respect to a Vector Measure Ricardo del Campo, Antonio Fern´andez, Irene Ferrando, Fernando Mayoral and Francisco Naranjo Abstract. We study properties of compactness of multiplication operators between spaces of p-power integrable scalar functions with respect to a vector measure m. Mathematics Subject Classification (2000). Primary 46G10, 46E30 Secondary 47B65, 47H07. Keywords. Vector measure, integrable function, multiplication operator, weakly compact operator.
1. Introduction The aim of the present paper is to study compactness and weak compactness properties of multiplication operators between spaces of p-power integrable functions with respect to a vector measure. In the following, m : Σ −→ X will be a countably additive vector measure defined on a σ-algebra Σ of subsets of a nonempty set Ω with values in a real Banach space X. We denote by X its dual space and by B(X) the unit ball of X. The properties of the Banach lattices Lp (m) and Lpw (m), for p ≥ 1, can be found in ([4] and [6]). In particular, neither Lp (m) nor Lpw (m) have to be reflexive spaces even if p > 1. For a given function g ∈ L0 (m), we can always consider the multiplication operator Mg : f ∈ L0 (m) −→ Mg (f ) := gf ∈ L0 (m) . This research has been partially supported by La Junta de Andaluc´ıa. The authors acknowledge the support of the Ministerio de Educaci´ on y Ciencia of Spain and FEDER, under projet MTM2006-11690-C02.
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In paper [2] we proved that if g ∈ Lq (m), then the multiplication operators Mg : Lp (m) −→ L1 (m) and Mg : Lpw (m) −→ L1 (m) are well defined, and if g ∈ Lqw (m), then the multiplication operators Mg : Lpw (m) −→ L1w (m) and Mg : Lp (m) −→ L1 (m) are also well defined. Moreover, we characterize the continuity of these operators. Furthermore, for g ∈ L∞ (m) the following multiplication operators Mg : Lp (m) −→ Lp (m) and Mg : Lpw (m) −→ Lpw (m) are well defined. Although the multiplication operator from Lpw (m) to Lp (m) is not always defined, we found, in the cited paper [2], conditions under which the multiplication from Lpw (m) to Lp (m) is continuous. We can notice that there exists an asymmetric behavior of multiplication operators depending on its domain when the continuity is considered. In the following section we will prove that this asymmetric behavior disappears when compactness and weak compactness are considered. In the sequel, if E and F are Banach lattices we denote by B (E, F ) the Banach space of all linear and continuous operators from E into F , and by K (E, F ) and W (E, F ) the ideals of compact and weakly compact operators, respectively. 1 1 Throughout the paper p, q > 1 will be conjugated exponents, that is, + = 1. p q
2. Compactness and weak compactness Next results will make evident that the behavior of multiplication operators must be the same when we deal with compactness-type conditions instead of continuity. Lemma 2.1. Let p, r ≥ 1 two real numbers and let g ∈ L0 (m) a function for which Mg ∈ W (Lp (m), Lr (m)). Then Mg ∈ B (Lpw (m), Lr (m)) and Mg (B(Lpw (m))) ⊆ Mg (Lp (m)) In particular, Mg ∈
Lr (m)
.
W (Lpw (m), Lr (m)).
Proof. First we are going to prove that Mg (Lpw (m)) ⊆ Lr (m). Let f ∈ Lpw (m). For each n = 1, 2, . . . , consider the set An := {w ∈ Ω : |f (w)| ≤ n}, and put fn := f χAn . Since fn ∈ Lp (m), and |fn | ≤ |f | we have fn Lp (m) ≤ f Lpw (m) ,
n = 1, 2, . . .
Now, from Mg ∈ W (L (m), L (m)), it follows that (gfn )n has a weakly convergent subsequence (gfnk )k in Lr (m). By [3, Corollary 2.2], there exists (ϕk )k such that ϕk ∈ co gfnj : j ≥ k , k = 1, 2, . . . p
r
and (ϕk )k converges to some h in Lr (m). Since (Ank )k ↑ Ω, it follows that ϕk χAnk = gf χAnk for all k = 1, 2, . . . and thus (ϕk )k → gf pointwise m-a.e. This gives gf = h m–a.e. and Mg (f ) = gf ∈ Lr (m). Next, we are going to prove that Mg (B(Lpw (m))) ⊆ Mg (B(Lp (m)))
Lr (m)
.
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Note that, gf − gfn Lr (m) = gf χΩ\An Lr (m) → 0 by the Dominated Convergence Theorem, and so, the convergence of (gfn )n to gf is, in fact, in the norm of Lr (m). Thus, given ε > 0 and f ∈ B(Lpw (m)), there exists n ∈ N such that gf − gfn Lr (m) < ε. Hence, Mg (f ) = gf = gfn + (gf − gfn ) ∈ Mg (B(Lp (m))) + εB(Lr (m)), which proves that Mg (B(Lpw (m))) ⊆ Mg (B(Lp (m)))
Lr (m)
.
With the help of the previous lemma we can prove the following proposition. Proposition 2.2. Let 1 < p < ∞, g ∈ L0 (m), r = 1 or r = p and assume that A ∈ {W, K}. Then the following conditions are equivalent: 1) 2) 3) 4)
Mg Mg Mg Mg
∈ A (Lpw (m), Lr (m)). ∈ A (Lpw (m), Lrw (m)). ∈ A (Lp (m), Lrw (m)). ∈ A (Lp (m), Lr (m)).
Proof. The implications 1) =⇒ 2) and 2) =⇒ 3) are evident. 3) =⇒ 4) If Mg ∈ A (Lp (m), Lrw (m)) then Mg ∈ B (Lp (m), Lrw (m)) and from [2, Theorem 4] we obtain that Mg ∈ B (Lp (m), Lr (m)). In particular, the inclusion Mg (Lp (m)) ⊆ Lr (m) holds, and hence Mg ∈ A (Lp (m), Lr (m)). 4) =⇒ 1) Apply Lemma 2.1. Next we are going to characterize the weak compactness of the multiplication operator Mg in terms of the function g. Theorem 2.3. Let g ∈ L0 (m). The following conditions are equivalent: 1) g ∈ Lq (m). 2) Mg ∈ W Lp (m), L1 (m) . 3) Mg ∈ B Lpw (m), L1 (m) . Proof. 1) =⇒ 2) Taking into account [5, Definition 3.6.1, Proposition 3.6.5] it is sufficient to prove that hn L1 (m) → 0 for every disjoint sequence (hn )n ⊆ Mg (B (Lp (m))). Consider the disjoint measurable sets An := {w ∈ Ω : hn (w) = 0}, for n = 1, 2, . . . . Thus, hn = Mg (fn ) = gfn = gfn χAn = gχAn fn for some sequence (fn )n ⊆ B (Lp (m)). From H¨ older’s inequality we deduce that hn L1 (m)
=
Mg (fn )L1 (m) = gχAn fn L1 (m)
=
gχAn Lq (m) fn Lp (m) ≤ gχAn Lq (m) ,
but gχAn Lq (m) → 0 since (gχAn )n is an order bounded disjoint sequence in Lq (m) and the space Lq (m) is order continuous. 2) =⇒ 3) It is a direct application of Lemma 2.1. Finally 3) =⇒ 1) is obtained from [2, Theorem 4].
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Next, we will consider, for a fixed set G ∈ Σ, the measurable space (G, ΣG ), where ΣG := {A ∈ Σ : A ⊆ G}, and the vector measure mG : A ∈ ΣG −→ mG (A) := m(A) ∈ X. We denote, respectively, by EG and RG the extension and restriction maps from L0 (mG ) to L0 (m). It is easy to establish that EG (RG (f )) = χG f , for all f ∈ L0 (m). 1 0 , for all Theorem 2.4. Let g ∈ L (m) and denote by Gn := w ∈ Ω : |g (w) | ≥ n n = 1, 2, . . . . The following assertions are equivalent: 1) g ∈ L∞ (m) and Lp (mGn ) is reflexive for all n = 1, 2, . . . . 2) Mg ∈ W (Lpw (m), Lp (m)). 3) Mg ∈ B (Lpw (m), Lp (m)). Proof. 1) =⇒ 2) Since g ∈ L∞ (m), by [2, Theorem 8] we have that Mg ∈ B (Lpw (m), Lpw (m)). Consider now the sequence MgχGn n of multiplication operators defined in Lpw (m). For each n = 1, 2, . . . we claim that MgχGn belongs to W (Lpw (m), Lp (m)). Indeed, note that the composition Mg
RG
EG
n n → Lpw (mGn ) = Lp (mGn ) −−−− → Lp (m) Lpw (m) −−−−→ Lpw (m) −−−−
is a weakly compact operator, since Lp (mGn ) is reflexive and, particularly, Lpw (mGn ) = Lp (mGn ). But, for each f ∈ Lpw (m), EGn RGn Mg (f ) = MgχGn (f ). Let us consider now the set G := {w ∈ Ω : g(w) = 0} and denote Cn := G \ Gn for all n = 1, 2, . . . Thus we have Mg − MgχG = MgχCn B(Lp (m),Lp (m)) p p n B(Lw (m),Lw (m)) w w = sup gχCn f Lpw (m) : f Lpw (m) ≤ 1 1 → 0. n Therefore Mg is the uniform limit in B (Lpw (m), Lpw (m)) of the operators MgχGn which belong to W (Lpw (m), Lp (m)). Since Lp (m) is closed in Lpw (m) we conclude that Mg ∈ W (Lpw (m) , Lp (m)). The implication 2) =⇒ 3) is evident. 3) =⇒ 1) We apply [2, Theorem 10], having in mind that Lp (m) is reflexive if and only if Lp (m) = Lpw (m), see [4, Corollary 3.10]. ≤
gχCn L∞ (m) ≤
Having in mind that the spaces Lp (m) are ideal spaces and that for a measurable function g the multiplication operator Mg is the same as the superposition operator F generated by the super-measurable function f (s, x(s)) = g(s)x(s), we obtain as a particular case of [1, Theorem 2.5], the following characterization for the operator Mg . Theorem 2.5. Let g ∈ L0 (m), then the following conditions are equivalent: 1) g ∈ Lq (m) and gχB = 0, where B is the non-atomic part of the vector measure m. 2) Mg ∈ K Lp (m), L1 (m) .
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1 , for all Theorem 2.6. Let g ∈ L (m) and denote by Gn := w ∈ Ω : |g (w) | ≥ n n = 1, 2, . . . . The following assertions are equivalent: 1) g ∈ L∞ (m) and dim (Lp (mGn )) < ∞, for all n = 1, 2, . . . . 2) Mg ∈ K (Lp (m), Lp (m)). 0
References [1] J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambridge University Press, Cambridge, 1990. [2] R. del Campo, A. Fern´ andez, I. Ferrando, F. Mayoral and F. Naranjo, Multiplication operators on spaces on integrable functions with respect to a vector measure, J. Math. Anal. Appl. 343 (2008), 514–524. [3] J. Diestel, W.M. Ruess and W. Schachermayer, On weak compactness in L1 (μ, X), Proc. Amer. Math. Soc. 118 (1993), 447–453. [4] A. Fern´ andez, F. Mayoral, F. Naranjo, C. S´ aez and E.A. S´ anchez–P´erez, Spaces of p-integrable functions with respect to a vector measure, Positivity 10 (2006), 1–16. [5] P. Meyer–Nieberg, Banach Lattices, Springer-Verlag. Berlin. 1991. [6] S. Okada, W. Ricker and E.A. S´ anchez–P´erez, Optimal domain and Integral extension of Operators acting in Function Spaces, Operator Theory: Advances and Applications, vol. 180, Birkh¨ auser Verlag, Basel, 2008. Ricardo del Campo Dpto. Matem´ atica Aplicada I EUITA, Ctra. de Utrera Km. 1 E-41013 Sevilla, Spain e-mail: [email protected] Antonio Fern´ andez, Fernando Mayoral and Francisco Naranjo Dpto. Matem´ atica Apda. II Escuela T´ecnica Superior de Ingenieros E-41092 Sevilla, Spain e-mail: [email protected] [email protected] [email protected] Irene Ferrando I.M.P.A., Universidad Polit´ecnica de Valencia E-46022 Valencia, Spain e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 115–124 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Some Applications of Nonabsolute Integrals in the Theory of Differential Inclusions in Banach Spaces Kinga Cicho´ n and Mieczyslaw Cicho´ n Abstract. In this paper we present a brief historical review about multivalued integrals and its relations with differential inclusions. Then a new theorem about existence of solutions (in some weak sense) for differential inclusions in Banach spaces is proved (by using some properties of nonabsolute integrals). Mathematics Subject Classification (2000). Primary 28B20; Secondary 34A60. Keywords. Aumann integrals, Pettis integrals, Henstock-Kurzweil-Pettis integrals, selections, differential inclusions, nonlocal Cauchy problems.
1. Introduction When we try to solve some problems for differential equations, we deal with different kind of “solutions”. Most of them are related to some integral equations with an appropriate definition of an interval. For instance, the Henstock-Kurzweil integral was constructed as a solution to the Cauchy problem. It was claimed that to recover a function from its derivative, the Lebesgue integral is not sufficient, so it is not sufficiently convenient for solving differential equations (Kurzweil, 1957). Similar problem for weak derivatives was solved by introducing the Henstock-KurzweilPettis integral (1999). Now, various types of integrals are used for solving such problems. But when we try to extend these results for differential inclusions, we need some new definitions of the multivalued integrals. The problem lies in not sufficiently investigated properties of multivalued integrals. In this paper we present some brief historical review about multivalued integrals. Due to lack of systematic study of different kind of solutions for differential inclusions we concentrate on this topic. To do this we present some results about multivalued nonabsolute integrals. Both single and multivalued integrals are really useful in theories of differential equations and inclusions and allow to unify separately considered cases (cf. [14]). Applicability of such integrals will be clarified in the next sections.
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Then a new theorem about existence of solutions (in some weak sense) for differential inclusions in Banach spaces is proved (by checking some properties of nonabsolute integrals). For a brief survey about different kind of solutions for differential equations see [14]. Let E be a Banach space and E ∗ be its dual space. The closed unit ball of ∗ E is denoted by B(E ∗ ). By Br we denote a closed ball {x ∈ E : x ≤ r}. In the whole paper I stands for a compact interval [0, α]. By cwk(X) we denote the family of all nonempty convex weakly compact subsets of X. For every bounded and convex set C the support function of C is denoted by s(·, C) and defined on E ∗ by s(x∗ , C) = supx∈C x∗ , x , for each x∗ ∈ E ∗ . A multifunction G : E → 2E with nonempty, closed values is called weakly sequentially upper hemi-continuous (w-seq uhc) iff for each x∗ ∈ E ∗ s(x∗ , G(·)) : E → R is sequentially upper semicontinuous from (E, w) into R. For a bounded subset A ⊂ E, we define the measure of weak noncompactness ω(A) (in the sense of DeBlasi [15]): ω(A) is the infimum of all ε > 0 s.t. there exists a weakly compact set K in E with A ⊆ K + ε · B1 . By ωC we will denote the measure of weak noncompactness in the space C(I, E). Let F : [a, b] → E and A ⊂ [a, b]. The function f : A → E is a pseudoderivative of F on A if for each x∗ in E ∗ the real-valued function x∗ F is differentiable almost everywhere on A and (x∗ F ) = x∗ f a.e. on A (Pettis, 1938). Unfortunately, we are unable to present in this short paper all necessary definitions. We leave the reader to remind the classical definitions of integrals ([22]). For the definitions and results about different concepts of solutions we refer the readers to [14].
2. Multivalued integrals We present below a brief survey about nonabsolute multivalued integrals. Such integrals were defined, in general, to solve some problems for differential (or integral) inclusions. For theorems about applications in the theory of differential inclusions we refer to the papers cited in this section. In the next section we will present a new example of application for multivalued integrals. It is rather unknown, that the study of multivalued integrals was begun by Alexander Dinghas in 1956 [17] who adapted the definition of the Riemann integral to the multivalued context (the Riemann-Minkowski integral). The same idea of Riemann sums was rediscovered by Hukuhara in 1967 [23] and developed by Artstein and Burns [4] or Jarnik-Kurzweil [24] (without using the Hausdorff distance in the definition). But the theory of integration for multivalued mappings is intensively studied since Aumann’s work of 1965 which is based on another idea. The Aumann integral is well suited for applications to various mathematical fields, in particular to solve some problems for differential inclusions. In fact, the next two treatments of the multivalued integrals is still considered. Aumann’s idea is connected with using
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some selection theorems. In the last concept it was defined the multivalued integral as a usual integral in the space of subsets (by using the Radstr¨ om embedding theorem or other isometries for spaces of subsets) – cf. [26] or [10], for instance. As claimed above, the first one was begun by Aumann [5] in 1965 and since then this idea is intensively developed. We concentrate on this kind of integrals as it seems to be really fruitful when we solve some problems for differential inclusions. For the second one we have the Debreu integral which stands for the Bochner integral for multifunctions or the Castaing integral for multivalued Pettis integrals. Let us remark, that by using the Radstr¨ om embedding theorem or due to the H¨ormander theorem it is possible to extend the other type of integrals for multifunctions with convex and compact values (the Debreu (Bochner) integral [16], the generalized McShane multivalued integral [8], the multivalued Birkhoff integral [9] or the (Debreu-)Pettis integral [10] or [9], for instance). The problem of integrability, properties of the integrals as well as comparison results between different kind of multivalued integrals are investigated as basic problems in many papers, including mentioned above. And now a few words about selected multivalued integrals. We denote by SF1 the set of all Bochner integrable selections of F, namely SF1 = {f ∈ L1 (E) : f (s) ∈ F (s) almost everywhere}. If F is a measurable multifunction and SF1 = ∅, then the Aumann integral (shortly (A)-integral) of F is given by (A) T F (s)ds = { T f (s)ds : f ∈ SF1 }. We will call a multifunction Aumann “integrable” if this set is not empty. One of the advantages for the Aumann integrals lies in the fact that the values of “integrable” multifunctions need be neither convex nor compact sets. Let us note, that the Debreu integral is also a multivalued extension of the Bochner integral and under classical assumptions these two mentioned integrals are equivalent. If we consider multivalued nonabsolute integrals then the Pettis integral is the oldest one. For instance, the results which are the most interesting for dealing with applications of multivalued integrals for differential inclusions can be found in the book of Castaing-Valadier [11] or in the paper of Tolstonogov [29]. The (Castaing-) Pettis integrability means the Lebesgue integrability of the function s(x∗ , F (·)) for multifunctions with convex compact [or: weakly compact] as well as “Pettis” integrability in the appropriate space of subsets (for each measurable subset A of the domain there exists a convex compact [weakly compact] set which realize the integral of s(x∗ , F (·)) over A). Cf. also [1]. A full theory for this topic (for many classes of values of F ), including more general definitions of the Pettis integral for multifunctions, can be found in [20]. The last paper deals also with Pettis integrability for multifunctions with possibly unbounded values and contains interesting examples of different kind of Pettis integrable multifunctions. Let us recall, that the Pettis integral for multifunctions was defined also via some isometries in [9] (the Debreu-type of integral) or via selection theorems (Valadier [30]). Then last concept (considered in [20]) is based on the idea of
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Aumann: (AP ) T F (s)ds = {(P ) T f (s)ds : f ∈ SFP e }, where SFP e denotes the set of all Pettis integrable selections of F provided that this set is not empty. As remarked above, an interesting discussion about all the types of Pettis integrals can be found for instance in [20]. The properties of multivalued Pettis integrals are intensively studied: [1], [10], [31], [2] or [20], for instance. Now, consider the multivalued integrals based on idea of Kurzweil and Henstock. If we deal with the integrals based on Riemann sums, we can treat such integrals as special cases of Henstock-Kurzweil integral. Thus, we can consider the integrals from [17], [23], [4], [24] or [8]. Although the application of such integrals for differential inclusions lies in a basis of the work of Jarnik and Kurzweil [24], the idea of using Aumann-type integrals is still better known. To facilitate some comparison of integrals, in the multivalued case, an Aumann-type integral will be also considered via Henstock-integrable selections. The really first definitions of multivalued Henstock-Kurzweil integrals can be found in [4] and [24]. For the Riemann-type definition and Debreu-type definition of the Henstock-Kurzweil multivalued integral see also [19]. A set-valued function F : [0, 1] → 2E is AumannHenstock-Kurzweil integrable if the collection of its HK-integrable selections SFHK is non-empty. In this case, the definition of Aumann-Henstock integrals of F is analogous to that of Aumann integral. As in a single-valued case it is possible to unify both concepts presented above by introducing a new integral. For multifunctions it can be found in [18]. If a multifunction has closed convex and bounded values we can define, similarly like in previous considerations, the multivalued Henstock-Kurzweil-Pettis integral by assuming scalar HK-integrability together with existence of a closed convex and bounded set IA in E such that s(x∗ , IA ) = (HK) A s(x∗ , F (t)) dt for each subinterval IA of I ([18], [19]). It is possible also to consider the integrability for different classes of subsets ([20] ). Some examples can be found in [18, Section 3].
3. Results Now, let us present a few definitions of solutions for the Cauchy problem (t ∈ I) and we show some dependencies between them: x (t) = f (t, x(t)) , x(0) = x0 ∗
(3.1)
∗
or a scalar problem considered for arbitrary x ∈ E : (x∗ x) (t) = x∗ f (t, x(t)) , x(0) = x0 .
(3.2)
For a more complete theory see [14]. Let us start with some assumptions. (1◦ ) Continuity of solutions: a) x is absolutely continuous (AC function), b) x is an ACG∗ function, c) x is weakly ACG∗ . (2◦ ) Differentiability hypotheses: a) x ∈ C 1 , b) x is weakly differentiable, c) x is differentiable a.e., d) x is pseudo-differentiable. A function x is a solution of (3.1) if it satisfies the initial condition together with: a) classical solution ⇔ (1◦ a) and (2◦ a), b) weak solution ⇔ (1◦ a) and
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(2◦ b), (Szep, 1971) c) Carath´eodory solution ⇔ (1◦ a) and (2◦ c), d) pseudosolution ⇔ (1◦ a) and (2◦ d), (Knight, 1974) e) Kurzweil solution (K-solution) ⇔ (1◦ b) and (2◦ c), (Kurzweil, 1957) f) pseudo-K-solution ⇔ (1◦ c) and (2◦ d), (Cicho´ n et al., 1999) and the derivative of x taken in the sense of (2◦ ) satisfies the equation (3.1) (for each t in the cases (2◦ a) and (2◦ b), almost everywhere in the cases (2◦ c) and (2◦ e) or satisfies (3.2) in the cases (2◦ d) and (2◦ f )). Let us note that under the appropriate conditions of integrability of the righthand side each solution of the problem (3.1) is equivalent to the solution of the integral problem t x(t) = x0 + f (s, x(s))ds, (3.3) 0
where the integral is depending on the type of considered solutions of (3.1). Theorem 3.1. ([14]) If the right-hand side of (3.1) is integrable in the sense considered below, respectively, then each solution x of the problem (3.1) is equivalent to the solution y of the integral equation (3.3) in the following cases: (a) (b) (c) (d) (e) (f)
x x x x x x
– – – – – –
classical solution ⇔ the Riemann integral, weak solution ⇔ the weak Riemann integral, Carath´eodory solution ⇔ the Bochner (Lebesgue) integral, pseudo-solution ⇔ the Pettis integral, K-solution ⇔ the Henstock-Kurzweil integral, pseudo-K-solution ⇔ the Henstock-Kurzweil-Pettis integral.
As a consequence of the above theorem we get the following fact ([14]): if we consider the following classes of solutions for the problem (3.1): (a) classical solutions, (b) Carath´eodory solutions, (c) weak solutions, (d) pseudo-solutions, (e) Kurzweil solutions, (f) pseudo-K-solutions. Then (a) ⊂ (c) ⊂ (b) ⊂ (d) ⊂ (f ) and (b) ⊂ (e) ⊂ (f ). Moreover, all these inclusions are proper. Let us turn to the differential inclusions. Consider the following problem: x (t) ∈ F (t, x(t)),
x(0) = g(x),
t ∈ [0, α] = I .
(3.4)
In contrast to the single-valued case only in a limited number of papers it was considered the other solutions than Carath´eodory ones. In the beginning, the Bochner integrability was replaced by the Pettis integrability, but due to the other assumptions as a solution was still considered a Carath´eodory (cf. Castaing, Valadier [11] Theorem VI-7). But in the paper of Tolstonogov both type of solutions were checked: Carath´eodory or weak one ([29], cf. also Maruyama [25]). The next result was obtained by Arino, Gautier and Penot [3] for weakly-weakly usc multifunctions F with almost weakly relatively compact images. In the last paper the existence of pseudo-solutions was proved, but under the assumptions from this paper they are in fact Carath´eodory solutions, too. Let us note that in all mentioned papers it was considered the local problem, i.e., g(x) = x0 . In the last years, thanks to the progress of the theory for multivalued integrals, this theory was rediscovered. Let us mention the papers of Godet-Thobie and Satco [21], Satco [27],
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[28], Azzam-Laouir, Castaing and Thibault [7] or Azzam-Laouir and Boutana [6] (results for the different kind of multivalued problems). Let us begin by proving the following important lemma (cf. [12], [6] or [27]): Lemma 3.2. Assume that E ∗ is separable. Let v ∈ C(I, E) and F : I × E → 2E is such that: (i) F (·, x) – has a weakly measurable selection for each x ∈ E, (ii) F (t, ·) – w-seq. uhc for each t ∈ I, (iii) F (t, x) are nonempty, closed and convex, (iv) F (t, x) ⊂ G(t) a.e., G has nonempty convex and weakly compact values and is Pettis uniformly integrable on I. Then there exists at least one Pettis integrable selection z0 of F (·, v(·)). Proof. Take a sequence of simple functions vn , such that vn → v uniformly on I. Thus by (i) there exists a weakly measurable selection zk s.t. zk (·) ∈ F (·, vk (·)). By our assumption on E the selection zk is, in fact, measurable. Put H(t) = conv{zk (t) : k ≥ 1}. Since zk (·) is measurable, {zk (·) : k ≥ 1} is measurable and hence H(·) is measurable. Moreover H(t) ⊂ convF (t, V (t)), where V (t) = {vk (t) : k ≥ 1}. But (vk ) is a convergent sequence, so V (t) is relatively compact. By Lemma 2 from [12] and Mazur’s lemma we have that the values convF (t, V (t)) are weakly compact. Our multifunction H is weakly measurable, H(t) ⊂ G(t) a.e. therefore by our assumption (iv) and due to Lemma 2.2. from [10] H is Pettis integrable. Since Pe SH = ∅, it is sequentially compact for the topology of pointwise convergence on L∞ (I, E)⊗E ∗ (cf. [2] Proposition 3.4 and [27] for the definition of Pettis uniformly integrable multifunctions). Thus we are able to extract a subsequence (znk ) of Pe (zn ) which is convergent in σ(PE1 , L∞ (I, E) ⊗ E ∗ ) topology to some z0 ∈ SH , i.e., Pe 1 znk −→ z0 ∈ SH . Here PE denotes the space of Pettis integrable functions from I to E. Denote by D a dense sequence for the Mackey topology in the unit ball in E ∗ . For each fixed x∗ ∈ D, Lemma 12 from [27] applied to the sequence x∗ znk gives us the existence of the “Mazur” sequence vk ∈ conv{x∗ znm : m ≥ k} such that vk is a.e. pointwisely convergent to some measurable x∗ z0 . The convergence theorem (Lemma 1 from [12]) can be applied in our situation (with some necessary changes in a suitable part of the proof: Lemma III.33 together with Corollary I.15 from [11] instead of the Separation Theorem). Thus x∗ z0 (t) ≤ s(x∗ , F (t, v(t))) a.e. and finally z0 (t) ∈ F (t, v(t)). Let us present an existence result for the problem (3.4). We extend some previous results by using an assumption expressed in terms of the measure of weak noncompactness instead of the strong one. This allows to cover the case of weakly compact mappings (or the sum of Lipschitz and weakly compact mappings). Moreover, this results remain true also for standard Cauchy problem, i.e., when g(x) = x0 as well as for classical functions g which are useful to describe some phenomena k by using nonlocal (nonstandard) conditions. For instance: g(x) = n=1 cn · x(tn ) T for some 0 ≤ t1 ≤ · · · ≤ tk ≤ T or g(x) = T10 · 0 0 x(s)ds for T0 < T .
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Theorem 3.3. Assume that E has separable dual E ∗ . Let F : I × E → 2E with nonempty convex and weakly compact values satisfies (a) F (t, ·) is weakly sequentially upper hemi-continuous for each t ∈ I, (b) for each continuous function x : I → E the multifunction F (·, x(·)) has Pettis-integrable selection, (c) F (t, x) ⊂ G(t) a.e., for some Pettis (uniformly) integrable multifunction G with values in cwk(E), (d) there exists a constant k < 1 such that for each bounded subset B ⊂ E and J ⊂ I we have ω(F (J × B)) ≤ k · ω(B), (e) the function g : E → E is convex, bounded say by N1 , i.e., g(x) ≤ N1 for each x, weakly-weakly sequentially continuous and ω(g(B)) ≤ d · ωC (B) for each bounded subset B of E and for some d such that d + k < 1. Then there exists at least one pseudo-solution of the Cauchy problem (3.4) on some J ⊂ I. Proof. By the hypothesis on the space E we get the uniqueness of the pseudoderivative accurate to a set of measure zero. From the definition of the pseudosolution it follows that (as in earlier papers about nonlocal problems) a pseudosolution of our problem (3.4) is, at the same time, a solution of the integral inclusion t x(t) ∈ g(x) + (P ) 0 F (s, x(s))ds. Then, we are interesting in finding a fixed point of R : C(I, E) → 2C(I,E) : t R(x)(t) = g(x) + (P ) 0 F (s, x(s))ds. Let W = {f ∈ PE1 : f (t) ∈ G(t) a.e. on I} and U = {xf ∈ C(I, E) : xf (t) = t g(xf ) + (P ) 0 f (s)ds, t ∈ I, f ∈ W }. For f ∈ W and x∗ ∈ E ∗ we have x∗ f ≤ s(x∗ , G). Then, by our assumptions, W is Pettis uniformly integrable (cf. [21]). Thus for arbitrary xf ∈ U and t, τ ∈ I there exists an appropriate f ∈ W and t τ ∗ ∗ x f (s)ds − x∗ f (s)ds) xf (t) − xf (τ ) = sup x (xf (t) − xf (τ )) = sup ( x∗ ∈B ∗
= sup
x∗ ∈B ∗
τ
t
x∗ f (s)ds ≤ sup
x∗ ∈B ∗ t
x∗ ∈B ∗
0
0
s(x∗ , G(s))ds.
τ
By uniform Pettis integrability of G it follows that U is an equicontinuous subset of C(I, E). The property of the multivalued Pettis integral gives us the convexity of U . Then U is nonempty, closed, convex, bounded and equicontinuous in C(I, E). As the set U is strongly equicontinuous then for each M > 0 there exists α ∈ I t such that for each t ∈ [0, α] and f ∈ SFP e we have (P ) 0 f (s)ds ≤ M. We fix M > 0. We have the following estimation: t R(x)(t) ≤ g(x) + (P ) 0 F (s, x(s))ds ≤ N1 + M. Denote by N = N1 + M and so by BN the ball {x ∈ C(I, E) : x ≤ N }. e Lemma 3.2 ensures us, that for each x ∈ C(I, E) SFP (·,x(·)) is nonempty, then R(x) = ∅ (for each x ∈ BN ). By the properties of multivalued Pettis integrals,
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the values of R are weakly compact and convex (because F has such values). Let e e ⊂ W for each x ∈ V . Indeed, for g ∈ SFP (·,x(·)) V = U ∩BN . It is clear that SFP (·,x(·)) g(t) ∈ F (t, x(t)) ⊂ G(t) a.e. In the next part of the proof we will consider V as a domain of R. Obviously R : V → 2V . Now, we are in a position to show, that R has a weakly-weakly sequentially closed graph. Let (xn , yn ) ∈ GrR, (xn , yn ) → (x, y) weakly in C(I, E) × C(I, E). From our assumptions it follows that g(xn ) tends weakly to g(x). Moreover, yn is of the following form t e yn (t) = g(xn ) + (P ) 0 fn (s)ds , fn ∈ SFP (·,x , t ∈ I. n (·)) But ω({fn (t) : n ≥ 1}) ≤ ω({F (t, xn (t)) : n ≥ 1} ≤ k · ω{xn (t) : n ≥ 1}) a.e. Since xn is weakly convergent in C(I, E), {xn (t) : n ≥ 1} is relatively weakly compact in E. Hence ω({xn (t) : n ≥ 1}) = 0 and finally ω({fn (t) : n ≥ 1}) = 0 a.e. on I. By redefining a new multifunction H on the set of measure zero: H(t) = conv{fn (t) : n ≥ 1} we can say that H(t) are nonempty closed convex and weakly compact. Pe As SH is nonempty, convex and is sequentially compact for the topology of pointwise convergence on L∞ ⊗ E ∗ ([2] Proposition 3.4), then we extract a subsequence (fnk ) of (fn ) such that (fnk ) converges σ(PE1 , L∞ ⊗ E ∗ ) to a function Pe f ∈ SH in such a way that (xnk ) converges weakly to a continuous function x. Fix an arbitrary x∗ ∈ E ∗ . By weak sequential hemi-continuity of F (t, ·) and weak convergence of xnk (t) in E we obtain weak convergence of (s(x∗ , F (t, xnk (t))). Remember, that C ∈ cwk(E) iff s(·, C) is τ (E ∗ , E)-continuous on E ∗ . Denote by D a dense sequence for the Mackey topology in the ball in E ∗ . Since (fnk ) unit 1 ∞ ∗ Pe ∗ converges σ(PE , L ⊗ E ) to f ∈ SH , we have A x fnk (s)ds → A x∗ f (s)ds t for each measurable A ∈ I and x∗ ∈ E ∗ . Thus ynk = g(xnk ) + (P ) 0 fnk (s)ds t and for each x∗ ∈ D we obtain that x∗ ynk tends to x∗ g(x) + 0 x∗ f (s)ds, i.e., t t e y = g(x) + (P ) 0 f (s)ds. Whence y(t) = g(x) + (P ) 0 f (s)ds , f ∈ SFP (·,x(·)) and (x, y) ∈ GrR. Take an arbitrary bounded subset B of V . For any selection f of F (·, B(·)) t we have (P ) 0 f (s)ds ∈ t · convf (I) ⊂ t · convF (I × B(I)). Thus R(B)(t) ⊂ g(B) + t · convF (I × B(I)). By using the properties of ωC we are able to prove that R is a contraction with respect to the measure of weak noncompactness: ω(R(B)(t)) ≤ ω(g(B))+ ω(F (I × B(I)) ≤ d·ωC (B)+ k ·ω(B(I)) ≤ (d+ k)·ωC (B). property. Thus ωC R(B) ≤ (d + k)ωC (B). As d + k < 1 we obtain the desired ∞ Define a sequence of sets: K0 = V , Kn+1 = convR(Kn ), and a set K = n=0 Kn . All the sets Kn are nonempty equicontinuous closed and convex. Moreover, it can be proved (by induction) that it is a decreasing sequence of sets. t We have ω(R(Kn (t)) ≤ ω(g(Kn )) + ω((P ) 0 F (s, Kn (s))ds). By the mean value theorem for the Pettis integral: t (P ) t−τ F (s, Kn (s)ds ∈ τ · conv{F (s, Kn (s)) : s ∈ [t − τ, t]}. Thus ω(R(Kn (t))) ≤ d·ωC (Kn )+ t·ω(convF ([0, t]× Kn ([0, t]))) ≤ d·ωC (Kn ) +t · k · ω(Kn ([0, t])). For sufficiently small t we have ω(R(Kn (t)) < ωC (Kn ) since ωC (R(Kn )) = supt∈I ω(R(Kn (t))). Finally ωC (R(Kn )) < ωC (Kn ). As ω(Kn (t)) =
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ω(convR(Kn−1 (t))) = ω(Kn−1 (t)) we obtain ωC (R(Kn )) = ωC (Kn−1 ) and consequently ωC (Kn−1 ) < ωC (Kn ). The sequence (ωC (Kn )) is decreasing and bounded below by zero, so is convergent. From the above consideration it follows also that ωC (Kn ) ≤ (d + k)n ωC (K0 ) and therefore the limit must be zero. The CantorKuratowski intersection lemma for the weak measure of noncompactness ([15]) ensures us that K is weakly compact (cf. [12]). By the properties of multivalued mappings we obtain that R is weakly-weakly upper semi-continuous on K. Then we have weakly-weakly upper semi-continuous multifunction R : K → cwk(K) (R(K) ⊂ K, cf. Lemma 2 in [12]), so the fixed point theorem of Kakutani type for weak topology ([3]) applies to the map R and we get a fixed point z of R. Of course, z is a pseudo-solution of problem (3.4).
References [1] A. Amrani, Lemme de Fatou pour l’int´ egrale de Pettis, Publ. Math. 42 (1998), 67–79. [2] A. Amrani, Ch. Castaing, Weak compactness in Pettis integration, Bull. Polish Acad. Sci. Math. 45 (1997), 139–150. [3] O. Arino, S. Gautier and J.P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkc. Ekvac. 27 (1984), 273–279. [4] Z. Artstein, J. Burns, Integration of compact set-valued functions, Pacific J. Math. 58 (1975), 297-3-7. [5] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12. [6] D. Azzam-Laouir, I. Boutana, Application of Pettis integration to differential inclusions with three-point boundary conditions in Banach spaces, Electron. J. Differential Equations 173 (2007), 1–8. [7] D. Azzam-Laouir, C. Castaing and L. Thibault, Three boundary value problems for second-order differential inclusions in Banach spaces. Well-posedness in optimization and related topics, Control Cybernet. 31 (2002), 659–693. [8] A. Boccuto, A.R. Sambucini, A McShane integral for multifunction, J. Concr. Appl. Math. 2 (2004), 307–325. [9] B. Cascales, J. Rodr´ıgues, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), 540–560. [10] B. Cascales, V. Kadets and J. Rodr´ıgues, The Pettis integral for multi-valued functions via single-valued ones, J. Math. Anal. Appl. 332 (2007), 1–10. [11] Ch. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer, Berlin, 1977. [12] M. Cicho´ n, Differential inclusions and abstract control problems, Bull. Austral. Math. Soc. 53 (1996), 109–122. [13] M. Cicho´ n, Convergence theorems for the Henstock-Kurzweil-Pettis integral, Acta Math. Hungar. 92 (2001), 75–82. [14] M. Cicho´ n, On solutions of differential equations in Banach spaces, Nonlin. Anal. Th. Meth. Appl. 60 (2005), 651–667.
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[15] F. DeBlasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259–262. [16] G. Debreu, Integration of correspondences, in: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability 1965/66, Berkeley, 1967, pp. 351–372. [17] A. Dinghas, Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Zeit. 66 (1956), 173–188. [18] L. Di Piazza, K. Musial, Set-valued Kurzweil-Henstock-Pettis integral, Set-Valued Anal. 13 (2005), 167–179. [19] L. Di Piazza, K. Musial, A decomposition theorem for compact-valued Henstock integral, Monatsh. Math. 148 (2006), 119–126. [20] K. El Amri, Ch. Hess, On the Pettis integral of closed valued multifunctions, SetValued Anal. 8 (2000), 329–360. [21] C. Godet-Thobie, B. Satco, Decomposability and uniform integrability in Pettis integration, Quaest. Math. 29 92006), 39–58. [22] R.A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Providence, Rhode Island, 1994. [23] M. Hukuhara, Int´egration des applications measurable dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 205–223. [24] J. Jarnik, J. Kurzweil, Integral of multivalued mappings and its connection with ˇ differential relations, Casopis pro Peˇstov´ ani Matematiky 108 (1983), 8–28. [25] T. Maruyama, A generalization of the weak convergence theorem in Sobolev spaces with application to differential inclusions in a Banach space, Proc. Japan Acad. Ser. A Math Sci. 77 (2001), 5–10. [26] A.R. Sambucini, A survey on multivalued integration, Atti Sem. Mat. Fis. Univ. Modena 50 (2002), 53–63. [27] B. Satco, Volterra integral inclusions via Henstock-Kurzweil-Pettis integral, Discuss. Math. Differ. Incl. Control Optim. 26 (2006), 87–101. [28] B. Satco, Second-order three boundary value problem in Banach spaces via Henstock and Henstock-Kurzweil-Pettis integral, J. Math. Anal. Appl. 332 (2007), 919–933. [29] A.A. Tolstonogov, On comparison theorems for differential inclusions in locally convex spaces. I. Existence of solutions, Differ. Urav. 17 (1981), 651–659 (in Russian). [30] M. Valadier, On the Strassen theorem, in: Lect. Notes in Econ. Math. Syst. 102, 203-215, ed. J.-P. Aubin, Springer, 1974. [31] H. Ziat, Convergence theorems for Pettis integrable multifunctions, Bull. Polish Acad. Sci. Math. 45 (1997), 123–137. Kinga Cicho´ n Institute of Mathematics, Faculty of Electrical Engineering Poznan University of Technology, Piotrowo 3a, 60-965 Pozna´ n, Poland e-mail: [email protected] Mieczyslaw Cicho´ n Faculty of Mathematics and Computer Science, Adam Mickiewicz University Umultowska 87, 61-614 Pozna´ n, Poland e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 125–133 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Equations Involving the Mean of Almost Periodic Measures Silvia-Otilia Corduneanu Abstract. We use the theory of Fourier-Bohr series for almost periodic measures to looking for a complex-valued function f which is almost periodic on R and satisfies equation f (x) = My [f (x − y)μ(y)] + ν ∗ h(x),
x ∈ R.
(E)
In this context h is an almost periodic function on R, μ is a positive almost periodic measure on R and ν is a bounded measure also on R. With a suitable choice of the measures μ and ν equation (E) becomes +∞ t 1 f (x) = lim f (x − y)g(y) dy + ϕ(y)h(x − y) dy, x ∈ R, t→∞ 2t −t −∞ where g is an almost periodic function on R and ϕ belongs to L1 (R). Mathematics Subject Classification (2000). Primary 42A05, 42A10, 42A16, 42A38, 43A25, 43A40, 43A60; Secondary 39B32. Keywords. Almost periodic function, almost periodic measure, functional equation, Fourier series, Fourier-Bohr coefficients.
1. Introduction Let AP (R) be the space of all almost periodic complex-valued functions defined on R, ap(R) the space of all almost periodic complex-valued measures on R and ap+ (R) the subspace of ap(R) containing positive measures. We denote by M (μ) or Mx [μ(x)] the mean of an almost periodic measure μ and by M (f ) or Mx [f (x)] the mean of an almost periodic function f . L.N. Argabright and J. Gil de Lamadrid proved that the measure f μ which is defined by an almost periodic function f as density and an almost periodic measure μ as base, is also an almost periodic measure (see [2], [9]). If f ∈ AP (R) and μ ∈ ap+ (R), the mean My [f (x − y)μ(y)] having the parameter x ∈ R is an almost periodic function as function of variable
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x ∈ R (see [4]). In this paper we solve equation f (x) = My [f (x − y)μ(y)] + ν ∗ h(x),
x ∈ R,
(1.1)
where h ∈ AP (R), μ ∈ ap+ (R) and ν is a bounded complex-valued measure on R. A solution is an almost periodic function f which satisfies the equation. Let λ be the Lebesgue measure on R, g ∈ AP (R), ϕ ∈ L1 (R). In the case of μ = gλ and ν = ϕλ, equation (1.1) becomes +∞ 1 t f (x) = lim f (x − y)g(y) dy + ϕ(y)h(x − y) dy, x ∈ R. (1.2) t→∞ 2t −t −∞ For solving equation (1.1) we use the theory of Fourier-Bohr series for almost 1 periodic measures and that of Fourier series for almost periodic functions. Let R be the dual group of the group R. Consider f ∈ AP (R) and μ ∈ R. For every 1 the Fourier coefficient of f corresponding to the character γ is denoted γ ∈ R, by cγ (f ) and is defined by cγ (f ) = M (γf ). On the other hand the Fourier-Bohr coefficient of μ corresponding to the character γ is denoted by cγ (μ) and is defined 1 | M (γf ) = 0} is at most a by cγ (μ) = M (γμ). It is proved that the set {γ ∈ R countable set (see Theorem 1.15 from [3]). Hence, we can denote the previous set 1 | n ∈ N} and we define the Fourier series of f as being by {γn ∈ R ∞
cγn (f )γn .
n=1
We calculate the mean of the almost periodic function Φ : R → C defined by Φ(x) = My [f (x − y)μ(y)],
x∈R
and we obtain the equality M (Φ) = M (f )M (μ) which induces the another one cγ (Φ) = cγ (f )cγ (μ),
1 γ ∈ R.
(1.3)
Taking into account (1.3) and the property that two almost periodic functions coincide if they have the same Fourier coefficients we solve equations (1.1) and (1.2).
2. Preliminaries Consider C(R) the set of all bounded continuous complex-valued functions on R and denote by · the supremum norm defined on C(R). Let λ be the Lebesgue measure on R. The space of Lebesgue measurable functions f on R, with R |f (x)| dλ(x) < ∞ will be denoted by L1 (R). We use mF (R) to denote the space of all bounded 1 to denote the dual group of the group R. Consider [R] 1 the measures on R and R 1 space of all trigonometric polynomials on R. If ϕ ∈ L (R), the Fourier transform of ϕ, denoted by ϕ, 1 is given by 1 ϕ(x)γ(x) dλ(x), γ ∈ R, ϕ(γ) 1 = R
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127
and if ν ∈ mF (R) the Fourier-Stieltjes transform of ν, denoted by ν1, is given by 1 ν1(γ) = γ(x) dν(x), γ ∈ R. R
For f ∈ C(R) and a ∈ R, the translate of f by a is the function fa (x) = f (x + a) for all x ∈ R. In [2], [3], [7], [8], [9], there are defined the almost periodic functions. Definition 2.1. A function f ∈ C(R) is called an almost periodic function on R, if the family of translates of f , {fa | a ∈ R} is relatively compact in the sense of uniform convergence on R. The set AP (R) of all almost periodic functions on R is a Banach algebra with 1 There exists respect to the supremum norm, closed to conjugation and contains R. a unique positive linear functional M : AP (R) → C such that M (fa ) = M (f ), for all a ∈ R, f ∈ AP (R) and M (1) = 1. We denote by 1 the constant function which is 1 for all x ∈ R. If f ∈ AP (R) we define the mean of f as being the 1 and we call above complex number M (f ), we put cγ (f ) = M (γf ) for all γ ∈ R 1 cγ (f ), the Fourier coefficient of f corresponding to γ ∈ R. Next, we recall the definition of the Fourier series of an almost periodic function. If f ∈ AP (R) the 1 | M (γf ) = 0} is at most a countable set (see Theorem 1.15 from [3]) set {γ ∈ R 1 | n ∈ N}. The Fourier series of f is and we denote it by {γn ∈ R ∞
cγn (f )γn .
n=1
If ϕ ∈ L1 (R) and h ∈ AP (R), their convolution, ϕ ∗ h, belongs to AP (R). We recall that ϕ ∗ h(x) = h(x − y)ϕ(y) dλ(y), x ∈ R. R
If ν ∈ mF (R) and h ∈ AP (R) we have that their convolution, ν ∗ h, belongs to AP (R). We remind the reader that ν ∗ h(x) = h(x − y) dν(y), x ∈ R. R
We denote by m(R) the space of complex Radon measures on R and by mB (R) the subspace of m(R) containing the translations-bounded measures (see [1]). Let K(R) be the subset of C(R) containing functions which have a compact support. We say that μ ∈ mB (R) is an almost periodic measure, and we denote it by μ ∈ ap(R), if for every ϕ ∈ K(R), ϕ ∗ μ ∈ AP (R) (see [2], [9]). Consider ϕ ∈ K(R) such that λ(ϕ) = 1. We can define the mean of an almost periodic measure μ ∈ ap(R) as being the number M (μ) ∈ C defined by M (μ) = M (ϕ ∗ μ). L.N. Argabright and J. Gil de Lamadrid proved that the measure f μ which is defined by an almost periodic function f as density and an almost periodic measure μ as base, is also an almost periodic measure (see [2], [9]). Denote by ap+ (R) the subspace of ap(R) containing positive measures. If f ∈ AP (R) and μ ∈ ap+ (R), the mean My [f (x − y)μ(y)] having the parameter x ∈ R is an almost periodic function as function of
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1 and we variable x ∈ R (see [4]). If μ ∈ ap(R) we put cγ (μ) = M (γμ) for all γ ∈ R 1 call cγ (μ), the Fourier-Bohr coefficient of μ corresponding to γ ∈ R.
3. Properties of the almost periodic functions Theorem 3.1. Consider f ∈ AP (R) and A ⊂ R a compact set. The function χ : R → C defined by χ(y) = f (x − y) dλ(x), y ∈ R, A
is an almost periodic function on R. 1 We obtain Proof. We first suppose that f = γ ∈ R. 1A (γ). γ(x)γ(y) dλ(x) = γ(y) 1A (x)γ(x) dλ(x) = γ(y)1 χ(y) = R
A
1 then χ ∈ AP (R). It immediately follows, that for f ∈ [R], 1 Therefore, if f = γ ∈ R, the function χ is an almost periodic function on R. Suppose that f ∈ AP (R) and is arbitrary. There exists a sequence (fn )n such that 1 ∧ (fn − f → 0). ((∀n ∈ N)(fn ∈ [R])) If we denote by
fn (x − y) dλ(x),
χn (y) =
y ∈ R, n ∈ N
A
we obtain that (∀n ∈ N)(χn ∈ AP (R)). For every n ∈ N and y ∈ R we have |χn (y) − χ(y)| ≤ |fn (x − y) − f (x − y)| dλ(x) ≤ λ(A)fn − f . A
It follows that χn → χ, uniformly, hence χ ∈ AP (R).
Theorem 3.2. Consider μ ∈ ap+ (R). For every f ∈ AP (R) the following equality is true 2 t 3 t My [f (x − y)μ(y)] dx = My f (x − y) dx μ(y) . (3.1) −t
−t
1 We obtain Proof. We first suppose that f = γ ∈ R. t My [γ(x)γ(y)μ(y)] dx = cγ (μ) −t
and
2 My
t
−t
3 γ(x)γ(y) dx μ(y) = cγ (μ)
t
γ(x) dx −t t
γ(x) dx. −t
1 it follows that the equality (3.1) So, because the equality (3.1) is valid for f ∈ R 1 is valid for f ∈ [R]. Consider f ∈ AP (R). There exists a sequence (fn )n such that 1 and fn → f in the sense of uniform convergence. Consider (∀n ∈ N)(fn ∈ [R])
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129
t > 0. For every n ∈ N we have the following inequalities 2 t 3 t My [f (x − y)μ(y)] dx − My f (x − y) dx μ(y) −t
≤
−t
My [f (x − y)μ(y)] dx − 2
+ My ≤
−t t
t
t
−t
−t
My [fn (x − y)μ(y)] dx
2 3 fn (x − y) dx μ(y) − My
t
−t
3 f (x − y) dx μ(y)
t
−t
+ My
My [(f (x − y) − fn (x − y))μ(y)] dx 2
t
−t
3 fn (x − y) − f (x − y) dx μ(y)
≤ 4tM (μ)fn − f . Therefore the equality (3.1) holds for every f ∈ AP (R).
Theorem 3.3. Consider μ ∈ ap+ (R) and f ∈ AP (R). If Φ : R → C is the almost periodic function defined by Φ(x) = My [f (x − y)μ(y)],
x∈R
then the following equality is true M (Φ) = M (f )M (μ). Proof. We have that 1 t→∞ 2t
(3.2)
t
M (Φ) = lim
−t
My [f (x − y)μ(y)] dx,
so, equality (3.2) is equivalent to 1 t lim My {[f (x − y) − f (x)]μ(y)} dx = 0. t→∞ 2t −t
(3.3)
From Theorem 3.2 we have that 1 t My {[f (x − y) − f (x)]μ(y)} dx 2t −t 2 t 3 1 = My (f (x − y) − f (x)) dx μ(y) . 2t −t According to Theorem 1.12 from [3] we obtain that 1 t [f (x − y) − f (x)] dx = 0, lim t→∞ 2t −t uniformly with respect to y ∈ R. Hence equality (3.3) follows.
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Corollary 3.4. Consider μ ∈ ap+ (R) and f ∈ AP (R). If Φ : R → C is the almost periodic function defined by Φ(x) = My [f (x − y)μ(y)],
x∈R
then the following equality is true cγ (Φ) = cγ (f )cγ (μ),
1 γ ∈ R.
(3.4)
1 We observe that Proof. Consider γ ∈ R. γΦ(x) = My [(γf )(x − y)γμ(y)],
x ∈ R.
Hence, using Theorem 3.3, it results that cγ (Φ) = M (γΦ) = M (γf )M (γμ) = cγ (f )cγ (μ).
4. Equations with almost periodic measures and functions We solve the following equation (4.1) f (x) = My [f (x − y)μ(y)] + ν ∗ h(x), x ∈ R, where h ∈ AP (R), μ ∈ ap+ (R) and ν ∈ mF (R). A solution of (4.1) is an almost periodic function f which satisfies the equation. Theorem 4.1. Consider a function h ∈ AP (R) such that its Fourier series is ∞ cγn (h)γn . n=1
Let μ ∈ ap+ (R) such that there exists δ > 0 with the property cγn (μ) − 1 > δ, and ν ∈ mF (R) such that
∞
n∈N
|1 ν (γn )|2 < ∞.
n=1
Then equation (4.1) has a solution f ∈ AP (R). 1 Let us suppose that f ∈ AP (R) is a solution of equation Proof. Consider γ ∈ R. (4.1). Then the function Φ : R → C defined by Φ(x) = My [f (x − y)μ(y)],
x∈R
is an almost periodic function and cγ (Φ) = cγ (f )cγ (μ). Taking into account that cγ (ν ∗ h) = ν1(γ)cγ (h), one obtains cγ (f ) = cγ (f )cγ (μ) + ν1(γ)cγ (h). We return to looking for a solution of (4.1). This previous observations suggest us to consider the series ∞ ν1(γn )cγn (h) γn . (4.2) 1 − cγn (μ) n=1
Equations Involving the Mean of Almost Periodic Measures It is obvious that ν1(γn )cγn (h) 1 γn (x) ≤ | ν1(γn ) | | cγn (h) |, 1 − cγn (μ) δ
131
x ∈ R.
On the other hand we have the Parseval equality 2
M (|h| ) =
∞
| cγn (h) |2 ,
n=1
hence the Cauchy inequality ∞
| ν1(γn ) | | cγn (h) | ≤
n=1
∞
12 | ν1(γn ) |
2
n=1
∞
12 | cγn (h) |
2
,
n=1
gets us that the series (4.2) is uniform convergent on R. We denote by f the sum of this series. It is obvious that f is an almost periodic function. Based on the property that two almost periodic functions coincide if they have the same Fourier series we conclude that f is a solution of equation (4.1). Next we discuss the equation ∞ 1 t f (x) = lim f (x − y)g(y) dy + h(x − y)ϕ(y) dy, t→∞ 2t −t −∞
x ∈ R.
(4.3)
In this context g ∈ AP (R), h ∈ AP (R) and ϕ ∈ L1 (R). A solution of (4.3) is an almost periodic function f which satisfies the equation. Corollary 4.2. Consider a function h ∈ AP (R) such that its Fourier series is ∞
cγn (h)γn .
n=1
Let g ∈ AP (R) a positive function such that there exists δ > 0 with the property cγn (g) − 1 > δ, and ϕ ∈ L1 (R) such that
∞
n∈N
|ϕ(γ 1 n )|2 < ∞.
n=1
Then equation (4.3) has a solution f ∈ AP (R). Proof. Consider the measures μ = gλ and ν = ϕλ. The Lebesgue measure λ is almost periodic on R, therefore, the measure μ is a positive almost periodic on R. On the other hand the measure ν is bounded. With this choice equation (4.1) becomes (4.3). It is easy to see that cγ (μ) = cγ (g) and ν1(γ) = ϕ(γ), 1
1 γ∈R
Therefore we can apply Theorem 4.1 and we find a solution f ∈ AP (R) for (4.3).
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Notation 4.3. Consider h ∈ AP (R) having the Fourier series ∞
cγn (h)γn .
n=1
For every n ∈ N there exists an ∈ R such that γn (x) = eian x ,
x ∈ R.
We denote Exp(h) = {an ∈ R | n ∈ N}. Application 4.4. Consider a function h ∈ AP (R) with the Fourier series ∞
cγn (h)γn ,
n=1
which satisfies the properties that Exp(h) \ Z is a finite set and 0 ∈ / Exp(h). Then the equation 2π 1 t f (x) = lim f (x − y) dy + h(x − y) dy, x ∈ R (4.4) t→∞ 2t −t 0 has a solution f ∈ AP (R). Proof. We observe that 2π h(x − y) dy = 1[0,2π] ∗ h(x),
x ∈ R,
0
$
where
1, x ∈ [0, 2π] 0, x ∈ / [0, 2π].
1[0,2π] (x) =
Choosing g ≡ 1 on R we can see that equation (4.4) is a particular case of (4.3). The set Exp(h) = {an ∈ R | n ∈ N} can be represented as {an ∈ R | n ∈ N} = {bn ∈ Z | n ∈ N} ∪ {cm ∈ R \ Z | m = 1, 2, . . . , p}. The previous partition of {an ∈ R | n ∈ N} induces the following partition for 1 | n ∈ N} in the following way {γn ∈ R 1 | n ∈ N} = {γ 1 ∈ R 1 | n ∈ N} ∪ {γ 2 ∈ R 1 | m = 1, 2, . . . , p} {γn ∈ R n m where for every n ∈ N
γn1 (x) = eibn x , and for every m ∈ {1, 2, . . . , p} 2 γm (x) = eicm x ,
x∈R x ∈ R.
It is easy to see that for every n ∈ N, cγn (g) = 0 and 2π 1 1 e−ibn x dx = 0. [0,2π] (γn ) = 0
Equations Involving the Mean of Almost Periodic Measures Therefore
∞
2 |1 [0,2π] (γn )| =
n=1
p
133
2 2 |1 [0,2π] (γm )| < ∞,
m=1
hence we can apply Corollary 4.2 and we conclude that equation (4.4) has a solution f ∈ AP (R).
References [1] L.N. Argabright and J. Gil de Lamadrid, Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups, Mem. Amer. Math. Soc. 145 (1974). [2] L.N. Argabright and J. Gil de Lamadrid, Almost Periodic Measures, Mem. Amer. Math. Soc. 428 (1990). [3] C. Corduneanu, Almost Periodic Functions, Interscience Publishers, New York, London, Sydney, Toronto, 1968. [4] S.O. Corduneanu, Inequalities for Almost Periodic Measures, Mathematical Inequalities & Applications, Volume 5, No. 1 (2002), 105–111. [5] S.O. Corduneanu, Inequalities for a Class of Means with Parameter, Buletinul Institutului Politehnic din Ia¸si, Tomul LIII (LVII), Fasc. 5 (2007), 77–84. [6] N. Dinculeanu, Integrarea pe Spat¸ii Local Compacte, Editura Academiei R.P.R., Bucure¸sti, 1965 (Romanian). [7] W.F. Eberlein, Abstract Ergodic Theorems and Weak Almost Periodic Functions, Trans Amer. Math. Soc. 67 (1949), 217–240. [8] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Vol. I, Springer-Verlag, Berlin, G¨ ottingen, Heidelberg, 1963 [9] J. Gil de Lamadrid, Sur les Mesures Presque P´eriodiques, Ast´erisque 4 (1973), 61–89. [10] W. Rudin, Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, Number 12, Interscience Publishers – John Wiley and Sons, New York, London, 1962. Silvia-Otilia Corduneanu Department of Mathematics Gh. Asachi Technical University of Ia¸si 11 Carol I Blvd. Ia¸si, Romania e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 135–148 c 2009 Birkh¨ auser Verlag Basel/Switzerland
How Summable are Rademacher Series? Guillermo P. Curbera Abstract. Khintchin inequalities show that a.e. convergent Rademacher series belong to all spaces Lp ([0, 1]), for finite p. In 1975 Rodin and Semenov considered the extension of this result to the setting of rearrangement invariant spaces. The space LN of functions having square exponential integrability plays a prominent role in this problem. Another way of gauging the summability of Rademacher series is considering the multiplicator space of the Rademacher series in a rearrangement invariant space X, that is, an rn ∈ X . Λ(R, X) := f : [0, 1] → R : f · an rn ∈ X, for all The properties of the space Λ(R, X) are determined by its relation with some classical function spaces (as LN and L∞ ([0, 1])) and by the behavior of the logarithm in the function space X. In this paper we present an overview of the topic and the results recently obtained (together with Sergey V. Astashkin, from the University of Samara, Russia, and Vladimir A. Rodin, from the State University of Voronezh, Russia.) Mathematics Subject Classification (2000). Primary 46E35, 46E30; Secondary 47G10. Keywords. Rademacher functions, rearrangement invariant spaces.
1. Introduction: a problem on vector measures The problem which originated the research that we are going to present arises from the study of vector measures and the space of scalar functions which are integrable with respect to them.
Partially supported D.G.I. #MTM2006-13000-C03-01 (Spain).
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In the note Sequences in the range of a vector measure, by R. Anantharaman and J. Diestel, [1], [2], the following vector measure was considered: A ∈ M([0, 1]) −→ ν(A) := rn (t) dt ∈ 2 , A
where M([0, 1]) is the σ-algebra of Lebesgue measurable sets of the interval [0,1] and (rn ) are the Rademacher functions (see Section 2). According to the integration theory of Bartle, Dunford, and Schwartz, [8] (see also [24]), the space L1 (ν) of functions which are integrable with respect to ν is the set of all f : [0, 1] → R such that, for every A ∈ M([0, 1]), the sequence of Rademacher–Fourier coefficients of f χA belongs to 2 , that is, f (t)rn (t) dt ∈ 2 . (1.1) A
The problem we were interested in was identifying the functions in L1 (ν). A similar problem, related to the Hausdorff-Young inequality, is to describe the space Fp (T), for 1 < p < 2, of all functions f ∈ L1 (T) such that for every A ∈ B(T) 1 f (t)e−int dt ∈ 1/p . 2π A Recently, Mockenhaupt and Ricker have shown that Lp (T) Fp (T), [22]. Thus answering a problem posed by R.E. Edwards in the 1960s. The underlying measure in this case is 1 A ∈ B(T) −→ ν(A) := e−int dt ∈ 1/p . 2π A
2. The Rademacher system We briefly recall the main properties of the system. It was defined by Hans Rademacher in 1922 in Section VI, Ein spezielles Orthogonalsystem, of [26]. The functions of the system are rn (t) := sign sin(2n πt),
t ∈ [0, 1], n ∈ N.
The system is uniformly bounded and has a strong orthogonality property: 1 rn1 (t)p1 rn2 (t)p2 . . . rnk (t)pk dt = 0, ni = nj , 1 ≤ i < j ≤ k, 0
unless all pj are even, in which case the integral is equal to 1. It follows that the closed linear subspace generated by (rn ) in L2 ([0, 1]) is isometric to the 2 , ∞ ∞ 1/2 an rn = a2n , n=1
2
n=1
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which write as Rad (L2 ) = 2 . It also follows that the system is not complete. In fact, the set of all finite products of different Rademacher functions constitute the Walsh system, which is complete. An important property of the Rademacher system is related to the almost everywhere convergence of Rademacher series. Namely, ∞
an rn (t) converges a.e. ⇐⇒
n=1
∞
a2n < ∞.
n=1
The reverse implication was proved by Rademacher in 1922, [26], and the direct implication by Khintchin and Kolmogorov in 1925, [15]. Regarding the closed linear subspace generated by (rn ) in other Lp spaces, it is easy to see that Rad (L∞ ) = 1 , since n n n ai ri = sup ai ri (t) = |ai |, i=1
∞
t∈I(a1 ,...,an ) i=1
i=1
where I(a1 , . . . , an ) is the dyadic interval where ri = sign ai for 1 ≤ i ≤ n. For other values of p, Khintchin proved in 1923, [14], that there exists constants Ap , Bp such that ∞ ∞ ∞ 1/2 1/2 Ap · a2n ≤ an rn ≤ Bp · a2n . n=1
n=1
p
n=1
(This formulation in terms of Lp -convergence and square summability was given by Paley and Zygmund in 1930, [25].) It follows that the closed linear subspace generated by (rn ) in Lp ([0, 1]), p = ∞, is isomorphic to 2 ; we write this as √ Rad (Lp ) ≈ 2 . Regarding the constants, Bp ≤ p. The best constants for these inequalities where found by Szarek in 1976, for p = 1, and for general p by Haagerup √ in 1982. Asymptotically, we have Bp ∼ p. Concerning best constants, it is worth mentioning [18], where they are discussed for Kahane’s inequalities, i.e., the vector version of Khintchin inequalities. The power series expansion of the exponential function together with Khintchin inequalities allow to prove that 1 a r 2 n n a2n < ∞ =⇒ exp < ∞, for some λ > 0. λ 0 It follows that Rad (LN ) ≈ 2 , where LN (= Lψ2 ) is the Orlicz space associated to 2 the function ψ2 (t) = et − 1 and consisting of all functions f such that 2 1 |f | exp < ∞, for some λ > 0. λ 0 The space LN is ‘close’ to L∞ in the sense that L∞ LN ⊂ Lp , for all 1 ≤ p < ∞. Are there any other function spaces on [0, 1] where the Rademacher functions generate a subspace isomorphic to 2 ? A precise answer was given by Rodin and Semenov in 1975, [27], in the context of rearrangement invariant (r.i.) spaces. These
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are Banach function spaces where the norm of a function f depends only on its distribution function, λ → m({t : |f (t)| > λ}). For the definition and properties of these spaces, see [10], [17], [19]. Rodin and Semenov proved that, if X is an r.i. space over [0,1], then Rad (X) ≈ 2 ⇐⇒ (LN )0 ⊂ X, where (LN )0 is the closure of L∞ in LN . The proof is based on the Central Limit Theorem and the role of the function log1/2 (2/t) in the space LN : f ∈ LN ⇐⇒ f ∗ (t) ≤ M · log1/2 (2/t), where f ∗ is the decreasing rearrangement of f (i.e., the right continuous inverse of its distribution function). The above equivalence is due to the fact that, as the 2 function ψ2 (t) = et − 1 increases very rapidly, the Orlicz space LN coincides with the Marcinkiewicz space (see Section 4.3) associated to log−1/2 (2/t), [21]. The situation when Rad (X) is complemented in X was characterized, in terms of (LN )0 , by Rodin and Semenov, [28], and independently by Lindenstrauss and Tzafriri, [19, Theorem 2.b.4].
3. A problem on function spaces The problem of identifying the space L1 (ν) of functions satisfying (1.1) can reformulated as follows. Describe the space of all functions f : [0, 1] → R such that 1# # # # an rn (t)# dt < ∞, (3.1) #f (t) · 0
for every (an ) ∈ 2 . At this stage, it is reasonable to abandon the original vector measure. Thus, we change notation and label this space by Λ(R). The space Λ(R) is Banach function space for the norm f Λ(R) := sup f · an rn . (3.2) (an )∈B2
1
Note that, due to Khintchin inequalities, the space Λ(R) satisfies ) Lp ⊂ Λ(R) L1 p>1
The attempts to identify the space Λ(R) with any of the classical Banach function spaces (Lp , Orlicz, Lorentz, Marcinkiewicz, Zygmund, Lorentz–Zygmund, . . . ) fail. The reason is revealed by the following result. Theorem 3.1 ([11, Theorem]). Λ(R) is not a rearrangement invariant space. The strategy for proving the result is building sequences of sets (Bn ) and (Dn ) with m(Bn ) = m(Dn ) = 2nn and such that χBn Λ(R) −→ 0. χDn Λ(R)
(3.3)
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This contradicts Λ(R) being r.i., since χBn and χDn have the same distribution function, so their norms in any r.i. space should be the same (or equivalent). Let us sketch how these sets are built. Note that, according to (3.2), n n ∞ sup χE · ai ri ≤ χE Λ(R) ≤ sup χE · ai ri + χE · ai ri . (ai )∈B2
i=1
1
(ai )∈B2
1
i=1
n+1
1
n
Let Δ1n , . . . , Δ2n be the dyadic intervals of order n. Let aij be the value of the Rademacher function ri on the interval Δjn . The values of r1 , r2 , . . . , rn over the n intervals Δ1n , . . . , Δ2n are shown in the following matrix: Δ1n r1 1 .. ⎜ .. . ⎜ ⎜ . ri ⎜ ⎜ ai1 .. ⎜ . . ⎝ .. 1 rn ⎛
Δ2n 1 .. .
... ... .. .
n
. . . Δ2n ⎞ . . . −1 .. ⎟ .. . . ⎟ ⎟ . . . ai2n ⎟ ⎟ .. ⎟ .. . . ⎠
Δjn a1j .. .
ai2 .. .
. . . aij .. .. . . −1 . . . anj . . . −1 ' Choose columns J1 so that, for Bn = j∈J1 Δjn , we have that sup
χBn
(ai )∈B2
n
ai ri Λ
i=1
is small; and choose columns J2 so that, for Dn = sup (ai )∈B2
χDn
n
' j∈J2
Δjn , we have that
ai ri Λ
i=1
is large. From this, together with an adequate control of the norm of the tails of the series, we deduce (3.3).
4. The Rademacher multiplicator space It was a suggestion of S. Kwapie´ n to the author to consider the above result substituting the role played by the space L1 ([0, 1]) in the definition of the space Λ(R), namely: an rn ∈ L1 , Λ(R) = f : f · an rn ∈ L1 , for all by Lp ([0, 1]). We went a step further and considered an arbitrary r.i. space. 4.1. The space Λ(R, X) Definition 4.1. Let X be an r.i. space on [0,1]. The Rademacher multiplicator space for X is Λ(R, X) := f : f · an rn ∈ X, for all an rn ∈ X .
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an rn ∈ X Note that when X = L1 , or X = Lp for finite p, the condition corresponds to (an ) ∈ 2 , due to Khintchin inequalities. However, this is not the case for a general r.i. space. Some examples are in order: ⎧ 2 ⎨ , for X = LN , due to Rodin and Semenov’s result. Rad (X) ≈ q ,∞ , for X = Lψq with ψq (t) = exp(tq ) − 1, q > 2. ⎩ 1 , for X = L∞ . In general, for X an arbitrary r.i. space Rad (X) is (isomorphic to) a sequence space which is an interpolation space between 2 and 1 (that is, for every bounded linear operator T satisfying T : i → i , i = 1, 2, we have T : Rad (X) → Rad (X)). The converse to this result is also true. Theorem 4.2 ([3]). Every interpolation space between 2 and 1 is a space Rad (X) for some r.i. space X. Theorem 3.1 can be extended to this more general setting. Note that the space Λ(R, X) is a Banach function space for the norm f Λ(R,X) := sup f · an rn : an rn ≤ 1 . X
X
Theorem 4.3 ([11, Theorem], [5, Theorem 2.1]). If X is an r.i. space such that the lower dilation index of its fundamental function ϕX satisfies γϕX > 0, then Λ(R, X) is not rearrangement invariant. Recall that the fundamental function of an r.i. space X is defined by ϕX (t) := χ[0,t] X for 0 ≤ t ≤ 1. In particular, for X = Lp , 1 ≤ p ≤ ∞, we have ϕX (t) = t1/p . The lower dilation index γϕ of a positive function ϕ is γϕ := lim
t→0+
log sup 0<s≤1 log t
ϕ(st) ϕ(s)
.
This index measures the situation of ϕ with respect to power type functions; see [17, II. §1.1]. In particular, γϕX = 1/p for X = Lp , 1 ≤ p ≤ ∞. The proof of Theorem 4.3 is modeled on that of Theorem 1.1 and follows from the estimate ∞ 1/2 χ[0,n2−n ] n+1 bi rr X χBn Λ(R,X) ϕX (1/2n ) + ≤ sup . χDn Λ(R,X) ϕX (n/2n ) n1/2 · ϕX (n/2n ) (bi )∈B2 Theorem 4.3 holds, for example, for Lp -spaces with 1 ≤ p < ∞; for Lorentz L -spaces, [10, IV.4.1]; for Orlicz spaces Lφ with φ satisfying the Δ condition globally, [16]; for r.i. spaces X whose lower Boyd index is strictly positive, αX > 0, [10, III.5.12]. p,q
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4.2. The symmetric kernel of Λ(R, X) Theorem 4.3 implies that for ‘most’ of the classical r.i. spaces the multiplicator space is not r.i. Thus, it becomes relevant to identify the largest r.i. space contained in Λ(R, X), which we call the symmetric kernel of Λ(R, X). Let us illustrate the concept of symmetric kernel with a simple example. The function space 1 1/p−1 Z= f: |f (t)|t dt < ∞ 0
is not r.i. (to see this, consider the functions f (t) = (1 − t)−1/p and g(t) = t−1/p , they have the same distribution function but, f ∈ Z and g ∈ Z). The largest r.i. space inside Z is easy to identify: it is the Lorentz space 1 p,1 ∗ 1/p−1 L = f: f (t)t dt < ∞ . 0
Note that the inclusion L ⊂ Z follows from a result of Hardy and Littlewood on rearrangements of functions; see [10, II.2.2]. The definition of the symmetric kernel of the multiplicator space follows. p,1
Definition 4.4. The symmetric kernel of Λ(R, X) is the space Sym (R, X) := f ∈ Λ(R, X) : if g f, then g ∈ Λ(R, X) , where g f means that g and f have the same distribution function. The norm in Sym (R, X) is f Sym (R,X) := sup gΛ(R,X) : g f For the identification of Sym (R, X) we need to recall the associate space of a Banach function space X. It is the space X of all measurable functions g such that g·f is integrable, for every f ∈ X. If X is r.i., then also X is r.i. The biassociate of X is the space defined by X := (X ) . Theorem 4.5 ([5, Theorem 2.8], [7, Proposition 3.1]). Let X be an r.i. space with LN ⊂ X, then Sym (R, X) = f : f ∗ log1/2 (2/t) X < ∞ . Recall that LN is the space functions with square exponential integrability. As could be expected, the proof relies on the Central Limit Theorem. Indeed, the proof is based on the inequalities: ∗ f ∗ (t) an rn (t) ≤ K (an )2 f ∗ (t) log1/2 (2/t), n
∗ ri 1/2 ∗ ∗ √ (t). f (t) log (2/t) ≤ C f (t)· lim n n 1 Some examples following from Theorem 4.5 are in order. In many problems in classical analysis the class of Lorentz–Zygmund spaces plays a prominent role;
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see [9], [10, IV.6.13]. For 0 < p, q ≤ ∞, α ∈ R, the Lorentz–Zygmund space Lp,q (log L)α consists of all measurable functions f : [0, 1] → R for which 1 q dt 1/q f p,q;α = < ∞, t1/p logα (2/t)f ∗ (t) t 0 with the usual modification in the case q = ∞. Note that the Lp spaces are Lorentz–Zygmund spaces for parameters p = q and α = 0. (Below, we denote by A " B the existence of constants C, c > 0 such that c·A ≤ B ≤ C·A.) • For X = Lp with 1 ≤ p < ∞, we have 1 p 1/p f Sym (R,X) " f ∗ (t) log1/2 (2/t) dt . 0
Hence, Sym (R, L ) is the Zygmund space Lp (logL)1/2 (these are Lorentz– Zygmund spaces with p = q, see [10, IV.6.11]). • For X = Lp,q (log L)α with either 1 < p < ∞, 1 ≤ q < ∞ and α ∈ R, or p = q = 1 and α ≥ 0, we have 1 q dt 1/q f Sym (R,X) " t1/p log1/2+α (2/t)f ∗ (t) . t 0 p
Hence, Sym(R,Lp,q (log L)α ) is the Lorentz–Zygmund space Lp,q (log L)1/2+α. • For X = Lp,∞ (log L)α with 1 < p < ∞ and α ∈ R we have f Sym (R,X) " sup t1/p log1/2+α (2/t)f ∗ (t). 0
Hence, Sym (R, L
p,∞
α
(log L) ) is the Lorentz–Zygmund space Lp,∞ (logL)1/2+α .
In particular, for α = 0 we have the weak Lp -spaces for which Sym(R,Lp,∞ ) = Lp,∞ (log L)1/2 . From the above examples it follows that, for example, there exists an r.i. space X whose symmetric kernel is Lp . Indeed, the Zygmund space X = Lp (logL)−1/2 has symmetric kernel Lp . This observation leads to a general question. What r.i. spaces Z arise as symmetric kernels of Rademacher multiplicator spaces? Note that, since Sym (R, L1 ) = L(logL)1/2 , such a space Z must satisfy Z ⊂ L log1/2 L. This question can be precisely answered. Recall that Banach function space has the Fatou property if X = X or, alternatively, if fn ∈ X with 0 ≤ fn ↑ f and supn fn X < ∞ imply f ∈ X and fn X → f X . Theorem 4.6 ([5, Theorem 2.17]). Let Z be an r.i. space with the Fatou property and such that Z is an interpolation space between Llog1/2 L and L∞ . Then there exists an r.i. space X such that Z = Sym (R, X). Of particular interest is the case when the symmetric kernel reduces to the smallest possible r.i. space, L∞ . The characterization in this case is the following.
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Theorem 4.7 ([5, Theorem 3.2]). Let X be an r.i. space and X0 denote the closure of L∞ in X. Then: / X0 Sym (R, X) = L∞ ⇐⇒ log1/2 (2/t) ∈ The spaces X satisfying Sym (R, X) = L∞ are ‘close’ to L∞ . For example, for X = LN , we have that f ∈ (LN )0 when limt→0 f ∗ (t) log−1/2 (2/t) = 0, [17, Lemma II.5.3]. Thus, Sym (R, LN ) = L∞ . 4.3. When is Λ(R, X) rearrangement invariant? The simplest case of the multiplicator space being r.i. is when Λ(R, X) = L∞ . This occurs, for example, when X = LN , [11, Example 1]. This follows from the fact n that, if Δ is a dyadic set of order n, we can choose signs ai = ±1 such that 1 ai ri = n on Δ. Thus, n √ √ a √i ri = n χΔ LN = n· log−1/2 (2−n ) " M. χΔ Λ(R,LN ) ≥ χΔ n LN i The above estimate can be extended to any measurable set A ∈ M([0, 1]) so, χA Λ(R,LN ) ≥ M . This implies that Λ(R, LN ) contains no unbounded functions. A similar strategy can be used to prove that for X = Lψq , the Orlicz space associated to the function ψq (t) = exp(tq )−1 with q > 2, we have Λ(R, Lψq ) = L∞ , [11, Example 3]. Observe that if Λ(R, X) = L∞ then, necessarily, Sym (R, X) = L∞ . The next results shows that, unexpectedly, both conditions are, in view of Theorem 4.7, equivalent. Partial results, still of interest, giving conditions for Λ(R, X) = L∞ were given in [13, Theorem 2] and [4, Theorem 1]. Theorem 4.8 ([6, Theorem 1]). Let X be an r.i. space and X0 denote the closure of L∞ in X. Then: / X0 Λ(R, X) = L∞ ⇐⇒ log1/2 (2/t) ∈ The proof uses a remarkable formula, due to Montgomery-Smith, [23], for the distribution function of a Rademacher series: # # 2 # # m t:# an rn # > K(a, t; 1 , 2 ) ≤ et /2 , # # 2 # # an rn # > c−1 ·K(a, t; 1 , 2 ) m t:# ≥ c−1 ·ect , where K(a, t; 1 , 2 ) is the K-functional of Petree for the sequence a = (an ) ∈ 2 with respect to the spaces 1 and 2 ; see [10, V.1.1]. Note that when X is separable we have X0 = X and so the condition in Theorems 4.7 and 4.8 becomes LN ⊂ X. Note also that if L∞ Λ(R, X) then Rad (X) ≈ 2 . Another consequence of Theorem 4.8 is that X ⊂ Y implies Λ(R, X) ⊂ Λ(R, Y ), [6, Corollary 3], which is far from obvious in view of Definition 4.1. It could be thought that LN is the largest r.i. space with Λ(R, X) = L∞ . This is not the case, as shown by Theorem 4.8 and Example 3.5 in [6].
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Can the multiplicator space Λ(R, X) be a r.i. space different from L∞ ? The answer is yes, and a first example is X = Lexp , the space of functions f of expo1 nential integrability: 0 exp(|f |/λ) < ∞, for some λ > 0. In this case we have, [11, Example 3], Λ(R, Lexp ) = LN . Other similar cases are X = Lψq , the Orlicz space associated to the function ψq (t) = exp(tq ) − 1 with 0 < q < 2, where, [13, Theorem 3], Λ(R, Lψq ) = Lψα ,
for α := 2q/(2 − q);
and X an exponential Orlicz space ExpLφ , that is, the Orlicz space associated to the function eφ(t) − 1. In this case we have, [5, Theorem 4.7], Λ(R, ExpLφ ) = ExpLΨ , √ where Ψ is the Orlicz function satisfying Ψ−1 (t) := φ−1 (t)/ t (whenever this last function is increasing). These examples have a common feature. Namely, the norm of the space X can be obtained via an extrapolation formula. For example, equivalent expressions for the norm in LN and Lexp are the following f p √ , p 1≤p<∞
f LN = sup
and
f p . p 1≤p<∞
f Lexp = sup
In proving the above identifications of the multiplicator spaces, the following lemma, of independent interest, is used. Lemma 4.9 ([11, Lemma 3]). For each p, 1 ≤ p < ∞, and for each function f ∈ Lp there exists a norm one sequence (an ) ∈ 2 such that f ≤ √3 · r a . f n n p p p Let us see how the extrapolation formula and Lemma 4.9 allow to prove that Λ(R, Lexp ) ⊂ LN . Let hp = an rn be the Rademacher series given by the lemma. Then, f LN = sup p−1/2 f p ≤ 3 sup p−1 f ·hp p 1≤p<∞
≤ 3 sup
1≤p<∞
sup
1≤p<∞ (an )∈B2
p
f an rn p =
−1
sup (an )∈B2
f an rn
Lexp
= 3·f Λ(R,Lexp ) Understanding rearrangement invariance of Λ(R, X) beyond the L∞ case requires a fine study of the behavior of logarithmic functions in X. In this regard, a useful sufficient condition is the following, [7, Theorem 3.3]:
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There exists C > 0 such that, for every n ∈ N and c1 ≥ c2 ≥ · · · ≥ c2n ≥ 0, we have 2n 2n 2 1/2 2 ck χΔkn · log ck χΔkn · log1/2 n (4.1) ≤ C , t X 2 t+1−k X k=1
where
k=1
Δkn
are the dyadic intervals of order n. 2 is the function log1/2 2t on [0,1] compressed and transNote that log1/2 2n t+1−k k lated to the interval Δkn = [ k−1 2n , 2n ]. Condition (4.1) allows to prove a general result on rearrangement invariance of multiplicator spaces. Theorem 4.10 ([7, Theorem 3.4]). Let X be an r.i. space. Consider the operator Q defined by Qf (x) := f (x2 ), x ∈ [0, 1]. Suppose that Q : X → X is bounded. Then Λ(R, X) is r.i. To see further results, recall that, given a function ϕ : [0, 1] → [0, +∞), increasing, concave, with ϕ(0) = 0, the associated Marcinkiewicz and Lorentz spaces are, respectively, ϕ(t) t ∗ M (ϕ) = f : f M(ϕ) = sup f (s) ds < ∞ , t 0
For example, if ϕ(t) = t for 1 ≤ p < ∞, we have Λ(ϕ) = Lp,1 and M (ϕ) = Lp,∞ . A consequence of Theorem 4.10 is the following, [7, Theorem 3.5]: 1/p
Suppose that X is an interpolation space between Λ(ϕ) and M (ϕ), where ϕ satisfies the Δ2 -condition: there exists C > 0 such that ϕ(t) ≤ Cϕ(t2 ),
0 < t ≤ 1.
Then Λ(R, X) is r.i. Let us present an interesting necessary condition for rearrangement invariance of Λ(R, X). Recall that, by Theorem 4.8, Λ(R, X) = L∞ precisely when log1/2 (2/t) ∈ X0 . Suppose that log1/2 (2/t) ∈ X0 and Λ(R, X) is r.i. Then, the following ‘logcondition’ holds: 2 1/2 2 χ(0,2−n ] ≤ C log1/2 n χ(0,2−n ] n ≥ 1. (4.2) log t 2 t X X Note that this condition, as occurs for the sufficient condition (4.1), is related to the behavior of the logarithm in X. For Marcinkiewicz spaces the log-condition (almost) gives a characterization of the rearrangement invariance of Λ(R, X).
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Theorem 4.11 ([7, Theorem 4.5]). Let X be a Marcinkiewicz space M (ϕ) such that log1/2 (2/t) ∈ X0 and δϕ < 1. Then Λ(R, X) is r.i. ⇐⇒ X satisfies the log-condition (4.2). Here, δϕ is the upper dilation index of ϕ, that is, log sup 0<s≤1/t δϕ := lim t→+∞ log t
ϕ(st) ϕ(s)
,
[17, II. §1.1]. The importance of condition δϕ < 1 in proving Theorem 4.11 is that it provides a simpler equivalent expression for the norm in the Marzinkiewicz space M (ϕ), [17, Theorem II.5.3], namely: f M(ϕ) " sup ϕ(t)f ∗ (t). 0
4.4. Head and tail behavior The proofs of the previous facts on the behavior of the results reveal interesting a heads (partial sums), n1 ai ri , and the tails, ∞ n+1 i ri , of Rademacher series in function spaces. There are occasions in which the norm in Λ(R, X) is attained (up to equivalence) at an appropriate Rademacher tail sum: • When log1/2 (2/t) ∈ X0 , the proof of Theorem 4.8 shows that, for Δ a dyadic interval of order n, ∞ ∞ χΔ Λ(R,X) " sup χΔ · ai ri : ai ri ≤ 1 . i=n+1
X
X
i=n+1
• When the operator Qf (t) = f (t2 ) is bounded on X, the proof of Theorem 4.10 shows that, for f a step dyadic function of order n, f = n1 ck rk , ∞ f Λ(R,X) " sup f · ai ri i=n+1
X
∞ : ai ri i=n+1
X
≤1 .
(4.3)
• When X is a Marcinkiewicz space M (ϕ) such that log1/2 (2/t) ∈ (M (ϕ))0 , δϕ < 1, and the log-condition (4.2) is satisfied, the proof of Theorem 4.11 shows that (4.3) holds. On other occasions, the norm in Λ(R, X) is attained (up to equivalence) at an appropriate Rademacher head (partial) sum. This occurs, for example, when X in an interpolation space between L∞ and LN . In this case, the proof of Theorem 4.8 shows that, for Δ is any dyadic interval of order n, n n χΔ Λ(R,X) " sup χΔ · ai ri : ai ri ≤ 1 . 1
X
1
X
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5. An open question There are many questions of interest open in the study of the multiplicator space Λ(R, X). For example, determining the precise role of the log-condition (4.2), and of the condition ϕ ∈ Δ2 . Or, finding characterizations for rearrangement invariance for Λ(R, X) in the case of X a Lorentz space Λ(ϕ). However, we are going to single out a concrete problem. Recall the space Λ(R) of all functions f : [0, 1] → R such that 1# # # # an rn (t)# dt < ∞, #f (t) · 0
for every (an ) ∈ 2 . We know that it is a Banach function space for the absolutely continuous norm (3.2), [11]; it has the Fatou property, [6, Proposition 2.6]; it is not r.i., Theorem 3.1; and the largest r.i. space inside Λ(R) is the Zygmund space L(logL)1/2 , Theorem 4.7, consisting in all functions f : [0, 1] → R such that 1 f ∗ (t) log1/2 (2/t) dt < ∞. 0
A question which remains open is to identify all functions from Λ(R). A less ambitious task, also open, is to give a concrete example of a function in Λ(R) but not in its symmetric kernel L(logL)1/2 .
References [1] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Department of Mathematical Sciences, Kent State University (1989). [2] R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), 221–235. [3] S.V. Astashkin, About interpolation of subspaces of rearrangement invariant spaces generated by Rademacher system, Int. J. Math. Math. Sci. 25 (2001), 451–465. [4] S.V. Astashkin, On multiplicator space generated by the Rademacher system, Math. Notes 75 (2004), 158–165. [5] S.V. Astashkin and G.P. Curbera, Symmetric kernel of Rademacher multiplicator spaces, J. Funct. Anal. 226 (2005), 173–192. [6] S.V. Astashkin and G.P. Curbera, Rademacher multiplicator spaces equal to L∞ , Proc. Amer. Math. Soc. 136 (2008), 3493–3501. [7] S.V. Astashkin and G.P. Curbera, Rearrangement invariance of Rademacher multiplicator spaces, J. Funct. Anal. 256 (2009), 4071–4094. [8] R.G. Bartle, N. Dunford and J. Schwartz, Weak compactness and vector measures, Canad. J. Math. 7 (1955), 289–305. [9] C. Bennett and K. Rudnick, On Lorentz–Zygmund spaces, Dissert. Math. 175 (1980) 1–67. [10] C. Bennett and R. Sharpley, Interpolation of Operators (Academic Press, Boston, 1988).
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[11] G.P. Curbera, Operators into L1 of a vector measure and applications to Banach lattices, Math. Ann. 293 (1992), 317–330. [12] G.P. Curbera, A note on function spaces generated by Rademacher series, Proc. Edinburgh. Math. Soc. 40 (1997), 119–126. [13] G.P. Curbera and V.A. Rodin, Multiplication operators on the space of Rademacher series in rearrangement invariant spaces, Math. Proc. Cambridge Phil. Soc. 134 (2003), 153–162. ¨ [14] A. Khintchin, Uber dyadische Br¨ uche, Math. Z. 18 (1923), 109–116. ¨ [15] A. Khintchin and A.N. Kolmogorov, Uber Konvergenz von Reihen, deren Glieder durch den Zufall bestimmt werden, Math. Sbornik 32 (1925), 668–677. [16] M.A. Krasnosel’skii and Ya.B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Gr¨ oningen, 1961. [17] S.G. Krein, Ju.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Amer. Math. Soc., Providence R.I., 1982. [18] R. Latala and K. Oleszkiewicz, On the best constant in the Khinchin-Kahane inequality, Studia Math. 109 (1994), 101–104. [19] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces vol. II, Springer-Verlag, Berlin, 1979. [20] D.R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157–165. [21] G.G. Lorentz, Relations between function spaces, Proc. Amer. Math. Soc. 12 (1961), 127–132. [22] G. Mockenhaupt, W.J. Ricker, Optimal extension of the Hausdorff-Young inequality, J. Reine Angew. Math. 620 (2008), 195–211. [23] S.J. Montgomery–Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990) 517–522. [24] S. Okada, W.J. Ricker and E.A. S´anchez P´erez, Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory Advances Applications, Birkh¨ auser Verlag, Basel-Berlin-Boston, 2008. [25] R.E.A.C. Paley and A. Zygmund, On some series of functions (I), Proc. Camb. Phil. Soc. 26 (1930), 337–357. [26] H. Rademacher, Einige S¨ atze u ¨ber Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 111–138. [27] V.A. Rodin and E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207–222. [28] V.A. Rodin and E.M. Semenov, The complementability of a subspace that is generated by the Rademacher system in a symmetric space, Functional Anal. Appl. 13 (1979), 150–151. [29] A. Zygmund, Trigonometric series, Cambridge University Press, Cambridge, 1977. Guillermo P. Curbera Facultad de Matem´ aticas Universidad de Sevilla, Aptdo. 1160 E-41080 Sevilla, Spain e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 149–158 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Rearrangement Invariant Optimal Domain for Monotone Kernel Operators Olvido Delgado Abstract. For a kernel operator T with values in a Banach function space X, we give monotonicity conditions on the kernel which allow us to describe the rearrangement invariant optimal domain for T (still with values in X). We also study the relation between this optimal domain and the space of integrable functions with respect to the X-valued measure canonically associated to T . Mathematics Subject Classification (2000). 47B34, 46E30, 28B05. Keywords. Optimal domain, kernel operator, rearrangement invariant space, vector measure.
1. Introduction Let K : [0, 1] × [0, 1] → [0, ∞] be a measurable function such that every x ∈ [0, 1] satisfies K(x, ·) < ∞ a.e. and consider the kernel operator T defined by K as 1 T f (x) = 0 f (y)K(x, y) dy, x ∈ [0, 1], (1.1) for any f ∈ L0 (the space of all measurable real functions on [0, 1], identifying functions which are equal a.e.) for which the integral exists a.e. x. Given a Banach function space (B.f.s.) X, an important problem is to find the optimal domain for T considered with values in X, that is the largest B.f.s. Y such that T : Y → X is well defined (and so continuous, since it is a positive linear operator between Banach lattices, see [11, p. 2]). The “largest” B.f.s. Y may be understood in the following sense: if Z is another B.f.s. such that T : Z → X is well defined then Z ⊂ Y . This problem has been studied for classical operators in numerous works as for instance [2], [4], [9], [12] and [13]. Throughout the paper, we will assume that K satisfies the condition 1 (1.2) 0 K(x, y) dx > 0 a.e. y ∈ [0, 1], Research supported by Generalitat Valenciana (project GVPRE/2008/312), MEC (TSGD-08), D.G.I. #MTM2006-13000-C03-01 (Spain) and FEDER.
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that is, T |f | = 0 a.e. implies f = 0 a.e., or equivalently, there exists no measurable set A of strictly positive Lebesgue measure such that T (f χA ) = 0 a.e. for all f ∈ L0 . Let us denote by [T, X] the optimal domain for T considered with values in X. This space has been studied in [3], where it is described in a natural way as [T, X] = {f ∈ L0 : T |f | ∈ X}
(1.3)
endowed with the norm f [T,X] := T |f | X . Note that (1.2) guarantees that · [T,X] is a norm. Moreover, conditions are given for obtaining a more precise description for [T, X] in terms of interpolation spaces. See also [8] for the case [0, ∞) instead of [0, 1]. On other hand, under appropriate conditions on X and K, the set function ν associated to T via ν(A) = T (χA ) is an X-valued vector measure which turns out to be a powerful tool for studying T . The spaces L1 (ν) and L1w (ν) of integrable and weakly integrable functions with respect to ν respectively, are closed related to the optimal domain [T, X] as shown in [3] and [5]. Indeed, the containments L1 (ν) ⊂ [T, X] ⊂ L1w (ν) always hold. In this paper we are interested in the rearrangement invariant (r.i.) optimal domain for T , that is the largest r.i. B.f.s. contained in [T, X], denoted by [T, X]r.i. . This space has been already studied for the kernel operator associated with the Sobolev’s inequality ([4], [7]) and the Hardy operator ([9]). In Section 3 we will see that [T, X]r.i. can be described in a similar way as (1.3) provided K satisfies that K(x, ·) is a monotone map for every x ∈ [0, 1]. Even more, we give conditions under which [T, X]r.i. can be more precisely described as an interpolation space. Section 4 is devoted to the study of the relation among all the spaces [T, X], [T, X]r.i. , L1 (ν) and L1w (ν).
2. Preliminaries A Banach function space (B.f.s.) is a Banach space X contained in L0 such that if f ∈ L0 , g ∈ X and |f | ≤ |g| a.e. then f ∈ X and f X ≤ gX . Note that a B.f.s. is a Banach lattice for the pointwise a.e. order. Given two B.f.s.’ X and Y , we will write X →c Y when X is continuously contained in Y with f Y ≤ cf X for all f ∈ X and X →i Y when the containment is isometric. By X ≡ Y we mean that X = Y and the norms coincide. A B.f.s. is order continuous (o.c.) if every order bounded increasing sequence is norm convergent. Let m denote the Lebesgue measure on [0, 1]. Note that, since m is finite, in the case when X contains the simple functions, X is o.c. if and only if every f ∈ X satisfies f χA X → 0 as m(A) → 0. A B.f.s. X has the Fatou property if for every sequence (fn ) ⊂ X such that 0 ≤ fn ↑ f a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X . A B.f.s. X is rearrangement invariant (r.i.) whenever f ∈ X if and ∗ ∗ decreasing only if f ∗ ∈ X, and in this case f X = f X. Here, f denotes the ∗ rearrangement of f , i.e., f (s) = inf r > 0 : m {x ∈ [0, 1] : |f (x)| > r} ≤ s for all s ∈ [0, 1]. A non trivial r.i. B.f.s. X satisfies L∞ ⊂ X ⊂ L1 , see [10, Theorem II.4.1]. Adding to X the Fatou property, we obtain an r.i. B.f.s. in the sense of
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Bennett and Sharpley [1, Definition I.1.1]. Then X can be generated by the Kmethod of interpolation of Peetre as (L1 , L∞ )X . Let us recall briefly this method. If (X0 , X1 ) are Banach spaces continuously embedded in a common Hausdorff topological vector space, then the K–functional of f ∈ X0 + X1 is defined as K(t, f ; X0 , X1 ) = inf f0 + tf1 : f = f0 + f1 ; f0 ∈ X0 , f1 ∈ X1 , t > 0. Assume X0 ∩ X1 is dense in X0 . Given an r.i. B.f.s. X having the Fatou property, (X0 , X1 )X denotes the space of all function f ∈ X0 +X1 such that K (·, f ; X0 , X1 ) ∈ X, where K is the derivative of the K-functional K. Note that K is a decreasing function. The interpolation space (X0 , X1 )X between X0 and X1 , is a B.f.s. endowed with the norm f (X0 ,X1 )X := K (·, f ; X0 , X1 )X . See [1, Chp. V] for further information. Given an increasing concave function ϕ : [0, 1] → [0, ∞) such that ϕ(0) = 0 1 and ϕ(0+ ) = 0, the Lorentz space Λϕ = {f ∈ L0 : f Λϕ = 0 f ∗ (t)ϕ (t) dt < ∞} with norm · Λϕ , is an o.c. r.i. B.f.s. having the Fatou property, see [10, §II.5]. Let B([0, 1]) be the σ-algebra of all Borel subsets of [0, 1], X a B.f.s. and ν : B([0, 1]) → X a vector measure (i.e., countably additive). Let us recall briefly the theory of integration of real functions with respect to ν, which will be used in Section 4. A set A ∈ B([0, 1]) is ν-null if ν(B) = 0 whenever B ∈ B([0, 1]) ∩ 2A . Assume that ν and m have the same null sets. A function f ∈ L0 is weakly integrable with respect to ν, if f ∈ L1 (|x∗ ν|) for every element x∗ in X ∗ (the topological dual of X), where |x∗ ν| is the variation of the real measure x∗ ν. If moreover f satisfies that for each A ∈ B([0, 1]) there exists xA ∈ X such that x∗ (xA ) = A f dx∗ m, for every x∗ ∈ X ∗ , f is said to be integrable with respect to ν. The vector xA is unique and will be written as A f dm. Let L1w (ν) denote the space of all weakly integrable function and L1 (ν) the space of all integrable function with respect to ν. In both spaces, functions which are equal a.e. are identified. The map · ν defined for f ∈ L0 as f ν = supx∗ ∈BX ∗ Ω |f | d|x∗ ν|, where BX ∗ denotes the unit ball of X ∗ , endows of B.f.s. structure the spaces L1w (ν) and L1 (ν). Of course, L1 (ν) is a closed subspace of L1w (ν). The space L1w (ν) has the Fatou property and L1 (ν) is order continuous containing the simple functions as a dense set. For more details see [6], [14, Ch. 3] and the references therein.
3. R.i. optimal domain for T Let T be the kernel operator given in (1.1) with kernel K satisfying (1.2). Depending on each particular B.f.s. X, the optimal domain [T, X] is or is not r.i. For instance, if T is the Volterra operator (i.e., K(x, y) = χ[0,x] (y)), then [T, L∞ ] ≡ L1 is r.i. while [T, L1 ] ≡ L11−y is not r.i. The equivalences follow directly from (1.3). So, a natural question arises: which is the largest r.i. space contained in [T, X], in other words, which is the r.i. optimal domain for T considered with values in X?
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Let us denote by [T, X]r.i. this r.i. optimal domain. A good candidate to describe [T, X]r.i. in a similar way as in (1.3) is ΓX = {f ∈ L0 : T f ∗ ∈ X}, since ΓX satisfies the r.i. and the ideal properties, and every r.i. space Y contained in [T, X] is inside of ΓX . Unfortunately, in general, ΓX is not a linear space. −1 χ[x,1] (y), we have that f ∈ ΓL1 if and only if Example. For K(x, y) = y(1 − y) 1 ∗ 1 1 ,g = χ 1 f (y) dy < ∞. Then, f = χ / ΓL 1 . [0, 2 ] ( 2 ,1] ∈ ΓL1 , while f + g ∈ 1−y 0 However, we can require K to satisfy an appropriate monotonicity condition guaranteeing the linearity of ΓX . Namely, for every fixed x ∈ [0, 1], the map K(x, ·) is decreasing. ∗
∗
(3.1)
∗
In this case, T (f +g) ≤ T f +T g (see [1, Theorem II.3.4 and Proposition II.3.6]) and T |f | ≤ T f ∗ (see [1, Theorem II.2.2]). Then, ΓX is a linear space contained in [T, X] and the functional ρ(f ) = T f ∗X is a norm on ΓX satisfying the Riesz Fischer property, see [15, Ch. 15, §64]. So, we obtain the following result. Proposition 3.1. If K satisfies (3.1), then [T, X]r.i. = {f ∈ L0 : T f ∗ ∈ X} with norm f [T,X]r.i. = T f ∗X . Moreover, [T, X]r.i. →1 [T, X]. From now on in this section we assume (3.1) holds. Let us see some cases in which [T, X]r.i. can be described more precisely. Consider the decreasing function 1 ω(y) = 0 K(x, y) dx. / L1 , [T, L1 ]r.i. = {0}. Proposition 3.2. If ω ∈ L1 , then [T, L1 ]r.i. ≡ Λ 0y ω(s)ds . If ω ∈ Proof. Given f ∈ L0 , we have that 1 1 ∗ 1 1 ∗ ∗ 0 T f (x) dx = 0 f (y) 0 K(x, y) dx dy = 0 f (y) ω(y) dy. Then, from Proposition 3.1, the conclusion follows for ω ∈ L1 . Note that 1 1 0 ω(y) dy = 0 T χ[0,1] (x) dx, then ω ∈ L1 if and only if χ[0,1] ∈ [T, L1 ]r.i. , or equivalently, [T, L1 ]r.i. = {0}.
The space [T, L∞ ]r.i. can be also described as a Lorentz space in the case of K being decreasing when fixing the second variable, i.e., K(·, y) decreases for all y ∈ [0, 1].
(3.2)
In this case, we consider the decreasing function ξ(y) = K(0, y). Proposition 3.3. Suppose K satisfies (3.2). If ξ ∈ L1 , then [T, L∞ ]r.i. = Λ 0y ξ(s)ds . In other case [T, L∞ ]r.i. = {0}.
R.i. Optimal Domain for Monotone Kernel Operators Proof. Given f ∈ L0 , we have that sup0≤x≤1 T f ∗ (x) = sup0≤x≤1
1 0
f ∗ (y)K(x, y) dy =
1 0
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f ∗ (y)ξ(y) dy.
Then, from Proposition 3.1, the conclusion follows for ξ ∈ L1 . Note that 1 ξ(y) dy = sup0≤x≤1 T χ[0,1] (x), 0 and so [T, L∞ ]r.i. = {0} if and only if ξ ∈ L1 .
Condition (3.2) also allows us to give a precise description for the r.i. optimal domain of T considered with values in a Lorentz space Λϕ . Under this condition, we consider the decreasing function 1 θϕ (y) = 0 ϕ (x)K(x, y) dx. Proposition 3.4. Suppose that (3.2) holds. Given a Lorentz space Λϕ , we have that [T, Λϕ ]r.i. = Λ 0y θϕ (s)ds whenever θϕ ∈ L1 and [T, Λϕ ]r.i. = {0} in other case. Proof. Condition (3.2) implies that T g decreases for all 0 ≤ g ∈ L0 . Given f ∈ L0 , 1 1 1 (T f ∗ )∗ (x)ϕ (x) dx = 0 T f ∗ (x)ϕ (x) dx = 0 f ∗ (y) θϕ (y) dy 0 and so the conclusion follows for θϕ ∈ L1 . Moreover, [T, Λϕ ]r.i. = {0} if and only ∗ 1 1 if θϕ ∈ L1 , as 0 θϕ (y) dy = 0 T χ[0,1] (x)ϕ (x) dx. Example. For 0 < α < 1, the kernel K(x, y) = min{ x1α , y1α } satisfies (1.2), (3.1) and (3.2). Note that the operator T defined by K is the sum of the kernel operator associated with the Sobolev’s inequality (see [4]) with its adjoint operator. Let us consider now a general r.i. B.f.s. X (non trivial) having the Fatou property. Then L∞ ⊂ X ⊂ L1 and X can be described as an interpolation space between L1 and L∞ , namely X = (L1 , L∞ )X . It is clear that [T, L∞ ]r.i. ⊂ [T, X]r.i. ⊂ [T, L1 ]r.i. . The question is the following: can [T, X]r.i. be describedas the corresponding interpolation space between [T, L1 ]r.i. and [T, L∞ ]r.i. , i.e., [T, L1 ]r.i. , [T, L∞ ]r.i. X ? Proposition 3.5. Suppose that [T, L∞ ]r.i. = {0}. Then [T, L1 ]r.i. , [T, L∞ ]r.i. X →1 [T, X]r.i. Proof. Since L∞ ⊂ [T, L∞ ]r.i. and L∞ is dense in [T, L1 ]r.i. (see Proposition 3.2), from [1, Proposition V.1.15] we have that t K(t, f ; [T, L1 ]r.i. , [T, L∞ ]r.i. ) = 0 K (s, f ; [T, L1]r.i. , [T, L∞ ]r.i. ) ds for every f ∈ [T, L1 ]r.i. and t > 0. Then, from [1, Theorem II.4.7] and since t 1 K(t, h; L1 , L∞ ) = 0 h∗ (s)ds for every V.1.6]), it is h 1∈ L (see∞[1, Proposition enough to prove that, for every f ∈ [T, L ]r.i. , [T, L ]r.i. X and t > 0, K(t, T f ∗ ; L1 , L∞ ) ≤ K(t, f ; [T, L1 ]r.i. , [T, L∞ ]r.i ) .
(3.3)
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Take f ∈ [T, L1 ]r.i. , [T, L∞ ]r.i. X and t > 0. For every f0 ∈ [T, L1 ]r.i. and f1 ∈ [T, L∞ ]r.i. such that f = f0 + f1 , it follows f0 [T,L1 ]r.i. + tf1 [T,L∞ ]r.i.
= T f0∗ L1 + tT f1∗L∞ ≥ K(t, T (f0∗ + f1∗ ); L1 , L∞ ) ≥ K(t, T f ∗ ; L1 , L∞ ) ,
where the last inequality holds as T f ∗ ≤ T (f0∗ + f1∗ ). Taking infimum on f0 , f1 we obtain the inequality (3.3). From Propositions 3.2, 3.3 and 3.5 we obtain the following corollary. Corollary 3.6. If K satisfies (3.2) and ξ ∈ L1 , then Λ 0y ω(s)ds , Λ 0y ξ(s)ds X →1 [T, X]r.i. We can require K to satisfy extra conditions under which the two spaces in Corollary 3.6 coincide. Theorem 3.7. Suppose that K satisfies (3.2), ξ ∈ L1 and there exists a constant C > 0 such that yt y y 1 K(x, s) dx ds ≥ C min 0 0 K(x, s) dx ds , t · 0 K(0, s) ds (3.4) 0 0 y −1 y is holds for all 0 < t, y < 1. Suppose also that h(y) = 0 ω(s)ds · 0 ξ(s)ds a monotone map. Then, [T, X]r.i. = Λ 0y ω(s)ds , Λ 0y ξ(s)ds X . Proof. Since h is monotone, arguments similar to the used in the proof of [10, Theorem II.5.9] lead to 1 K(t, f ; Λ 0y ω(s)ds , Λ 0y ξ(s)ds ) = 0 f ∗ (y) dφt (y) (3.5) for all f ∈ Λ 0y ω(s)ds and 0 < t < 1, where y y φt (y) = min 0 ω(s) ds , t· 0 ξ(s) ds . Let f ∈ [T, X]r.i. and 0 < t < 1. From (3.4) and (3.5) it follows t t K(t, T f ∗ ; L1 , L∞ ) = 0 (T f ∗ )∗ (x) dx = 0 T f ∗ (x) dx t 1 = 0 f ∗ (y) 0 K(x, y) dx dy y t 1 = 0 f ∗ (y) d 0 0 K(x, s) dx ds (y). ≥ C · K(t, f ; Λ 0y ω(s)ds , Λ 0y ξ(s)ds ). Then f ∈ Λ 0y ω(s)ds , Λ 0y ξ(s)ds X with C · f (Λ y ω(s)ds ,Λ y ξ(s)ds )X ≤ f [T,X]r.i. . 0 0 From this and Corollary 3.6 the conclusion follows. Example. For 0 < α < 1 < β, the kernel K(x, y) = min{ x1β , y1α } satisfies the hypothesis of Theorem 3.7.
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Remark 3.8. If K satisfies that K(·, y) increases for all y ∈ [0, 1] instead of (3.2), Proposition 3.3, 3.4, Corollary 3.6 and Theorem 3.7 hold replacing K(x, y) by K(1 − x, y) in the definition of ξ, θϕ and in (1.2). Note that under this condition, T g increases for all 0 ≤ g ∈ L0 and so (T g)∗ (x) = T g(1 − x) for all x ∈ [0, 1]. Moreover, in the case when K(x, ·) is increasing for all x ∈ [0, 1], the r.i. optimal domain for T can be described as [T, X]r.i. = {f ∈ L0 : T (τ f ∗ ) ∈ X}, where τ is the operator which takes f ∈ L0 into the function defined by τ f (t) = f (1 − t). Similar results can be obtained by taking τ ω and τ ξ.
4. Vector integral representation for T Let ν be the set function given by A ∈ B([0, 1]) → ν(A) = T (χA ), where T is as in (1.1) with K satisfying (1.2). Depending on the B.f.s. on which ν takes values, ν will be or not a vector measure. Consider a B.f.s. X satisfying 1 K(·, y) dy ∈ X and lim K(·, y) dy (4.1) = 0. m(A)→0 A 0 X
Then, ν : B([0, 1]) → X is a vector measure which will be denoted by νX to indicate the space where values are taken. Indeed, the first condition in (4.1) guarantees that 1 νX is well defined (as T (χ[0,1] ) = 0 K(·, y) dy) and the second one implies that νX is countably additive. Note that actually the conditions in (4.1) are equivalent to νX being a well-defined vector measure. The next result which has been proved in [3] and [4] under stronger conditions on X and K, remains hold in our context. Proposition 4.1. The following containments always hold: L1 (νX ) →i [T, X] →1 L1w (νX ). Moreover, (a) T f = f dνX for all f ∈ L1 (νX ). (b) L1 (νX ) is the largest o.c. B.f.s. contained in [T, X]. (c) L1w (νX ) is the smallest B.f.s. with the Fatou property containing [T, X]. Assume (3.1) holds. Then [T, X]r.i. is described as in Proposition 3.1 and [T, X]r.i. →1 [T, X] →1 L1w (νX ).
(4.2)
1
But what is the relation between L (νX ) and [T, X]r.i. ? Proposition 4.2. The containment L1 (νX ) ⊂ [T, X]r.i. holds if and only if f ∈ L1 (νX ) implies f ∗ ∈ L1 (νX ). Moreover, in this case, L1 (νX ) is r.i. endowed with the norm · [T,X]r.i. , which is equivalent to · νX . Proof. Suppose that L1 (νX ) ⊂ [T, X]r.i. . Then, there exists a constant c > 0 such that L1 (νX ) →c [T, X]r.i. . Given f ∈ L1 (νX ), it follows that f ∗ ∈ [T, X]r.i. . Taking simple functions ϕn such that 0 ≤ ϕn ↑ |f |, we have that ϕn → |f | in norm · νX (as L1 (νX ) is o.c.) and 0 ≤ ϕ∗n ↑ f ∗ , where ϕ∗n are also simple functions. From
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(4.2) and since T (f ∗ − g ∗ ) ≤ T (|f | − |g|)∗ (see the comment after (3.1) and note that f ∗ = |f |∗ ), we have that f ∗ − ϕ∗n νX ≤ f ∗ − ϕ∗n [T,X] ≤ |f | − ϕn [T,X]r.i. ≤ c |f | − ϕn νX → 0 and so f ∗ ∈ L1 (νX ). Conversely, suppose that f ∗ ∈ L1 (νX ) whenever f ∈ L1 (νX ). Then, given f ∈ L1 (νX ), from Proposition 4.1, f ∗ ∈ [T, X] and so f ∈ [T, X]r.i. (as T f ∗ ∈ X). Note that, in the case when L1 (νX ) ⊂ [T, X]r.i. , there exists c > 0 such that f νX = f [T,X] ≤ f [T,X]r.i. ≤ c f νX , for all f ∈ L1 (νX ). That is, ·[T,X]r.i. is equivalent to ·νX on L1 (νX ). Moreover, L1 (νX ) is an r.i. B.f.s. with the norm · [T,X]r.i. , since for every f ∈ L0 with f ∗ ∈ L1 (νX ), we have that f ∈ [T, X]r.i. and, from (4.2) and Proposition 4.1, f χA νX ≤ f χA [T,X]r.i. ≤ f ∗ χ[0,m(A)) [T,X] = f ∗ χ[0,m(A)) νX → 0 as m(A) → 0, from which it follows that f ∈ L1 (νX ).
Remark 4.3. The space L1 (νX ) is r.i. if and only if L1 (νX ) →i [T, X]r.i. . Indeed, if L1 (νX ) is r.i., from Proposition 4.2, L1 (νX ) ⊂ [T, X]r.i. and for every f ∈ L1 (νX ), f [T,X]r.i. = f ∗ [T,X] = f ∗ νX = f νX . Conversely, if L1 (νX ) →i [T, X]r.i. , by Proposition 4.2, we have that f ∈ L1 (νX ) if and only if f ∗ ∈ L1 (νX ). Moreover, in this case, f ∗ νX = f ∗ [T,X]r.i. = f [T,X]r.i. = f νX . Proposition 4.4. The containment [T, X]r.i. ⊂ L1 (νX ) holds if and only if [T, X]r.i. is o.c. Moreover, in this case, [T, X]r.i. →1 L1 (νX ). Proof. From Proposition 4.1(b), it follows that if [T, X]r.i. is o.c. then [T, X]r.i. ⊂ L1 (νX ). Moreover, f νX = f [T,X] ≤ f [T,X]r.i. for every f ∈ [T, X]r.i. . Suppose that [T, X]r.i. ⊂ L1 (νX ). For every f ∈ [T, X]r.i. , it follows that (f χA )∗ ∈ L1 (νX ), and then f χA [T,X]r.i. = (f χA )∗ [T,X] = (f χA )∗ νX ≤ f ∗ χ[0,m(A)) νX → 0 as m(A) → 0. Hence, [T, X]r.i. is o.c.
Note that if [T, X] is o.c. then [T, X]r.i. is also o.c., since for f ∈ [T, X]r.i. , f χA [T,X]r.i. = (f χA )∗ [T,X] ≤ f ∗ χ[0,m(A)) [T,X] → 0 as m(A) → 0. In this case, from Proposition 4.1(b), [T, X]r.i. →1 L1 (νX ) ≡ [T, X]. The space [T, X] is o.c. for instance if X is o.c., see [5, Proposition 3.1(i)]. Another interesting fact is that L∞ [T, X]r.i. . Indeed, it is not difficult to prove that an r.i. B.f.s. Y coincides with L∞ if and only if there exists c > 0 such that χA Y ≥ c
R.i. Optimal Domain for Monotone Kernel Operators for all A ∈ B([0, 1]) with m(A) > 0, and by (4.1), m(A) K(·, y) dy χA [T,X]r.i. = T χ[0,m(A))X = 0
X
157
→ 0 as m(A) → 0.
This also follows from Proposition 4.4, since L∞ ⊂ L1 (νX ) but L∞ is not o.c. The next result shows simple conditions on K and X guaranteeing that [T, X]r.i. is the whole of the space L1 , in particular it is o.c. Lemma 4.5. Suppose that K is strictly bounded, i.e., there exists C > 0 such that K(x, y) ≤ C for all x, y ∈ [0, 1], and L∞ ⊂ X. Then, [T, X]r.i. = L1 →1 L1 (νX ). Proof. We only have to see that L1 ⊂ [T, X]r.i. . Given f ∈ L1 , for every x ∈ [0, 1], 1 1 1 T f ∗ (x) = 0 f ∗ (y)K(x, y) dy ≤ C 0 f ∗ (y)dy = C 0 |f (y)| < ∞, that is, T f ∗ ∈ L∞ ⊂ X. So, f ∈ [T, X]r.i. .
x
Example. Let V be the Volterra operator, i.e., Vf (x) = 0 f (y) dy. Its kernel K(x, y) = χ[0,x] (y) satisfies (1.2), (3.1) and (4.1) for X containing the simple functions. Moreover, K is strictly bounded. So, [V, X]r.i. = L1 →1 L1 (νX ). In general, there is no containment relation between [T, X]r.i. and L1 (νX ). x Example. Let H be the Hardy operator, i.e., Hf (x) = x1 0 f (y) dy. Its kernel K(x, y) = x1 χ[0,x] (y) satisfies (1.2), (3.1) and (4.1) for X being a Lorentz space Lp,∞ with 1 < p < ∞ (see for instance [1, Definition 4.4.1]). Consider the functions f (y) = y −1/p and g(y) = (1 − y)−α for 1p < α < 1. It can be checked that f ∈ [H, Lp,∞ ]r.i. \ L1 (νLp,∞ ) and g ∈ L1 (νLp,∞ )\ [H, Lp,∞ ]r.i. . Acknowledgment The author thanks Prof. G.P. Curbera for useful discussions on this topic.
References [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988. [2] G.P. Curbera, Volterra convolution operators with values in rearrangement invariant spaces, J. London Math. Soc. 60 (1999), 258–268. [3] G.P. Curbera and W.J. Ricker, Optimal domains for kernel operators via interpolation, Math. Nachr. 244 (2002), 47–63. [4] G.P. Curbera and W.J. Ricker, Optimal domains for the kernel operator associated with Sobolev’s inequality, Studia Math. 170 (2005), 217–218; Studia Math. 158 (2003), 131–152. [5] G.P. Curbera and W.J. Ricker, Banach lattices with the Fatou property and optimal domains of kernel operators, Indag. Math. (N.S.) 17 (2006), 187–204. [6] G.P. Curbera and W.J. Ricker, Vector measures, integration and applications, Positivity, Trends in Mathematics, Birkh¨auser Basel (2007), 127–160.
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[7] G.P. Curbera and W.J. Ricker, Can optimal rearrangement invariant Sobolev imbeddings be further extended?, Indiana Univ. Math. J. 56 (2007), 1479–1497. [8] O. Delgado, Optimal domains for kernel operators on [0, ∞) × [0, ∞) , Studia Math. 174 (2006), 131–145. [9] O. Delgado and J. Soria, Optimal domain for the Hardy operator, J. Funct. Anal. 244 (2007), 119–133. [10] S.G. Kreˇın, Ju.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Amer. Math. Soc., Providence, R.I., 1982. [11] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, vol.II, Springer-Verlag, Berlin, 1979. [12] G. Mockenhaupt and W.J. Ricker, Optimal extension of the Hausdorff–Young inequality, J. Reine Angew. Math. 620 (2008), 195–211. [13] S. Okada and W.J. Ricker, Optimal domains and integral representations of convolution operators in Lp (G), Integral Equations Operator Theory 48 (2004), 525–546. [14] S. Okada, W.J. Ricker and E.A. S´anchez P´erez, Optimal Domain and Integral Extension of Operators acting in Function Spaces, Operator Theory Advances and Applications, vol. 180, Birkh¨ auser, Basel, 2008. [15] A.C. Zaanen, Integration, 2nd rev. ed. North Holland, Amsterdam; Interscience, New York, 1967. Olvido Delgado Instituto Universitario de Matem´ atica Pura y Aplicada Universidad Polit´ecnica de Valencia E-46071 Valencia, Spain e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 159–170 c 2009 Birkh¨ auser Verlag Basel/Switzerland
The Fubini and Tonelli Theorems for Product Local Systems Luisa Di Piazza and Valeria Marraffa Abstract. The notion of product local system and of the Kurzweil-Henstock type integral related to a product local system is introduced. The main result is a version of the Fubini and Tonelli theorems for product local systems. Mathematics Subject Classification (2000). Primary 26A39, 26A42, 26A45, 28A12. Keywords. Local system, product of local systems, Henstock integral.
1. Introduction In this paper we proceed to investigate Riemann type integrals related to local systems. The notion of local system in R was extensively studied in [9] and [4]. Properties of the associated integrals in the real line have been considered in [1, 2, 3, 4, 5]. Here we introduce on the plane the notion of product local system and of integral related to a product local system S (S-integral). The classical Henstock integral on the plane with respect to the Kurzweil basis (see [6]) is an example of integral with respect to a product local system. In Section 5 the Cauchy test for the S-integral is proved and properties of S-integrable functions and their primitives are obtained. In Section 3 a monotone convergence theorem for the S-integral in the real line is shown (Theorem 3.1). Such a result plays an important role in proving a Tonelli type theorem (Theorem 6.4) for a product local system. The main result of the paper is a Fubini type theorem (Theorem 6.3) for the S-integral in the plane.
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2. Preliminaries Unless stated otherwise, the following conventions and notation will be used. The set of real numbers is denoted by R, and the cartesian product R × R by R2 . The Lebesgue measure of a subset A of R is denoted by λ1 (A), while λ(E) denotes the outer Lebesgue measure of E ⊂ R2 . An interval in R2 is always the Cartesian product of two non-degenerate compact intervals in R. A collection of intervals in R or in R2 is called nonoverlapping whenever their interiors are disjoint. If I is an interval in R or R2 , we shall write λ1 (I) or λ(I) as |I|. We recall the following definition (see [9]). Definition 2.1. A family S1 = {S(x) : x ∈ IR} is said to be a local system on R if each S(x) is a collection of sets of R with the following properties: • • • •
{x} ∈ / S(x), for all x; if si ∈ S(x) and si ⊆ sj , then sj ∈ S(x); if s ∈ S(x), then x ∈ s; if s ∈ S(x) and δ > 0, then s ∩ (x − δ, x + δ) ∈ S(x).
We observe that this notion is a modification of that of filter in topology and has been used in formulating a variety of general notions of limit, continuity, derivative and even outer measure. Examples of local systems are: (a) the interval local system: for all x, S(x) contains all the neighborhoods on R of x; (b) the approximate local system: for all x, S(x) contains all the sets Ex such that x ∈ Ex and x is a density point of Ex ; (c) the path systems: for all x, there is Ex R such that x ∈ Ex , x is an accumulation point of Ex , and S(x) is the filter generated by {Ex ∩ (x − η, x + η), η > 0}. If E ⊂ R, a function ρ : E → 2R with ρ(x) ∈ S(x), is called an S1 -choice or simply a choice on the set E. A local system S1 is said to be bilateral if, for each x ∈ R, every set s ∈ S(x) contains points on each side of x. A local system S1 is said to be filtering if at each point x ∈ R we have si ∩ sj ∈ S(x) whenever si and sj belong to S(x). A local system S1 is said to have the intersection condition if corresponding to any choice ρ on R there exists a positive function δ such that whenever 0 < y − x < min{δ(x), δ(y)}, then ρ(x) ∩ ρ(y) ∩ [x, y] = ∅. Given a local system S1 we associate to each S1 -choice ρ a family βρ = {([u, v], x) : x = u, v ∈ ρ(x) or x = v, u ∈ ρ(x); x ∈ R}. Note that if a system S1 is filtering then the set of all families βρ forms a differentiation basis in terms of the Kurzweil-Henstock integration theory (see [6, 10]). For a set E put βρ [E] = {([u, v], x) ∈ βρ : x ∈ E}.
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A finite subset π ⊂ βρ , π = {(Ii , xi ) : i = 1, . . . , p}, is called a βρ -partition if the intervals I1 , . . . , Ip are nonoverlapping. A partition ' ' π in βρ [E] is called a βρ -partition in E (resp. of E) if (I,x)∈π I ⊂ E (resp. (I,x)∈π I = E). If S1 is a bilateral local system having the intersection condition, then for any S1 -choice ρ there exists a βρ -partition of any non degenerate compact interval of R ([9], p. 37). So for this basis we can define in the usual way the corresponding Kurzweil-Henstock type integral (see [3] and [2]) on an interval. Definition 2.2. Let [a, b] be a compact subinterval of R. A function f : [a, b] → R is said to be S1 -integrable on [a, b] with integral A, if for every ε > 0 there exists a choice ρ on [a, b] such that # # # # # #<ε, f (x)|I| − A # # (I,x)∈π
for any βρ -partition π of [a, b]. We write A = (S1 )
b a
f.
For this integral many of the usual properties, known also for other classes of bases, hold (see, for example, [1, 3, 4]).
3. A convergence theorem for the S1 -integral on the real line In this section all functions are assumed to be real-valued functions on [a, b] ⊂ R and S1 is a local system on [a, b]. Theorem 3.1. Let (fn ) be a monotone sequence of S1 -integrable functions on [a, b] b is bounded. Then and let fn → f everywhere in [a, b]. Assume that (S1 ) a fn b nb the function f is S1 -integrable on [a, b] and (S1 ) a fn → (S1 ) a f . Proof. We may assume that the sequence (fn ) is nondecreasing. Then also the b b sequence (S1 ) a fn is nondecreasing. Let M = sup{(S1 ) a fn : n ∈ N} and 1 < ε > 0. Then there is a natural number r such that 2r−1 b ε fr < . 0 ≤ M − (S1 ) 3 a
ε 3
and (3.1)
As for each n ∈ N, fn is S1 -integrable, there is a choice ρn on [a, b] such that (see [3]) # ## # #fn (x)|I| − (S1 ) fn # < 1 , (3.2) # # 2n I (I,x)∈π
for any βρn -partition π of [a, b]. Since limn fn (x) = f (x), then for each x ∈ [a, b], there is an integer k(x) ≥ r with ε . (3.3) 0 ≤ f (x) − fk(x) < 3(b − a)
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Define a choice ρε on [a, b] as ρε (x) = ρk(x) (x). In order to prove that the function f is S1 -integrable with S1 -integral M , let π = {(Ii , xi ) : i = 1, . . . , p} be a βρε -partition. We have # # # # p # p # # p # # # # f (xi )|Ii | − M # ≤ # f (xi )|Ii | − fk(xi ) (xi )|Ii |## # i=1
i=1
i=1
# # p # # p p # # # # + ## fk(xi ) (xi )|Ii | − (S1 ) fk(xi ) ## + ## (S1 ) fk(xi ) − M ## i=1
Ii
i=1
(3.4)
Ii
i=1
= A1 + A2 + A3 . Let us consider the three terms separately. By (3.3) we have A1 ≤
p p # # #f (xi ) − fk(x ) (xi )# |Ii | ≤ i i=1
i=1
ε ε |Ii | = . 3(b − a) 3
(3.5)
To estimate A2 , let s = max{k(x1 ), . . . , k(xp )} ≥ r and observe that for each q = r, . . . , s the collection {(Ii , xi ) : k(xi ) = q} is in βρq . Then by (3.2), we get # p # # # # fk(xi ) ## A2 ≤ #fk(xi ) (xi )|Ii | − (S1 ) Ii
i=1
=
s
# # s # # 1 #fk(x ) (xi )|Ii | − (S1 ) # ≤ f k(xi ) # i # q 2 Ii q=r
(3.6)
q=r k(xi )=q
1 ε < . 2r−1 3 We now estimate A3 . Since the sequence (fn ) is nondecreasing it follows (S1 ) fr ≤ (S1 ) fk(xi ) ≤ (S1 ) fs . <
Ii
Ii
Ii
By the additivity of the S1 -integral we get b p (S1 ) fr ≤ (S1 ) fk(xi ) ≤ (S1 ) a
Ii
i=1
b
fs , a
hence by (3.1) we obtain ε < (S1 ) 3 i=1 p
M− which implies
A3 <
fk(xi ) ≤ M, Ii
ε . 3
(3.7)
By (3.5), (3.6) and (3.7) we infer A1 + A2 + A3 < and the assertion follows.
ε ε ε + + =ε 3 3 3
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4. Product local system From now on our ambient space is R2 endowed with the norm ||x|| = max{|x1 |,|x2 |}, for x = (x1 , x2 ) in R2 . Let S1 and S2 be two local systems in R. The family S is said to be the product local system of S1 and S2 if S = {S(z) : z = (z1 , z2 ), S(z) = S1 (z1 ) × S2 (z2 ) with S1 (z1 ) ∈ S1 and S2 (z2 ) ∈ S2 }. A product local system S is said to be filtering if both S1 and S2 are filtering. Examples (i) the product of two interval local systems: for all z = (z1 , z2 ), S(z) = S1 (z1 ) × S2 (z2 ) where S1 (z1 ) and S2 (z2 ) contain respectively all the neighborhoods on R of z1 and z2 ; (ii) the product of two approximate local systems: for all z = (z1 , z2 ), S(z) = S1 (z1 ) × S2 (z2 ) contains all the sets E = Ez1 × Ez2 such that z = (z1 , z2 ) ∈ E and z1 and z2 are linear density points respectively of Ez1 and Ez2 ; (iii) the product of two path systems: for all z = (z1 , z2 ), there are Ez1 , Ez2 R such that z1 ∈ Ez1 , z2 ∈ Ez2 , z1 and z2 are linear accumulation points respectively of Ez1 and Ez2 , and Si (zi ) (i = 1, 2) is the filter generated by {Ezi ∩ (zi − η, zi + η), η > 0}. Throughout S = S1 × S2 is a product local system. If E = X × Y ⊂ R2 , we call 2 an S-choice or simply a choice on E any function γ : E → 2R such that, for x = (x1 , x2 ) ∈ E, γ(x) ∈ S(x). By the definition of S it follows that if γ(x) ia a choice on E, then γ(x) = ρ1 (x1 ) × ρ2 (x2 ), where ρ1 and ρ2 are respectively a choice on X and on Y . From now on the Greek letter ρ will denote a choice on a subset of R while the Greek letter γ will denote a choice on a subset of R2 . Given an S-choice γ, we associate a family of intervals in R2 as follows: βγ = {([a,b] × [c,d],x) : x = (a,c), b ∈ ρ1 (x1 ) and d ∈ ρ2 (x2 ) or x = (b,c), a ∈ ρ1 (x1 ) and d ∈ ρ2 (x2 ) or x = (b,d), a ∈ ρ1 (x1 ) and c ∈ ρ2 (x2 ) or x = (a,d), b ∈ ρ1 (x1 ) and c ∈ ρ2 (x2 ); x = (x1 ,x2 ) ∈ R2 }. Note that if a local system S is filtering then the set of all families βγ forms a differentiation basis in terms of the Henstock integration theory (see [6]), which we call S-basis. For a set E ⊂ R2 put βγ [E] = {([a, b] × [c, d], x) ∈ βγ : x ∈ E}. A finite subset π ⊂ βγ , π = {(Ii , xi ) : i = 1, . . . , p}, is called a βγ -partition if the intervals I1 , . . . , Ip are nonoverlapping. A partition ' ' π in βγ [E] is called a βγ -partition in E (resp. of E) if (I,x)∈π I ⊂ E (resp. (I,x)∈π I = E). From now on Δ denotes a fixed interval in R2 and I the family of all subintervals of Δ. Proposition 4.1. Assume that S1 and S2 are local systems on R which are bilateral and have the intersection condition and set S = S1 × S2 . For any choice γ on Δ and for all subintervals I = [a, b] × [c, d] ⊂ Δ there is a βγ -partition of I.
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Proof. For x = (x1 , x2 ) ∈ Δ, let γ(x) = ρ1 (x1 ) × ρ2 (x2 ) be a choice on Δ. Since both S1 and S2 have the intersection condition, denote respectively by π 1 = {(Ii1 , x1i ), i = 1, . . . , p} a βρ1 -partition of the interval [a, b], and by π 2 = {(Ij2 , x2j ), j = 1, . . . , l} a βρ2 -partition of the interval [c, d]. Then π = {(Ii1 × Ij2 , (x1i , x2j )), i = 1, . . . , p , j = 1, . . . , l} is a βγ -partition of I.
5. S-integral for a product local system From now on let S = S1 ×S2 be a bilateral, filtering product local system such that both S1 and S2 satisfy the intersection condition. Then, according to Proposition 4.1, S has the partitioning property: for any S-choice γ there exists a βγ -partition of any interval [a, b] × [c, d] ⊂ Δ. So for the associated basis we can define a Kurzweil-Henstock type integral on the interval Δ. Definition 5.1. A function f : Δ → R is said to be S-integrable on Δ, with Sintegral A, if for every ε > 0 there exists a choice γ on Δ such that # # # # # f (x)|I| − A## < ε , # (I,x)∈π
for any βγ -partition π of Δ. In this case we write A = (S)
Δ
f.
Remark 5.2. Since on R the interval local system, the approximate local system and a path system are bilateral, filtering and have the intersection condition (see [9]), then the definition of S-integral is well posed for the product local systems in the (i)–(iii) Examples of Section 4. Moreover observe that, when S is the product of two interval local systems, then the S-integral coincides with the Henstock integral in the plane with respect to the Kurzweil basis (see [6]). In order to investigate some properties for the S-integral we need the following Cauchy test. Proposition 5.3. A function f : Δ → R is S-integrable on Δ, if and only if for each ε > 0 there exists a choice γ on Δ such that # # # # # f (x)|I| − f (y)|J|## < ε # (I,x)∈π
(J,y)∈π
whenever π, π are βγ -partitions of Δ. Proof. Necessity can be proved in a standard way. We are proving sufficiency. For n = 1, 2, . . . , there is a choice γn on Δ such that # # # # 1 # #< f (x)|I| − f (y)|J| # # n (I,x)∈π
(J,y)∈π
whenever π, π are βγn -partitions of Δ. Since the local system S is filtering, replacing eventually γn by γ1 ∩ γ2 ∩ · · · ∩ γn , we may assume that γ1 γ2 · · · If πn is a βγn -partition of Δ, then the sequence (I,x)∈πn f (x)|I| is Cauchy and hence
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convergent. Let A = limn (I,x)∈πn f (x)|I|, then A = (S) Δ f . Indeed let ε > 0 and find an integer k such that k > 2ε and # # # ε # #< . # (5.1) f (x)|I| − A # 2 # (I,x)∈πk
Set γ = γk and let π be any βγ -partition of Δ. Then by (5.1) we infer # # # # # # # # # # # # # # # # # f (x)|I| − A# ≤ # f (x)|I| − f (x)|I|# + # f (x)|I| − A## # (I,x)∈π
(I,x)∈π
(I,x)∈πk
(I,x)∈πk
ε 1 < + <ε k 2 and the assertion is shown.
The following lemma follows easily from the Cauchy test. Lemma 5.4. Let f : Δ → R be an S-integrable function. Then, for any subinterval I of Δ, f χI is S-integrable. Corollary 5.5. Let f : Δ → R be an S-integrable function. Then the map F : I → (S) f I
is an additive function on I. The interval function F is called the S-indefinite integral of f . In a standard way, it is possible to prove the following properties: Q1 ) If f = 0 a.e. on Δ, then f is S-integrable on Δ and the S-integral of f is equal to 0. Q2 ) If f and g are S-integrable on Δ and α and β are two constants, then also the function αf + βg is S-integrable on Δ and (S) αf + βg = α(S) f+ Δ Δ β(S) Δ g. Q3 ) If f is an S-integrable function, then to each ε > 0 there corresponds a choice γ such that |f (x)|I| − F (I)| < ε (I,x)∈π
for each βγ -partition π in Δ (Henstock Lemma). The definition of S-continuity as well as that of S-derivative and S-variational measure follows as in the case of an S1 -system in R (see [9]). As a consequence of Henstock Lemma we get that the S-indefinite integral F of f is S-continuous. Remark 5.6. Observe that, in case of an S1 -system in the real line, the S1 -indefinite integral possesses the S1 -derivative a.e. The main tool for such a proof relies upon the Vitali Covering Theorem. In general the latter is no longer true in higher dimension for a general basis. On the other hand from the classical “Saks Rarity Theorem” ([8]) it is well known that there are on R2 Lebesgue integrable functions
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whose primitive is not differentiable with respect to the Kurzweil basis. Since each Lebesgue integrable function on the plane is also Henstock integrable and then S-integrable with respect to the product of two interval local systems (see Remark 5.2), the existence of the S-derivative for an S-indefinite integral, in general, may fail ([6] p. 38). Proposition 5.7. Let F be an additive function on I. If F is S-derivable a.e on Δ and the S-variation VFS of F is absolutely continuous, then F is the S-indefinite integral of DS F (x). Proof. Let N = {x ∈ Δ : DS F (x) does not exist}, then λ(N ) = 0. Define f (x) = DS F (x) in Δ \ N and f (x) = 0 in N . Fix > 0. There is a choice γ0 on N such that p |F (Ii )| < 2 i=1 for any βγ0 [N ] -partition π = {(Ii , xi ) : i = 1, . . . , p}. Moreover there is a choice γ1 on Δ\N such that |I| |F (I) − f (x)|I|| < 2|Δ| for each I such that (I, x) ∈ βγ1 [Δ\N ] . Let γ(x) = γ0 (x) if x ∈ N and γ(x) = γ1 (x) if x ∈ Δ \ N , then γ is a choice on Δ. If π = {(Ii , xi ) : i = 1, . . . , p} ⊂ βγ , it follows p
|f (xi )|Ii | − F (Ii )| =
<
xi ∈Δ\N
|F (Ii )|
xi ∈N
xi ∈Δ\N
i=1
|f (xi )|Ii | − F (Ii )| + |Ii | + ≤ . 2|Δ| 2
Thus f is S-integrable and F is the S-indefinite integral of f .
6. The Fubini Theorem for a product local system Let f be a real-valued function defined on Δ = [a, b] × [c, d] and S = S1 × S2 . If x1 ∈ [a, b] and x2 ∈ [c, d], we define two real-valued functions fx1 : [c, d] → R and fx2 : [a, b] → R as follows fx1 (·) = f (x1 , ·)
and
fx2 (·) = f (·, x2 ).
Our aim is to relate the S-integral of f with the iterated S-integrals in the real line of fx1 and fx2 . To prove the Fubini theorem for the S-integral we develop ideas similar to that in [7] for the Henstock integral on the plane. Proposition 6.1. Let f : Δ → R be an S-integrable function on Δ. Then fx1 : [c, d] → R is S2 -integrable for almost all x1 ∈ [a, b] and fx2 : [a, b] → R is S1 integrable for almost all x2 ∈ [c, d].
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Proof. Let N = {u ∈ [a, b] : fu is not S2 -integrable}. We will prove that λ1 (N ) = 0. Let u ∈ N . Then by the Cauchy test for the S1 -integral in the real line ([3] Theorem 3) there is an εu > 0 such that for each S2 -choice ρ2 on [c, d] there are two βγ2 -partitions π1u and π2u of [c, d], such that fx1 (v)|I| − fx1 (z)|J| ≥ εu . (6.1) (I,v)∈π1u
(J,z)∈π2u
Define a nonnegative function h on [a, b] as follows εu if u ∈ N h(u) = 0 if u ∈ [a, b] \ N . In order to prove that h = 0 a.e., at first we show that h is S1 -integrable. Since the function f is S-integrable, by Proposition 5.3, fixed ε > 0, there exists a choice γ = ρ1 × ρ2 on Δ such that # # # # # f (x)|I| − f (y)|J|## < ε # (I,x)∈π
(J,y)∈π
whenever π and π are βγ -partitions of Δ. For u ∈ N , select two βρ2 -partitions π1u = {(Kju , vju ) : j = 1, . . . , qu } and π2u = {(Luk , zku ) : k = 1, . . . , tu } of [c, d] such that (6.1) is satisfied. For u ∈ [a, b] \ N , let π1u = π2u be any βρ2 -partition. Let π = {(I1 , u1 , ), . . . , (Ip , up )} ⊂ βρ1 . Now, if Q = {(Ii × Kjui , (ui , vjui )) : i = 1, . . . , p; j = i, . . . , qui }, and T = {(Ii × Luk i , (ui , zkui )) : i = 1, . . . , p; k = i, . . . , tui }, then Q and P are βγ -partitions of Δ. Therefore p
h(ui )|Ii | =
h(ui )|Ii | +
ui ∈N
i=1
≤
qui 2 ui ∈N j=1
h(ui )|Ii | =
εui |Ii |
ui ∈N
ui ∈[a,b]\N
fui (vjui )|Kjui | −
tui
3 fui (zkui )|Luk i | |Ii |
k=1
# # qui tui p # p # ui ui ui ui # # ≤# f (ui , vj )|Ii ||Kj | − f (ui , zk )|Ii ||Lk |# < ε. i=1 j=1
i=1 k=1
b Since h ≥ 0, it follows that the function h is S1 -integrable and (S1 ) a h = 0. If t ∈ [a, b], by Lemma 5.4 the function f is S-integrable on the interval [a, t]×[c, d]. t Then with the same computation we get that H(t) = (S1 ) a h = 0, that is the S1 primitive H of h is equal to zero. So the S1 -derivative of H exists and DS1 H(t) = h(t) a.e (see [3] and [4]). Since DS1 H(t) = 0 we get that h = 0 a.e. Therefore λ1 (N ) = 0 and the function fx1 : [c, d] → R is S2 -integrable for almost all u ∈ [a, b]. Equivalently the function fx2 : [a, b] → R is S1 -integrable for almost all v ∈ [c, d] and the assertion holds true.
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Proposition 6.2. Let f : Δ → R be an S-integrable function on Δ. Then the functions d b F1 : x1 → (S2 ) fx1 and F2 : x2 → (S1 ) fx2 c
a
are, respectively S1 -integrable on [a, b] and S2 -integrable on [c, d]. Moreover, the following holds b d (S) f = (S1 ) F1 = (S2 ) F2 . Δ
a
c
Proof. Observe that from property Q1 ), if g is a function which is a.e. equal to an S-integrable function, then also g is S-integrable. Therefore modifying eventually the function f in a null set, we may assume that fx1 is S2 -integrable for all x1 ∈ [a, b]. Let ε > 0 and find a choice γ = ρ1 × ρ2 on Δ such that # # # # ε # f (x)|I| − (S) f ## < , # 2 Δ (I,x)∈π
for any βγ -partition π of Δ. For each u ∈ [a, b] fix a βρu -partition π1u = {(Kju , vju ) : j = 1, . . . , qu } of [c, d] such that # # qu # # ε u u # . (6.2) fu (vj )|Kj | − F1 (u)## < # 2(b − a) j=1 Since the local system S is filtering, we may assume that ρu ⊂ ρ2 . Let π = {(I1 , u1 ), . . . , (Ip , up )} be a βρ1 -partition of [a, b]. Now, if Q = {(Ii × Kjui , (ui , vjui )) : i = 1, . . . , p; j = i, . . . , qui }, then Q is a βγ -partition of Δ. Then by (6.2) # # qui # # p # # p # # p ui ui # # # # F (u )|I | − (S) f ≤ F (u )|I | − f (u , v )|I ||K | 1 i i 1 i i i j i j # # # # i=1
Δ
i=1
i=1 j=1
# qui # # p # + ## f (ui , vjui )|Ii ||Kjui | − (S) f ## i=1 j=1
Δ
# p 2 qui 3 # # # ε ≤ ## F1 (ui ) − fui (vjui )|Kjui | |Ii |## + 2 i=1 j=1 ≤ <
# qui p # # # ε ui ui # #F1 (ui ) − f (v )|K | ui j j #|Ii | + # 2 i=1 j=1 p i=1
|Ii |
ε ε + = ε. 2(b − a) 2
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Therefore the function F1 is S1 -integrable and b f = (S1 ) F1 . (S) Δ
a
Analogously the function F2 is S2 -integrable and d f = (S2 ) F2 , (S) Δ
c
and the assertion follows.
By Propositions 6.1 and 6.2 we obtain the following version of the Fubini Theorem for the S-integral. Theorem 6.3. Let f : Δ → R be an S-integrable function on Δ. Then the function fx1 : [c, d] → R is S2 -integrable on [c, d] for almost all x1 ∈ [a, b] and the function fx2 : [a, b] → R is S1 -integrable on [a, b] for almost all x2 ∈ [c, d]. Moreover the functions d b F1 : x1 → (S2 ) fx1 and F2 : x2 → (S1 ) fx2 c
a
are, respectively S1 -integrable and S2 -integrable and b f = (S1 ) F1 = (S2 ) (S) Δ
a
d
F2 .
c
As an application of Theorem 3.1 we obtain the following version of the Tonelli Theorem. Theorem 6.4. Let f : Δ → R, f ≥ 0 be a measurable function. Assume that d either the function F1 : x1 → (S2 ) c fx1 is S1 -integrable on [a, b] or the function b F2 : x2 → (S1 ) a fx2 is S2 -integrable on [c, d]. Then f is Lebesgue integrable on Δ. d Proof. Assume that F1 : x1 → (S2 ) c fx1 is S1 -integrable. For each n, let [fn ] = min{f, n}. The function [fn ], being measurable and bounded is McShane integrable. Therefore it is S-integrable and by Proposition 6.1 the same is true for d b the function [Fn ]1 : x1 → (S2 ) c [fn ]x1 . By Proposition 6.2 (S1 ) a [Fn ]1 = Δ [fn ]. Since the sequence ([fn ]x1 )n is nondecreasing, the same is true for the sequence d ((S2 ) c [fn ]x1 )n . The function F1 is S-integrable and [Fn ]1 # F1 . Therefore by Theorem 3.1 we infer b b [Fn ]1 = (S1 ) F1 . lim(S) [fn ] = lim(S1 ) n
Δ
n
a
a
Since f = limn→∞ [fn ], by the Monotone Convergence Theorem for the Lebesgue integral it follows that f is Lebesgue integrable.
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References [1] B. Bongiorno, L. Di Piazza and V.A. Skvortsov, On dyadic integrals and some other integrals associated with local systems, J. Math. Anal. Appl., 271 (2002), 506–524. [2] D. Bongiorno, L. Di Piazza and V.A. Skvortsov, Variational measures related to local systems and the Ward property of P-adic path bases, Czech. Math. J., 56, No 131 (2006), 559–578. [3] W. Cai-shi and D. Chuan-Song, An integral involving Thomson’s local systems , Real Anal. Exchange 19, No. 1 (1993/94), 248–253. [4] V. Ene, “Real functions – Current Topics”, Lecture Notes in Math., Vol. 1603, Springer-Verlag, 1995. [5] S. Lu, Notes on approximately continuous Henstock integral, Real Anal. Exchange, 22, No. 1 (1996/7), pp. 377–381. [6] K.M. Ostaszewski, “Henstock integration in the plane”, Memoirs of the Amer. Math. Soc., Providence, 353, 1986. [7] W.F. Pfeffer, “The Riemann approach to integration”, Cambridge University Press, 1993. [8] S. Saks, Remarks on the differentiability of the Lebesgue indefinite integral, Fund. Math., 22, (1934), 257–261. [9] B.S. Thomson, “Real functions”, Lecture Notes in Math., Vol. 1170, Springer-Verlag, 1980. [10] B.S. Thomson, Derivation bases on the real line, Real Anal. Exchange 8, No. 1–2 (1982/83), 67–207 and 278–442. Luisa Di Piazza and Valeria Marraffa Department of Mathematics University of Palermo via Archirafi 34 I-90123 Palermo, Italy e-mail: [email protected] [email protected]
Operator Theory: Advances and Applications, Vol. 201, 171–182 c 2009 Birkh¨ auser Verlag Basel/Switzerland
A Decomposition of Henstock-Kurzweil-Pettis Integrable Multifunctions Luisa Di Piazza and Kazimierz Musial Abstract. We proved in our earlier paper [9] that in case of separable Banach space-valued multifunctions each Henstock-Kurzweil-Pettis integrable multifunction can be represented as a sum of one of its Henstock-Kurzweil-Pettis integrable selectors and a Pettis integrable multifunction. Now, we prove that the same result can be achieved in case of an arbitrary Banach space. Applying the representation theorem we describe the multipliers of the HenstockKurzweil-Pettis integrable multifunctions. Then we use this description to obtain a characterization of the Henstock-Kurzweil-Pettis integrability in terms of subadditive operators. Mathematics Subject Classification (2000). Primary 28B20; Secondary 26A39, 28B05, 46G10, 54C60. Keywords. Multifunctions, Pettis set-valued integral, Kurzweil-Henstock integral, Kurzweil-Henstock-Pettis integral, support function, Denjoy-KhintchinePettis integral, selectors.
Introduction The Lebesgue integral, introduced by Lebesgue himself, is a powerful tool playing very important role in mathematics. However, because of its abstract character, it does not have the intuitive appeal of the Riemann integral. Moreover, as already Lebesgue observed in his thesis [20], his integral does not integrate all unbounded derivatives and so it does not provide a complete solution for the problem of primitives, i.e., for the problem of recovering a function from its derivative. Also non-absolutely convergent integrals are beyond the Lebesgue integration theory. In 1957 Kurzweil [19] and, independently, in 1963 Henstock [17], by a simple modification of Riemann’s method, gave a new definition of an integral (called now the Henstock-Kurzweil integral), that retains the intuitive appeal of the Riemann The authors were partially supported by the grant N 201 00932/0243.
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definition and, at the same time, coincides with the Lebesgue integral on the set of Lebesgue integrable functions. Moreover, it integrates all derivatives. In the last years a number of mathematicians became interested in the Pettis integral of Banach space-valued multifunctions (see for instance [5], [12], [26], [27]). The definition of such integral involves the Lebesgue integrability of the support functions. It is our motivation to consider, also in case of multifunctions, the Henstock-Kurzweil integral in places where the Lebesgue integral used to be applied. In this paper we continue the study of such a generalization of the Pettis integral of a multifunction obtained by replacing the Lebesgue integrability of the support functions by the Henstock-Kurzweil integrability (we call such an integral Henstock-Kurzweil-Pettis). In [9] we proved that in case of separable Banach space-valued multifunctions each Henstock-Kurzweil-Pettis integrable multifunction can be represented as a sum of one of its Henstock-Kurzweil-Pettis integrable selectors and of a Pettis integrable multifunction. A similar result in case of Henstock integrable multifunctions was proved in [11]. A very important tool for the study of set-valued integration is the Kuratowski and Ryll-Nardzewski theorem guaranteeing the existence of measurable selectors. Its drawback is the requirement of separability for the range space. Recently Cascales, Kadets and Rodriguez in two remarkable papers ([6], [7]) have got rid of the separability constraints of the range space. In [6] they proved the existence of scalarly measurable selectors for Pettis integrable multifunctions taking their values in non separable Banach spaces and in [7] they improved the previous result by showing the existence of scalarly measurable selectors of arbitrary weakly compact valued scalarly measurable multifunctions. Applying the selection results from [6] and [7], we prove in this paper that the decomposition theorem for Henstock-Kurzweil-Pettis integrable multifunctions can be achieved also in case of an arbitrary Banach space. So for a general Banach space the Henstock-Kurzweil-Pettis integral of a multifunction is, in some way, a translation of the Pettis integral of a multifunction. As applications of such decomposition we present a characterization of Henstock-Kurzweil-Pettis integrable multifunctions (see Proposition 6) and we show that a real function is a multiplier of Henstock-Kurzweil-Pettis integrable multifunctions if and only if it is almost everywhere equal to a function of bounded variation (see Corollary 1). At last we observe that all the results and the proofs here are also valid for the Denjoy-Khintchin-Pettis integrable multifunctions (see Remark 3).
1. Notations and preliminaries Throughout this paper X is an arbitrary Banach space with dual X ∗ and τ (X ∗ , X) denotes the Mackey topology on X ∗ , that is the topology of uniform convergence on weakly compact subsets of X. L is the family of all Lebesgue measurable subsets
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of [0, 1] and I the family of all closed subintervals of [0, 1]. If I ∈ I, then |I| denotes its Lebesgue measure. A partition P of [0, 1] is a collection {(I1 , t1 ), . . . , (Ip , tp )}, where I1 , . . . , Ip are non-overlapping subintervals of [0, 1], ti is a point of Ii and ∪pi=1 Ii = [0, 1]. A gauge on [0, 1] is a positive function on [0, 1]. For a given gauge δ on [0, 1], we say that a partition {(I1 , t1 ), . . . , (Ip , tp )} of [0, 1] is δ-fine if Ii ⊂ (ti − δ(ti ), ti + δ(ti )), i = 1, . . . , p. Definition 1. A function f : [0, 1] → R is said to be Henstock-Kurzweil integrable (or simply HK-integrable) on [0, 1] if there exists a real number z with the following property: for every > 0 there exists a gauge δ on [0, 1] such that # # p # # # # f (ti )|Ii | − z # < ε , # # # i=1 1 for each δ-fine partition {(I1 , t1 ), . . . , (Ip , tp )} of [0, 1]. We set z : = (HK) 0 f (t) dt. 1 If I ∈ I, then we set (HK) I f (t) dt : = (HK) 0 f (t)χI (t) dt. We suggest [16] as the proper reference to the theory of Henstock-Kurzweil integration of real functions. Definition 2. A function f : [0, 1] → X is said to be Henstock-Kurzweil-Pettis integrable (or HKP -integrable) on [0, 1] if for every x∗ ∈ X ∗ the function x∗ f is HK-integrable and for each I ∈ I there exists a vector wI ∈ X such that x∗ , wI = (HK) I x∗ f (t)dt, for every x∗ ∈ X ∗ . wI is called the Henstock-Kurzweil-Pettis integral of f over I and we set wI : = (HKP ) I f (t)dt. For a detailed account on Henstock-Kurzweil-Pettis integral we refer the reader to [10]. Throughout cwk(X) will denote the family of all nonempty convex weakly compact subsets of X and ck(X) will denote the family of all nonempty compact convex subsets of X. For every C ∈ cwk(X) the support function of C is defined for each x∗ ∈ X ∗ by s(x∗ , C) = sup{ x∗ , x : x ∈ C}. A map Γ : [0, 1] → cwk(X) is called a multifunction. A multifunction Γ is said to be scalarly measurable if for every x∗ ∈ X ∗ , the map s(x∗ , Γ (·)) is measurable. Γ is said to be scalarly integrable (resp. scalarly HK-integrable) if, for every x∗ ∈ X ∗ , the function s(x∗ , Γ (·)) is integrable (resp. HK-integrable). A function f : [0, 1] → X is called a selector of Γ if, for every t ∈ [0, 1], one has f (t) ∈ Γ (t). Definition 3. A multifunction Γ : [0, 1] → cwk(X) is said to be Pettis integrable in cwk(X) (or in ck(X)), if Γ is scalarly integrable and for each A ∈ L there exists a set WA ∈ cwk(X) (or WA ∈ ck(X), respectively) such that ∗ s(x∗ , Γ (t)) dt for all x∗ ∈ X ∗ . s(x , WA ) = A We call WA the Pettis integral of Γ over A and we set WA : = (P ) A Γ (t)dt. Replacing L by I and the Lebesgue integrability by HK-integrability, we obtain the definitions of HKP-integrability of Γ .
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Given a multifunction Γ , the symbol SHKP (Γ ) denotes the family of all selectors of Γ that are HKP-integrable. Definition 4. A scalarly measurable multifunction Γ : [0, 1] → cwk(X) is said to be Aumann-Henstock-Kurzweil-Pettis integrable if SHKP (Γ ) = ∅. Then we define for each J ∈ I (AHKP ) Γ (t) dt := {(HKP ) f (t)dt : f ∈ SHKP (Γ )} . J J To simplify further notations let ISΓ (J) := {(HKP ) J f (t)dt : f ∈ SHKP (Γ )}. It is clear that the values of the AHKP-integral are closed and convex sets. We suggest [12] and [6] as the proper references to the theory of Pettis integrable multifunctions with values respectively in separable Banach spaces and in arbitrary Banach spaces, and [22] and [23] as the proper references to the theory of Pettis integrable functions. In case of Pettis integrable multifunctions we have the following result that is essentially inclosed in [6]. Proposition 1. (i) Let Γ : [0, 1] → cwk(X) be a scalarly integrable multifunction. Then Γ is Pettis integrable in cwk(X) if and only if for each A ∈ L the mapping x∗ −→ A s(x∗ , Γ (t)) dt is τ (X ∗ , X)-continuous. (ii) Let Γ : [0, 1] → ck(X) be a scalarly integrable multifunction. Then Γ is if and only if for each A ∈ L the mapping x∗ −→ Pettis ∗integrable in ck(X) ∗ ∗ A s(x , Γ (t)) dt is σ(X , X)-continuous on B(X ). Proof. (i) is Lemma 2.1 in [6]. The proof of (ii) follows as in [6, Lemma 2.1], taking into account this time the Banach-Dieudonn´e theorem and the characterization of the support functions of ck(X) sets via weak∗-continuity (cf. [12, Proposition 1.5]). In order to prove our main result we need the following Henstock-KurzweilPettis version of the previous proposition. We omit the proof because it is ideologically identical with the original. Proposition 2. (i) Let Γ : [0, 1] → cwk(X) be a scalarly HK-integrable multifunction. Then Γ is HKP-integrable in cwk(X) if and only if for each I ∈ I the mapping x∗ −→ (HK) I s(x∗ , Γ (t)) dt is τ (X ∗ , X)-continuous. (ii) Let Γ : [0, 1] → ck(X) be a scalarly HK-integrable multifunction, then Γ is HKP-integrable in ck(X) if and only if for each I ∈ I the mapping x∗ −→ ∗ (HK) I s(x , Γ (t)) dt is σ(X ∗ , X)-continuous on B(X ∗ ). We need also the next result which is the Henstock-Kurzweil-Pettis version of [6, Theorem 2.5] concerning the existence of a Pettis integrable selector for a Pettis integrable multifunction. We would like to mention that the proof of existence of
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selectors in [6] is based on an idea of using exposed points. Exposed points were earlier used to find selectors of multivalued measures by Hiai [18]. Proposition 3. Let Γ : [0, 1] → cwk(X) be HKP-integrable in cwk(X). Then the family of scalarly measurable selectors of Γ is nonempty and each scalarly measurable selector of Γ is HKP-integrable. Proof. If Γ is HKP-integrable in cwk(X), then Γ is scalarly measurable. So by [7, Theorem 3.8] we have the existence of a scalarly measurable selector f of Γ . Then we apply [9, Lemma 2] (whose proof works in an arbitrary Banach space as in the separable case) to get the HKP-integrability of f .
2. A decomposition theorem for HKP-integrable multifunctions We start with a lemma that can be considered as a generalization of the wellknown fact that real-valued non-negative Henstock-Kurzweil integrable functions are Lebesgue integrable. Lemma 1. If G : [0, 1] → cwk(X) is HKP-integrable in cwk(X) and for every x∗ ∈ X ∗ the inequality s(x∗ , G(t)) ≥ 0 holds true a.e., then G is Pettis integrable in cwk(X). Proof. The HK-integrability of s(x∗ , G(·)) with the hypothesis s(x∗ , G(·)) ≥ 0 a.e., yield the Lebesgue integrability of s(x∗ , G(·)) (see [16, Theorem 9.13]). Moreover by Proposition 2(i) the mapping x∗ → I s(x∗ , G(t)) dt is τ (X ∗ , X)-continuous, for every I ∈ I. In particular, for each τ (X ∗ , X)-convergent net x∗α → 0, we have 1 1 s(x∗α , G(t)) dt → s(0, G(t)) dt = 0. 0
0
Now, if A ∈ L, then we have 0≤ s(x∗α , G(t)) dt ≤ A
1
s(x∗α , G(t)) dt → 0.
0
Thus, due to subadditivity of the support functions, for each A ∈ L the mapping x∗ → A s(x∗ , G(t)) dt is τ (X ∗ , X)-continuous and so by Proposition 1(i) the multifunction G is Pettis integrable in cwk(X). The following theorem is a generalization of our earlier characterization of HKP-integrable multifunctions with values in a separable Banach space [9]. The proof is similar to that in the separable case. Theorem 1. Let Γ : [0, 1] → cwk(X) be a scalarly measurable multifunction. Then the following conditions are equivalent: (i) Γ is HKP-integrable in cwk(X); (ii) SHKP (Γ ) = ∅ and for every f ∈ SHKP (Γ ) the multifunction G : [0, 1] → cwk(X) defined by Γ (t) = G(t) + f (t), is Pettis integrable in cwk(X);
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(iii) there exists f ∈ SHKP (Γ ) such that the multifunction G : [0, 1] → cwk(X) defined by Γ (t) = G(t) + f (t) is Pettis integrable in cwk(X); (iv) Γ is scalarly HK-integrable and for each I ∈ I, the set (AKHP ) I Γ (t) dt belongs to cwk(X) and s x∗ , (AHKP ) Γ (t) dt = (HK) s(x∗ , Γ (t)) dt I ∗
I
∗
for all x ∈ X ; (v) each scalarly measurable selector of Γ is HKP-integrable. Proof. (i)⇒(ii) Let f ∈ SHKP (Γ ) be arbitrary (existing by Proposition 3). Define G : [0, 1] → cwk(X) by setting G(t) : = Γ (t) − f (t). Then s(x∗ , G(t)) ≥ 0 for all x∗ ∈ X ∗ and t ∈ [0, 1]. Moreover, s(x∗ , Γ (t)) = s(x∗ , G(t)) + x∗ f (t) , and so the HK-integrability of s(x∗ , Γ (·)) and of x∗ f yields the HK-integrability of s(x∗ , G(·)). Thus, for each x∗ ∈ X ∗ and for each I ∈ I we have ∗ ∗ (HK) s(x , G(t)) dt = (HK) s(x , Γ (t)) dt − (HK) x∗ f (t) dt I I I ∗ = s(x , (HKP ) Γ (t) dt) − (HK) x∗ f (t) dt I I ∗ = s x , (HKP ) Γ (t) dt − (HKP ) f (t) dt .
I
I
And since (HKP ) I Γ (t) dt belongs to cwk(X), also the set (HKP ) I Γ (t) dt − (HKP ) I f (t) dt belongs to cwk(X). Therefore G is HKP-integrable. Since for all x∗ ∈ X ∗ we have s(x∗ , G(t)) ≥ 0, by Lemma 1 we infer that G is Pettis integrable in cwk(X). The implications (ii)⇒(iii), (iii)⇒(i) and (iv)⇒(i) are obvious. (i)⇒(v) follows from Proposition 3. (v)⇒(i) and (v)⇒(iv) The proof mimics that of Theorem 4.2 from [6] but we present it for convenience of readers. Given an interval J ∈ I the set ISΓ (J) is closed and convex. Applying James’ characterization of weak compactness (cf. [13, §6]) we are going to prove that ISΓ (J) is weakly compact. So let x∗ ∈ X ∗ be an arbitrary functional and let G : [0, 1] → cwk(X) be defined by G(t) := {x ∈ Γ (t) : x∗ (x) = s(x∗ , Γ (t))}. G is scalarly measurable (by [25, Lemma 3]) and so, in view of [7, Theorem 3.8] it has a scalarly measurable selector g that is also a selector of Γ and s(x∗ , Γ ) = x∗ g is HK-integrable. By assumption, g is HKP-integrable. In particular (HKP ) J g(t) dt ∈ ISΓ (J).
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If f is any HKP-integrable selector of Γ , then ∗ ∗ = (HK) x g(t) dt = (HK) s(x∗ , Γ (t)) dt (1) x (HKP ) g(t) dt J J J ∗ ∗ ≥ (HK) x f (t) dt = x (HKP ) f (t) dt . J
J
It follows that
s(x∗ , ISΓ (J)) = sup{x∗ (x) : x ∈ ISΓ (J)} = x∗ (HKP ) g(t) dt .
(2)
J
This proves that x∗ attains its supremum on the set ISΓ (J) forcing its weak compactness. Moreover, it follows immediately from (1) and (2) that ∗ s(x , ISΓ (J)) = (HK) s(x∗ , Γ (t)) dt J
and so Γ is HKP-integrable in cwk(X).
It is proved in Theorem 4.2 from [6] and Theorem 3.8 from [7] that if (Ω, Σ, μ) is a finite measure space and Γ : Ω → cwk(X) is scalarly measurable and each its scalarly measurable selector is Pettis integrable, then Γ is Pettis integrable in cwk(X). Applying Lemma 1 and Theorem 1 we obtain the following generalization of that result in case when the measure space coincides with [0, 1] endowed with the Lebesgue measure and support functions of Γ are non-negative: Proposition 4. Let Γ : [0, 1] → cwk(X) be scalarly measurable. If s(x∗ , Γ ) ≥ 0 a.e. for every x∗ ∈ X ∗ and every scalarly measurable selector of Γ is HKP-integrable, then Γ is Pettis integrable in cwk(X). Proof. According to Theorem 1 if Γ : [0, 1] → cwk(X) is scalarly measurable and its scalarly measurable selectors are HKP-integrable, then Γ is HKP-integrable in cwk(X). Applying Lemma 1, we obtain the Pettis integrability of Γ . In case of ck(X)-valued multifunctions, applying, respectively, the (ii) part of Proposition 2 and of Proposition 1 and Proposition 3 we obtain Lemma 2. If G : [0, 1] → ck(X) is HKP-integrable in ck(X) and s(x∗ , G(t)) ≥ 0 a.e., for all x∗ ∈ X ∗ , then G is Pettis integrable in ck(X). Theorem 2. Let Γ : [0, 1] → ck(X) be a scalarly measurable multifunction. Then the following conditions are equivalent: (i) Γ is HKP-integrable in ck(X); (ii) SHKP (Γ ) = ∅ and for every f ∈ SHKP (Γ ) the multifunction G : [0, 1] → ck(X) defined by Γ (t) = G(t) + f (t) is Pettis integrable in ck(X); (iii) there exists f ∈ SHKP (Γ ) such that the multifunction G : [0, 1] → ck(X) defined by Γ (t) = G(t) + f (t) is Pettis integrable in ck(X);
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(iv) Γ is scalarly HK-integrable and for each I ∈ I, the set (AKHP ) belongs to ck(X) and ∗ s x , (AHKP ) Γ (t) dt = (HK) s(x∗ , Γ (t)) dt I ∗
I
Γ (t) dt
I
∗
for all x ∈ X . Question 1. One may ask if condition (v) of Theorem 1 is equivalent to conditions (i)–(iv) of Theorem 2 (provided Γ is ck(X)-valued)? The answer is affirmative in case of a separable X (see [9]). Unfortunately, we do not know the answer in the general case. If Γ : [0, 1] → ck(X) is HKP-integrable in ck(X) and s(x∗ , Γ ) ≥ 0 a.e., for every x∗ ∈ X ∗ , then the inclusion ISΓ (I) ⊃ (HKP ) f (t) dt : I % J ⊂ I J
holds true for every I ∈ I and f ∈ SHKP (Γ ). In particular integrals of Γ -selectors must have norm relatively compact ranges of their integrals. So the question can be formulated in the following way: Assume that Γ : [0, 1] → ck(X) is scalarly HK-integrable and s(x∗ , Γ ) ≥ 0 a.e., for each x∗ ∈ X ∗ . Moreover all scalarly measurable selectors are HKP-integrable. Is Γ HKP-integrable in ck(X)? We know it is HKP-integrable in cwk(X). Due to Proposition 4 this is in fact a question on Pettis integrability of Γ . For the cwk(X)-valued multifunctions that are Pettis integrable in cwk(X) the following result concerning the multipliers holds true: Proposition 5. Let G : [0, 1] → cwk(X) be a Pettis integrable multifunction in cwk(X). If g : [0, 1] → R belongs to L∞ [0, 1], then also gG is a Pettis integrable multifunction in cwk(X). Proof. According to [6, Theorem 2.5], there exists a Pettis integrable selector f of G. Then G = H+f , where H := G−f . Because the measurable essentially bounded functions are multipliers of Pettis integrable functions (see [8, Proposition 1]), it is enough to show that the multifunction gH is Pettis integrable in cwk(X). Let g = g + − g − be the standard decomposition of g. Due to the fact that s(x∗ , H) ≥ 0 for each x∗ ∈ X ∗ , it follows from Proposition 1(i) that s(x∗α , H) → s(x∗0 , H) in the weak topology of L1 [0, 1], whenever x∗α → x∗0 in τ (X ∗ , X). In particular, we have ∗ + s(xα , g (t)H(t)) dt = lim g + (t)s(x∗α , H(t)) dt lim α α E E g + (t)s(x∗0 , H(t)) dt = s(x∗0 , g + (t)H(t)) dt = E
E
for every E ∈ L. It follows that g H is Pettis integrable. Concerning the multifunction (−g − )H, it is enough to observe that s(x∗ , (−g − )H) = s(−x∗ , (g − )H) ≥ 0. Hence (−g − )H is Pettis integrable as well. +
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We have now g(t)H(t) = g + (t)H(t) − g − (t)H(t) and as the sum of two multifunctions Pettis integrable in cwk(X), the multifunction gH is also Pettis integrable in cwk(X). In [10, Theorem 1] we have proven that the functions that are a.e. equal to functions of bounded variation are multipliers of HKP-integrable functions. Now as an immediate consequence of Theorem 1 and Proposition 5 we can extend that result to the cwk(X)-valued multifunctions. Corollary 1. Let Γ : [0, 1] → cwk(X) be HKP-integrable in cwk(X). If g is a.e. equal to a function of bounded variation, then gΓ is also HKP-integrable in cwk(X). And conversely, let g : [0, 1] → R be such that for each multifunction Γ : [0, 1] → cwk(X) that is HKP-integrable in cwk(X), the multifunction gΓ is HKP-integrable in cwk(X). Then g is a.e. equal to a function of bounded variation. Proof. Let a multifunction Γ : [0, 1] → cwk(X) be HKP-integrable in cwk(X) and let g : [0, 1] → R be of bounded variation. According to Theorem 2, Γ = G + f , where G is Pettis integrable in cwk(X) and f is an HKP-integrable selector of Γ . By Proposition 5 the multifunction gG is Pettis integrable in cwk(X) and by [10, Theorem 1] the function gf is HKP-integrable. Consequently, applying once again Theorem 2 we obtain the HKP-integrability of gΓ = gG + gf in cwk(X). As long as the second part is considered, then each g being a multiplier of cwk(X)-valued HKP-integrable multifunctions is also a multiplier of HKPintegrable functions (and of HK-integrable real functions), and so by [10, Theorem 1] (or by [21, Theorem 12.9]) g must be a.e. equal to a function of bounded variation. Remark 1. Proposition 5 and Corollary 1 hold true also for ck(X)-valued multifunctions, what follows from part (ii) of Proposition 1 and of Theorem 2. By the previous results we infer a characterization of the HKP-integrability of multifunctions that is a generalization of that proved in [10, Theorem 2] for HKP-integrable functions. In the following the symbol BV [0, 1] denotes the family of all real functions of bounded variation on [0, 1], and HK[0, 1] is the family of all real-valued HK-integrable functions on [0, 1], where functions a.e. equal are identified. The space HK[0, 1] is endowed with the Alexiewicz norm (see [1]) # # α # # gA = sup ##(HK) g(t) dt## . 0<α≤1
0
6 1] of HK[0, 1] is isomorphic to the space of all distribuThe completion HK[0, tions, each one of which is the distributional derivative of a continuous function (see [2]). As it is known, HK-integrability coincides with Denjoy-Perron integrability (see [16]). The Denjoy-Perron integral is also called the Denjoy integral in the restricted sense (see [24]). If D[0, 1] is the (quotient) space of Denjoy-Perron integrable functions endowed with the Alexiewicz norm, then its conjugate space is linearly isometric to the space BV [0, 1] (see [1]). Consequently, the conjugate
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space HK ∗ [0, 1] is linearly isometric to the space BV [0, 1]. If g ∈ BV [0, 1], then 1 its value on a function f ∈ HK[0, 1] is given by g, f = (HK) 0 f (t)g(t) dt. Proposition 6. Let Γ : [0, 1] → cwk(X) be scalarly HK-integrable. Then Γ is HKP-integrable in cwk(X) if and only if the mapping TΓ : X ∗ → HK[0, 1] given by TΓ (x∗ ) = s(x∗ , Γ (·)) is τ (X ∗ , X)-weakly continuous. Proof. If TΓ is τ (X ∗ , X)-weakly continuous, then the HKP-integrability of Γ follows from Proposition 2. Assume now that Γ : [0, 1] → X is HKP-integrable in cwk(X). We are going to prove the τ (X ∗ , X)-weak continuity of TΓ . According to Corollary 1, if g ∈ BV [0, 1], then gΓ is HKP-integrable. Fix g ∈ BV [0, 1]. Since each BV function can be written as a difference of two bounded non-decreasing functions, we have the following representation g = g1 − g2 = [g1 − g1 (0)] − [g2 − g2 (0)] + [g1 (0) − g2 (0)], where g1 and g2 are bounded non-decreasing. Notice that g1 − g1 (0) ≥ 0 and g2 − g2 (0) ≥ 0. By Corollary 1 the multifunctions [gi (·) − gi (0)]Γ (·), i = 1, 2, are HKP-integrable. Now we have the following equality, for each x∗ ∈ X ∗ :
TΓ (x∗ ), g = TΓ (x∗ ), g1 − g1 (0) − TΓ (x∗ ), g2 − g2 (0) + TΓ (x∗ ), g1 (0) − g2 (0) 1 s (x∗ , [g1 (t) − g1 (0)]Γ (t)) dt = (HK) 0
1
s (x∗ , [g2 (t) − g2 (0)]Γ (t)) dt + TΓ (x∗ ), g1 (0) − g2 (0) 1 ∗ ∗ ∗ Γ (t) dt , = s(x , V1 ) − s(x , V2 ) + [g1 (0) − g2 (0)] s x , (HKP ) − (HK)
0
0
where V1 , V2 ∈ cwk(X) are the HKP-integrals of [g1 − g1 (0)]Γ and [g2 − g2 (0)]Γ , and the equality 1 ∗ ∗
TΓ (x ), g1 (0) − g2 (0) = [g1 (0) − g2 (0)]s x , (HKP ) Γ (t) dt 0
is just the HKP-integrability of Γ . Hence TΓ (·), g is τ (X ∗ , X)-continuous on X ∗ what proves the τ (X ∗ , X) weak continuity of TΓ . Remark 2. Proposition 6 holds true also for ck(X)-valued multifunctions if τ (X ∗ , X)-weak continuity is replaced by σ(X ∗ , X)-weak continuity on B(X ∗ ). Remark 3. If in the definition of the Henstock-Kurzweil-Pettis integral we replace the Henstock-Kurzweil integrability of the scalar functions by the DenjoyKhintchine integrability, we obtain the Denjoy-Khintchine-Pettis integral, introduced in [15] and studied in [15, 14] under the name Denjoy-Pettis integral. Denote
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by DK[0, 1] the set of all real-valued Denjoy-Khintchine integrable functions on [0, 1] (we identify a.e. equal functions) and endow it with the Alexiewicz norm # # α # # gA = sup ##(DK) g(t) dt## , 0<α≤1 0
where (DK) stands for Denjoy-Khintchine. Then, the space HK[0, 1] is dense in DK[0, 1], the completion of DK[0, 1] in the Alexiewicz norm coincides with the 6 1] of HK[0, 1] and the conjugate space DK ∗ [0, 1] is linearly completion HK[0, isometric to BV [0, 1] (cf. [4]). In particular, the duality is given by the same formula as in case of HK[0, 1]. Moreover the classical theorem on integration by parts holds true also for the Denjoy-Khintchine integral (cf. [16], Theorem 15.14). Using the above facts instead of the corresponding ones for the Henstock-Kurzweil integral, it is easy to see that the Denjoy-Khintchine-Pettis versions of Lemmata 1–2, Propositions 2–4 and 6, Theorems 1–2 and Corollary 1 also hold true. Question 2. Are there analogs of the decomposition theorems for Henstock integrals of multifunctions in case of an arbitrary Banach space? We proved such a result in [11], for compact-valued multifunctions with values in a separable Banach space. Acknowledgment The authors are grateful to the referee for useful comments and suggestions concerning the initial version of the paper.
References [1] A. Alexiewicz, Linear functionals on Denjoy integrable functions, Coll. Math. 1 (1948), 289–293. [2] B. Bongiorno, Relatively weakly compact sets in the Denjoy space, J. Math. Study, 27 (1994), 37–43. [3] B. Bongiorno, L. Di Piazza and K. Musial, Approximation of Banach space-valued non-absolutely integrable functions by step functions, Glasgow Math. J. 50(2008), 583–593. [4] D. Bongiorno and L. Di Piazza, Convergence of Foran integrals, Math. Japonica, 49 (2) 1999, 251–263. [5] B. Cascales, V. Kadets, and J. Rodriguez, The Pettis integral for multi-valued functions via single-valued ones, J. Math. Anal. Appl. 332 (1), (2007), 1–10. [6] B. Cascales, V. Kadets, and J. Rodriguez, Measurable selectors and set-valued Pettis integral in non-separable Banach spaces, J. Funct. Anal. 256 (3), (2009), 673–699. [7] B. Cascales, V. Kadets, and J. Rodriguez, Measurability and selections of multifunctions in Banach spaces, to appear in J. Convex Anal. [8] S.D. Chatterji, Sur l’int´egrabilit´e de Pettis, Math. Z. 136(1974), 53–58. [9] L. Di Piazza and K. Musial, Set-valued Kurzweil-Henstock-Pettis integral, Set-valued Analysis 13(2005), 167–179. [10] L. Di Piazza and K. Musial, Characterizations of Kurzweil-Henstock-Pettis integrable functions, Studia Math. 176(2006), 159–176.
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[11] L. Di Piazza and K. Musial, A decomposition theorem for compact-valued Henstock integral, Monatsh. Math. 148 (2), (2006), 119–126. [12] K. El Amri and C. Hess, On the Pettis integral of closed-valued multifunctions, Set-valued Anal. 8 (2000), 329–360. [13] K. Floret, Weakly compact sets, Lecture Notes in Math. 801 (1980), Springer-Verlag. [14] J.L. Gamez and J. Mendoza, On Denjoy-Dunford and Denjoy-Pettis integrals, Studia Math. 130 (1998), 115–133. [15] R.A. Gordon, The Denjoy extension of the Bochner, Pettis and Dunford integrals, Studia Math. 92 (1989), 73–91. [16] R.A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Math. vol. 4 (1994), AMS. [17] R. Henstock, Theory of integration, Butterworths, London (1963). [18] F. Hiai, Radon-Nikodym theorems for set-valued measures, J. Multivariate Anal. 8 (1978), 96–118. [19] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J., 7 (1957), 418–446. [20] H. Lebesgue, Int´egral, longueur, maire, Annali Mat. Pura Applic., 7 (1902), 231–359. [21] P.Y. Lee, Lanzhou Lectures on Henstock Integration, World Scientific, Singapore, 1989. [22] K. Musial, Topics in the theory of Pettis integration, Rend. Istit. Mat. Univ. Trieste 23 (1991), 177–262. [23] K. Musial, Pettis Integral, Handbook of Measure Theory I, 532–586. Elsevier Science B. V. 2002. [24] S. Saks, Theory of the integral, 2nd revised ed., Hafner, New York 1937. [25] M. Valadier, Multi-applications mesurables ` a valeurs convexes compactes, J. Math. Pures Appl. 50(1971), 265-297. [26] H. Ziat, Convergence theorems for Pettis integrable multifunctions, Bull. Polish Acad. of Sciences, Mathematics 45 (1997), 123–137. [27] H. Ziat, On a characterization of Pettis integrable multifunctions, Bull. Polish Acad. of Sciences, Mathematics 48 (2000), 227–230. Luisa Di Piazza Department of Mathematics University of Palermo via Archirafi 34 I-90123 Palermo, Italy e-mail: [email protected] Kazimierz Musial Wroclaw University Department of Mathematics and Informatics Institute of Mathematics Pl. Grunwaldzki 2/4 PL-50-384 Wroclaw, Poland e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 183–198 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Non-commutative Yosida-Hewitt Theorems and Singular Functionals in Symmetric Spaces of τ -measurable Operators Peter G. Dodds and Ben de Pagter Abstract. We present a new approach to non-commutative Yosida-Hewitt decompositions in the setting of normed M-bimodules of τ -measurable operators affiliated with a semifinite von Neumann algebra M. Our principal theorem permits the systematic study of the linear spaces of normal and singular linear functionals on symmetrically normed M-bimodules. We present some applications and give a decomposition into normal and singular parts for weakly compact operators on such spaces. Mathematics Subject Classification (2000). Primary 46L52; Secondary 46E30, 47A30. Keywords. Measurable operators, normal functionals, singular functionals.
1. Introduction and preliminaries The classical theorem of Yosida and Hewitt [19] asserts that each bounded additive measure can be uniquely represented as the sum of a countably additive measure and a purely finitely additive measure, the so-called singular part. The singular part is characterised by the fact that its absolute value does not dominate any nonzero positive countably additive measure. This classical theorem, which goes back to the Lebesgue decomposition theorem, admits many generalisations in different settings. Within the framework of vector lattices, it emerges as a very special case of the well-known theorem of F. Riesz that a band (= every order closed ideal) in an order-complete vector lattice is a projection band. On the other hand, in the study of the Banach dual space of a general (noncommutative) von Neumann algebra, it was first shown by Takesaki [14],[15] that This work was partially supported by the Australian Research Council.
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each continuous linear functional φ on a von Neumann algebra M could be represented uniquely as the sum of an element φn in the von Neumann predual M∗ (which may be identified as the linear space of all completely additive linear functionals on M; see [16]) and an element φs which is singular in the sense that it is a linear combination of positive linear functionals, each of which dominates no non-zero positive normal linear functional on M. Moreover, this decomposition of the dual of a von Neumann algebra into the direct sum of its predual and the space of singular functionals is implemented by the action of a central projection in the von Neumann algebra bidual M∗∗ . In the present paper, we wish to describe recent work which shows the existence of the Yosida-Hewitt decomposition in the dual space of a normed Mbimodule E of τ -measurable operators affiliated with a semi-finite von Neumann algebra M. Our approach goes well beyond that of [9], and removes the assumption made there that the space E is an interpolation space for the pair M and its predual. Further, the present approach does not rely on the weak compactness arguments on which the results of [9] are based and consequently brings much greater conceptual clarity. In addition, our methods permit a very natural avenue to showing that the linear space of normal (= order-continuous) linear functionals En∗ on a symmetric space E of τ -measurable operators may be identified with the K¨ othe dual E × and that the notions of normality and complete additivity coincide. While this latter equivalence has been established in [7], the present approach is far more transparent. We give a number of further applications to the study of singular functionals. In particular, the space of singular functionals Es∗ is left and right invariant under the action of M. Moreover, the inverse annihilator of Es∗ in E consists precisely of the set E oc of elements of order continuous norm, and, provided E oc separates the points of the K¨ othe dual, then the annihilator of E oc in E ∗ is precisely the linear space of singular functionals Es∗ . Finally, we show that a Yosida-Hewitt decomposition continues to hold for weakly compact mappings from a symmetric space of τ -measurable operators to a Banach space.This complements earlier results from [2] for weakly compact mappings from a general von Neumann algebra to an arbitrary Banach space. Detailed proofs of the principal results are are quite lengthy, and will appear elsewhere.
2. Preliminaries and notation Throughout this paper M will denote a von Neumann algebra on some Hilbert space H. Unless otherwise stated, it will be assumed throughout that M is equipped with a fixed semifinite faithful normal trace τ . For standard facts concerning von Neumann algebras, we refer to [3], [16]. The identity in M is denoted by 1 and we denote by P (M) the complete lattice of all (self-adjoint) projections
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in M. A linear operator x : D(x) → H, with domain D(x) ⊆ H, is said to be affiliated with M if ux = xu for all unitary u in the commutant M of M. For any self-adjoint operator x on H, its spectral measure is denoted by ex . A self-adjoint operator x is affiliated with M if and only if ex (B) ∈ P (M) for any Borel set B ⊆ R. The closed and densely defined operator x, affiliated with M, is called τ measurable if and only if there exists a number s ≥ 0 such that τ e|x| (s, ∞) < ∞. The collection of all τ -measurable operators is denoted by S(τ ). With the sum and product defined as the respective closures of the algebraic sum and product, it is well known that S(τ ) is a *-algebra. For , δ > 0, we denote by V (, δ) the set of all x ∈ S(τ ) for which there exists an orthogonal projection p ∈ P (M) such that p(H) ⊆ D(x), xpB(H) ≤ and τ (1 − p) ≤ δ. The sets {V (, δ) : , δ > 0} form a base at 0 for a metrizable Hausdorff topology on S(τ ), which is called the measure topology. Equipped with this topology, S(τ ) is a complete topological ∗-algebra. These facts and their proofs can be found in the papers[12], [17] For x ∈ S(τ ), the singular value function μ(·; x) = μ(·; |x|) is defined by μ (t; x) = inf s ≥ 0 : τ e|x| (s, ∞) ≤ t , t ≥ 0. It follows directly that the singular value function μ(x) is a decreasing, rightcontinuous function on the positive half-line [0,∞). Moreover, μ(uxv) ≤ uvμ(x) for all u, v ∈ M and x ∈ S(τ ) and and μ(f (x)) = f (μ(x)) whenever 0 ≤ x ∈ S(τ ) and f is an increasing continuous function on [0, ∞) which satisfies f (0) = 0. It should be observed that a sequence {xn }∞ n=1 in S (τ ) converges to zero for the measure topology if and only if μ (t; xn ) → 0 as n → ∞ for all t > 0. The measure topology may be localised as follows. For ε, δ > 0 and e ∈ P (M) satisfying τ (e) < ∞, define N (ε, δ, e) = {x ∈ S (τ ) : exe ∈ V (ε, δ)} . The family of all sets N (ε, δ, e) forms a neighbourhood base at 0 for a Hausdorff linear topology in S (τ ), called the local measure topology. If {xn }∞ n=1 is a sequence in S (τ ), then xn → 0 for the local measure topology if and only if μ (t; exn e) → 0 as n → ∞ for all t > 0 and all e ∈ P (M) satisfying τ (e) < ∞. If m denotes Lebesgue on the semiaxis [0, ∞), and if we consider L∞ (m) as an Abelian von Neumann algebra acting via multiplication on the Hilbert space H = L2 (m), with the trace given by integration with respect to m, then S(m) consists of all measurable functions on [0, ∞) which are bounded except on a set of finite measure, and that for f ∈ S(m), the generalized singular value function μ(f ) is precisely the classical decreasing rearrangement of the function |f |, which is usually denoted by f ∗ . In this setting, convergence for the measure topology coincides with the usual notion of convergence in measure, while convergence for the local measure topology reduces to the familiar notion of convergence in measure
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on every set of finite measure. If M = L(H) and τ is the standard trace, then S(τ ) = M, the measure topology coincides with the operator norm topology, and the topology of local convergence in measure coincides with the weak operator topology. If x ∈ S(τ ), then x is compact if and only if limt→∞ μ(t; x) = 0; in this case, μn (x) = μ (t; x) , t ∈ [n, n + 1), n = 0, 1, 2, . . . , and the sequence {μn (x)}∞ is just the sequence of eigenvalues of |x| in nonn=0 increasing order and counted according to multiplicity. The real vector space Sh (τ ) = {x ∈ S (τ ) : x = x∗ } is a partially ordered vector space with the ordering defined by setting x ≥ 0 if and only if xξ, ξ ≥ 0 for + all ξ ∈ D (x). The positive cone in Sh (τ ) will be denoted by S (τ ) . If 0 ≤ xα ↑α ≤ x + + holds in S (τ ) , then supα xα exists in S (τ ) . The trace τ extends to S (τ )+ as a non-negative extended real-valued functional which is positively homogeneous, additive, unitarily invariant and normal. This extension is given by ∞ + τ (x) = μ (t; x) dt, x ∈ S (τ ) , 0
and satisfies τ (x∗ x) = τ (xx∗ ) for all x ∈ S (τ ). It should be observed that if f is an increasing continuous function on [0, ∞) satisfying f (0) = 0, then ∞ ∞ τ (f (|x|)) = μ (t; f (|x|)) dt = f (μ (t; x)) dt (2.1) 0
0
for all x ∈ S (τ ). p If 1 ≤ p < ∞, we set Lp (τ ) = {x ∈ S (τ ) : τ (|x| ) < ∞}. Note that it follows from (2.1) that Lp (τ ) is also given by Lp (τ ) = {x ∈ S (τ ) : μ (x) ∈ Lp (m)} , where m denotes Lebesgue measure on [0, ∞). The space Lp (τ ) is a linear subspace p 1/p of S (τ ) and the functional x −→ xLp (τ ) = τ (|x| ) , x ∈ Lp (τ ), is a norm. It should be observed that xLp (τ ) = μ (x)Lp (m) for all x ∈ Lp (τ ). Equipped with this norm, Lp (τ ) is a Banach space. In this setting, we also have that L∞ (τ ) = M. In the commutative setting, the spaces Lp (τ ) are the familiar Lebesgue spaces. In the special case that M is B(H) equipped with standard trace, the corresponding Lp -spaces are the Schatten classes Sp . As is well known, the space L1 (τ ) may be identified with the von Neumann algebra predual of M with respect to trace duality. If x ∈ S(τ ), then the projection onto the closure of the range of |x| is called the support of x and is denoted by s(x). We set F (τ ) = {x ∈ M : τ (s(x)) < ∞}. If (N , σ) is a semifinite von Neumann algebra, if x ∈ S(τ ) and y ∈ S(σ) then x is said to be submajorised by y (in the sense of Hardy, Littlewood and Polya) if and only if t
t
μ(s; x)ds ≤ 0
μ(s; y)ds 0
for all t ≥ 0. We write x ≺≺ y, or equivalently, μ(x) ≺≺ μ(y). For further details and proofs, we refer the reader to [10], [4], [7].
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3. Normed spaces of τ -measurable operators In this section, we present a brief overview of some of the basic theory of normed spaces of τ -measurable operators and their K¨ othe duals. 3.1. Normed M-bimodules A linear subspace E ⊆ S(τ ) is called an M-bimodule if uxv ∈ E whenever x ∈ E and u, v ∈ M. If the M-bimodule E is equipped with a norm · E which satisfies uxvE ≤ uB(H)uB(H) xE ,
x ∈ E, u, v ∈ M,
then E is called a normed M-bimodule (of τ -measurable operators). If E ⊆ S(τ ) is an M-bimodule, then 1. If x ∈ S(τ ), then x ∈ E if and only if |x| ∈ E. 2. If x ∈ S(τ ), then x ∈ E if and only if x∗ ∈ E. 3. If x ∈ S(τ ) and y ∈ E are such that |x| ≤ |y| then x ∈ E. Further, if E is a normed M-bimodule, then |x| E = xE and x∗ E = xE for all x ∈ E and xE ≤ yE whenever x, y ∈ E satisfy |x| ≤ |y|. A normed M-bimodule which is a Banach space is called a Banach M-bimodule. It is easily seen that F (τ ) is an M-bimodule and that each of the Banach spaces M, L1 (τ ), L1 (τ ) ∩ M, L1 (τ ) + M are Banach M-bimodules. If E ⊆ S(τ ) is a normed M-bimodule, then the embedding of E into S(τ ) is continuous from the norm topology on E to the topology of local convergence in measure on S(τ ), that is, whenever {xn }∞ n=1 ⊆ S(τ ) satisfies xn E →n 0, it follows that μ(t; exn e) →n 0 for each t > 0 and all e ∈ P (M) satisfying τ (e) < ∞. 3.2. Symmetrically normed M-bimodules and their K¨ othe duals It will be convenient to adopt the following terminology. If E ⊆ S (τ ) is a normed M-bimodule, then E will be called: (i) symmetrically normed if x ∈ E, y ∈ S (τ ) and μ (y) ≤ μ (x) imply that y ∈ E and yE ≤ xE ; (ii) strongly symmetrically normed if E is symmetrically normed and its norm has the additional property that yE ≤ xE whenever x, y ∈ E satisfy y ≺≺ x. In the present paper we shall only consider strongly symmetrically normed spaces. It should be pointed out that all results are also valid for symmetrically normed spaces if one assumes in addition that the von Neumann algebra M is either non-atomic (that is, does not contain any minimal projections) or atomic and all minimal projections have equal trace. Furthermore, most of the examples of symmetrically normed spaces are actually strongly symmetrically normed. It is not easy to exhibit examples of symmetrically normed spaces which are not strongly symmetrically normed. If a (strongly) symmetrically normed space is Banach, then it will be simply called a (strongly) symmetric space.
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P.G. Dodds and B. de Pagter It may be shown that any strongly symmetrically normed space satisfies F (τ ) ⊆ E ⊆ L1 (M) + L∞ (M) ,
with continuous inclusions (where F (τ ) is equipped with the L1 ∩ L∞ -norm). If, in addition, E is a Banach space, then L1 (M) ∩ L∞ (M) ⊆ E, with continuous embedding. Furthermore, we denote the closure of F (τ ) in E by E b , that is, E
E b = F (τ ) . If E is Banach, then it is easy to see that E
E b = L1 (τ ) ∩ L∞ (τ ) . Remark 3.1. We would like to warn the reader that the terminology introduced above differs from that which has been used elsewhere and is an attempt to unify some of the terminology in the literature. We point out explicitly that the terms “symmetrically normed” and “strongly symmetrically normed” as defined above are used in the present paper instead of the earlier terminology of “rearrangement invariant” and “symmetrically normed”, respectively, used in the papers [6], [7], [9]. If M is L∞ (m), with m Lebesgue measure on the semiaxis [0, ∞), then a symmetrically normed M-bimodule E ⊆ S(m) will be called, for simplicity, a symmetrically normed space on [0, ∞). If E ⊆ S(τ ) is a strongly symmetrically normed M-bimodule, then the embedding of E into S(τ ) is continuous from the norm topology of E to the measure topology on S(τ ). A wide class of strongly symmetrically normed M-bimodules may be constructed as follows. If E ⊆ S(m) is a strongly symmetrically normed space on [0, ∞), set E (τ ) = {x ∈ S (τ ) : μ (x) ∈ E} ,
xE(τ ) := μ(x)E
It may be shown as in [4] (see [5]) that (E(τ ), · E(τ ) is a strongly symmetrically normed M-bimodule and is a Banach M-bimodule if E is a Banach space. Now suppose that E ⊆ S(τ ) is a strongly symmetrically normed M-bimodule and let If E ⊆ S(m) a strongly symmetrically normed space on [0, ∞), set E × = {y ∈ S (τ ) : sup {τ (|xy|) : x ∈ E, xE ≤ 1} < ∞} and yE × = sup {τ (|xy|) : x ∈ E, xE ≤ 1} . If y ∈ S(τ ), then y ∈ E × if and only if μ(x)μ(y)dm : x ∈ E, xE ≤ 1} < ∞, sup{ [0,∞)
in which case, the latter quantity is equal to yE × . The space (E × , · E × ) is a normed Banach M-bimodule. If y ∈ E × , set φy : E → C, φy (x) = τ (xy), x ∈ E
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The Banach M-bimodule E × has the following properties. φy ∈ E ∗ and the map y → φy ∈ E ∗ , y ∈ E × is an isometry. 0 ≤ yα ↑α ⊆ E × , supα yα E × < ∞ =⇒ y = supα yα exists in E × and yE × = supα yα E × . The closed unit ball in E × is closed in S(τ ) for the local measure topology. x ∈ E × , y ∈ S(τ ), y ≺≺ x =⇒ y ∈ E × and yE ≤ xE . If E is a strongly symmetrically normed space on [0, ∞),then E × (τ ) = E(τ )× .
3.3. Normal and singular functionals on a normed M-bimodule If E ⊆ S(τ ) is a normed M-bimodule, then the positive cone E + = {x ∈ E : x ≥ 0} is closed. If φ is a linear functional on E, then φ is called positive if φ(x) ≥ 0 for all 0 ≤ x ∈ E. If φ is a linear functional on E, then the adjoint functional φ∗ is defined by setting φ∗ (x) = φ(x∗ ) for all x ∈ E. The linear functional φ is said to be self-adjoint if φ = φ∗ . If Re φ = (φ + φ∗ )/2,
Im φ = (φ − φ∗ )/2i,
then it is readily seen that Re φ and Im φ are self-adjoint and φ = Re φ + iIm φ. If E is a Banach M-bimodule, then each positive linear functional on E is necessarily continuous. If E is a normed M-bimodule, then it may be shown that the dual cone (E ∗ )+ generates E ∗ and that, if 0 ≤ φ ∈ E ∗ , then φE ∗ = sup{φ(x) : 0 ≤ x ∈ E, xE ≤ 1} Let E ⊆ S(τ ) be an M-bimodule. If 0 ≤ φ ∈ (E ∗ )+ , then the absolute kernel N (φ) of φ is defined by setting N (φ) = {x ∈ E : φ(|x|) = φ(|x∗ |) = 0}. A linear subspace J of an M-bimodule E ⊆ S(τ ) is called an order ideal if J is ∗-closed, if x ∈ J implies that |x| ∈ J and if x ∈ E and y ∈ J satisfy 0 ≤ x ≤ y then x ∈ J. Any order ideal J ⊆ E is generated by J + as a linear subspace. Moreover, any intersection of order ideals is again an order ideal. The absolute kernel of any positive linear functional on E is an order ideal. An order ideal J in the M-bimodule E ⊆ S(τ ) is said to be order dense if and only if, for every 0 < x ∈ E + , there exists y ∈ J + such that 0 < y ≤ x. By a standard argument, an order ideal J is order dense in E if and only if for every 0 < x ∈ E + , there exists an upwards directed system {xα } ⊆ J + such that 0 ≤ xα ↑α x. Equivalently, J is order dense if and only if for each projection e ∈ E, there exists an upwards directed system {eα } ⊆ J ∩P (M) of projections such that 0 ≤ eα ↑α e. To establish the setting for the discussion which follows, it will be convenient to first recall some basic facts from the duality theory of general von Neumann algebras. A continuous linear functional φ on the von Neumann algebra M is said to be normal if and only xα ↓α 0 in M implies φ(xα ) →n 0. The functional φ is normal if and only if φ is ultraweakly continuous, and the collection M∗n all normal functionals on M may be identified with the predual M∗ of M. There
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˜ exists a central projection z0 in the universal enveloping von Neumann algebra M ∗ ∗ ∗ such that Mn = M z0 . The functionals in M (1 − z0 ) are called singular and M∗ (1 − z0 ) is denoted by M∗s . Consequently, each continuous linear functional φ on M has a unique decomposition φ = φn + φs ,
φn ∈ M∗n , φs ∈ M∗s .
(3.1)
Each singular linear functional is a linear combination of positive singular functionals. Further (see [15]), a positive linear functional φ is singular if and only if for every non-zero projection 0 = p ∈ P (M), there exists a non-zero projection q ∈ M such that 0 ≤ q ≤ p and φ(q) = 0. In turn, each of these statements is equivalent to the assertion that φ dominates no non-zero positive normal linear functional on M. Using the terminology introduced above, the singularity of a functional on M may be characterised as follows. 1. A positive functional 0 ≤ φ ∈ M∗ is singular if and only if the absolute kernel N (φ) is order dense. 2. A functional φ ∈ M ∗ is singular if and only if φ vanishes on some order dense ideal in M. This discussion provides the motivation for the following definitions. If E ⊆ S(τ ) is a normed M-bimodule and φ ∈ E ∗ , then 1. φ is called normal if xα ↓α 0 implies φ(xα ) →n 0; 2. φ is called completely additive if φ x = ei φ(xei ) and φ ei x = φ(ei x) i
i
i
i
for every collection {ei } ⊆ P (M ) of mutually orthogonal projections and for every x ∈ E; 3. φ is called singular if φ vanishes on some order dense ideal in E. The collections of all normal, completely additive and singular functionals on ∗ the normed M-bimodule E ⊆ S(τ ) will be denoted by En∗ , Eca and Es∗ respectively. ∗ ∗ ∗ It is evident that En , Eca ⊆ E are linear subspaces; and since the intersection of finitely many order dense ideals is again an order dense ideal, it follows readily that Es∗ ⊆ E is a linear subspace. Further, it is not difficult to see that En∗ ∩ Es∗ = {0}.
4. The Yosida-Hewitt decomposition in M-bimodules Throughout this section, E ⊆ S(τ ) will denote a strongly symmetrically normed M-bimodule. Theorem 4.1. If ψ ∈ E ∗ , then there exists a unique decomposition ψ = ψn + ψs Es∗
En∗
where ψs ∈ and ψn ∈ is given by ψn (x) = τ (xy), x ∈ E, for some unique element y ∈ E × . Moreover if ψ is self-adjoint, then y = y ∗ and if ψ ≥ 0, then y ≥ 0 and ψs ≥ 0.
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One particular consequence of the preceding theorem is an alternative characterisation of singularity for positive functionals. Corollary 4.2. If 0 ≤ ψ ∈ E ∗ , then ψ is singular if and only if ψ does not dominate any non-zero positive normal functional. The theorem which follows is an important consequence of the preceding Theorem 4.1 Theorem 4.3. If ψ ∈ E ∗ , then the following statements are equivalent. (i) ψ is normal. (ii) ψ is completely additive. (iii) There exists a unique element y ∈ E × such that ψ(x) = τ (xy),
x ∈ E.
If any of the preceding equivalent assertions are valid, then ψ is self-adjoint if and only if y = y ∗ and ψ is positive if and only if y ≥ 0. It follows from Theorem 4.1 that E ∗ = En∗ ⊕ Es∗ and it follows from Theorem 4.3 that the space En∗ may be identified with the K¨ othe dual E × of E via ∗ trace duality. For ψ ∈ E , the decomposition ψ = ψn + ψ,
ψn ∈ En∗ ,
ψs ∈ Es∗
will be referred to as the Yosida-Hewitt decomposition. Let Pn : E ∗ → En∗ be the linear projection of E ∗ on En∗ along Es∗ given by setting Pn ψ = ψn , ψ ∈ E ∗ and set Ps = I − Pn . If follows that both Pn , Ps are positive projections and that 0 ≤ Pn ψ ≤ ψ and 0 ≤ Ps ψ ≤ ψ for each 0 ≤ ψ ∈ E ∗ . Since E ∗ is positively generated, it follows that Pn , Ps are bounded projections and this implies that the subspaces En∗ , Es∗ are norm-closed subspaces of E ∗ . Indeed, the subspaces En∗ , Es∗ are even sequentially σ(E ∗ , E)-closed, as the following proposition shows. ∗ ∗ Proposition 4.4. If {ψk }∞ k=1 is a sequence in E and if ψ ∈ E is such that ψk →k ψ ∗ for the σ(E , E)-topology then Pn ψk → Pn ψ (and hence also Ps ψk →k Ps ψ for the σ(E ∗ , E)-topology).
Proposition 4.4 preceding together with Theorem 4.1 yields the following interesting consequence. Theorem 4.5. The following statements are equivalent. (i) E is a KB-space, that is, E ∗ = En∗ and E = E ×× . (ii) E is weakly sequentially complete. The following special case is, of course, well known. Corollary 4.6. The space L1 (τ ) is weakly sequentially complete. If ψ ∈ E ∗ and if w ∈ M, then the linear functionals wψ, ψw are defined by setting (wψ)(x) = ψ(xw) and (ψw)(x) = ψ(wx), x ∈ E.
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Proposition 4.7. If ψ ∈ Es∗ , then wψ, ψw ∈ Es∗ for all w ∈ M. This section ends with a further consequence of Theorem 4.3. It has been observed earlier the norm closed unit ball in E × is closed in S(τ ) for the local measure topology. Indeed, it is the case that the norm closed unit ball in E × is complete in S(τ ) for the local measure topology. This is an immediate consequence of the following. Proposition 4.8. The following statements are equivalent. (i) The closed unit ball of E is closed in S(τ ) for the local measure topology. (ii) The closed unit ball of E is complete in S(τ ) for the local measure topology.
5. Elements of order-continuous norm and singular functionals If E ⊆ S(τ ) is a normed M-bimodule, then the norm · E is called order continuous if xα E ↓α 0 whenever 0 ≤ xα ↓α 0 ⊆ E. Standard arguments yield the following alternative characterisation of order continuity of the norm. Proposition 5.1. If E ⊆ S(τ ) is a normed M-bimodule, then the following statements are equivalent (i) The norm on E is order-continuous. (ii) E ∗ = En∗ . (iii) xn E ↓n 0 whenever {xn }∞ n=1 ⊆ E is a decreasing sequence for which 0 ≤ xn ↓n 0 ⊆ E. Although the norm on E may not be order-continuous, it may well be the case that E contains non-trivial subspaces to which the restriction of the norm is order-continuous. If E ⊆ S(τ ) is a normed M-bimodule, the subset E oc ⊆ E is defined by setting E oc = {x ∈ E : |x| ≥ xα ↓α 0 =⇒ xα E ↓α 0} Equivalently, in terms of sequences, E oc = {x ∈ E : |x| ≥ xn ↓n 0 =⇒ xn E ↓n 0} The set E oc may be characterised as the inverse annihilator of the singular part Es∗ of the Banach dual E ∗ . Recall that, if ∅ = A ⊆ E and if ∅ = B ⊆ E ∗ , then the annihilator A⊥ of A and the inverse annihilator ⊥ B are defined by setting A⊥ = {ψ ∈ E ∗ : ψ(x) = 0 ∀x ∈ A}, ⊥
∗
⊥
⊥
B = {x ∈ E : ψ(x) = 0 ∀ψ ∈ B}.
It is clear that A ⊆ E , B ⊆ E are closed subspaces. In the remaining part of the section it will be assumed that E ⊆ S (τ ) is a strongly symmetrically normed M-bimodule.
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Proposition 5.2. The set E oc of elements of order continuous norm is given by the equality E oc = ⊥ Es∗ . oc b oc Moreover, E ⊆ E and E is an M-bimodule which is norm-closed in E. It may well be the case that E oc = {0}, which is the case if M = L∞ (ν) for some non-atomic measure ν. Proposition 5.3. The following statements are equivalent. (i) (E oc )⊥ = Es∗ . (ii) Es∗ is σ(E ∗ , E)-closed. (iii) E oc separates the points of En∗ . (iv) The map φ → φ |E oc , φ ∈ En∗ is a linear isometry from En∗ onto (E oc )∗ Recall that if ψ ∈ E ∗ , then ψ is singular if and only if ψ vanishes on some order dense ideal in E. It is worth noting therefore, that if E oc separates the points of En∗ , then the preceding proposition implies that the linear functional ψ ∈ E ∗ is singular if and only if ψ vanishes on the fixed order dense ideal E oc . In the case that the von Neumann algebra M is non-atomic, or if M is atomic and all minimal projections have equal trace, then some improvements may be made in the preceding proposition. Proposition 5.4. If the von Neumann algebra M is non-atomic, then the following statements are equivalent. (i) E oc = {0}. (ii) E oc separates the points of En∗ . (iii) E oc = E b . Proposition 5.5. If the von Neumann algebra M is atomic, and all minimal projections are of equal trace, then E oc = E b and E oc is the closed linear span of the minimal projections. Corollary 5.6. Suppose that M is atomic and that all minimal projections have equal trace. The linear functional ψ ∈ E ∗ is singular if and only if ψ vanishes on all minimal projections. If, for each y ∈ E × , the linear functional φy on E b is defined by setting φy (x) = τ (xy), x ∈ E b , then the map y → φy is a linear isometry from E × onto (E b )∗ .
6. A vector-valued Yosida-Hewitt theorem The purpose of this section is to add some remarks concerning an operator version of the Yosida-Hewitt theorem. We assume throughout that E ⊆ S(τ ) is a strongly symmetrically normed Banach M-bimodule. A continuous linear mapping T : E → X, where X is a Banach space is said to be weakly normal (respectively, weakly singular) if and only if, for each x∗ ∈ X ∗ , T ∗ x∗ is normal (respectively, singular).
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Proposition 6.1. Let X be a Banach space. If T : E → X is weakly compact, then there exists a unique decomposition T = Tn + Ts where Tn , Ts : E → X are weakly compact and Tn (respectively, Ts ) is weakly normal (respectively, weakly singular). Proof. Let π : E → E ∗∗ be the natural embedding and let Pn : E ∗ → E ∗ be the Yosida-Hewitt projection. Since T : E → X is weakly compact, note that T ∗∗ (E ∗∗ ) ⊆ X in the sense that T ∗∗ maps the Banach bidual E ∗∗ into the canonical image of X in the bidual X ∗∗ . Define Tn , Ts : E → X respectively by setting Tn = T ∗∗ Pn∗ π,
Ts = T ∗∗ (I − Pn∗ )π
and observe that Tn , Ts : E → X are weakly compact and clearly satisfy T = Tn + Ts . ∗
Now observe that, for all z ∈ E, x ∈ X ∗ ,
Tn z, x∗ = T ∗∗ Pn∗ π(z), x∗ = π(z), Pn T ∗ x∗ = z, Pn T ∗ x∗ and
Ts z, x∗ = z, (I − Pn )T ∗ x∗ .
In particular,
Tn∗ = Pn T ∗ , Ts∗ = (I − Pn )T ∗ , and this clearly implies that Tn is weakly normal and Ts is weakly singular. The uniqueness of the decomposition follows immediately from the uniqueness of the Yosida-Hewitt decomposition in the scalar case. In order to strengthen the conclusion of the preceding Proposition 6.1, some additional considerations are needed. The following result is a refinement of the equivalence (i)⇐⇒(ii) of [6], Proposition 2.8. Proposition 6.2. If K ⊆ E × is bounded, then the following statements are equivalent. (i) K is relatively σ(E × , E)-compact. (ii) For every system {xα } ⊆ E with xα ↓α 0, sup{|τ (yxα )| : y ∈ K} →α 0. The continuous linear mapping T : E → X will be called normal if and only if whenever xα ↓α 0 ⊆ E, it follows that T xα X →α 0. Proposition 6.3. Let X be a Banach space. If T : E → X is weakly compact and weakly normal, then T is normal.
Proof. Since T is weakly compact and weakly normal, it follows that {T x∗ : x∗ ∈ X ∗ , x∗ X ∗ ≤ 1} ⊆ E × is relatively σ(E ∗ , E ∗∗ )-compact, and hence relatively σ(E × , E)-compact, since E × is σ(E ∗ , E ∗∗ )-closed, and the σ(E × , E)-topology is weaker that the σ(E × , E ∗∗ )-topology. By the implication (i)=⇒(ii) of Proposition 6.2, it follows that, whenever xα ↓α 0 ⊆ E, T xα E = sup{| xα , T x∗ | : x∗ ∈ X ∗ , x∗ X ∗ ≤ 1} →α 0, and so T is normal.
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In the case that E = M (and even in the case that M is a general von Neumann algebra) the preceding Proposition (as well as the decomposition given by Proposition 6.1) may be found in [2], Theorem 4.1. The continuous linear mapping T : E → X will be called singular if and only if T vanishes on some order dense ideal in E. It seems to be an interesting question as to when a weakly compact, weakly singular mapping T : E → X is singular. Proposition 6.4. Suppose that E oc =⊥ (Es∗ ). If T : E → X is weakly singular, then T is singular. Proof. Since T is weakly singular, it follows from Proposition 5.5 that T ∗ φ vanishes on the fixed order dense ideal E oc , for every φ ∈ E ∗ . This implies immediately that T x = 0 for all x ∈ E oc and so T is singular since E oc ⊆ E is an order dense ideal. The following refinement of Proposition 6.1 now follows immediately from Proposition 6.4 and Proposition 6.3. Corollary 6.5. Suppose that E oc =⊥ (Es∗ ). If T : E → X is weakly compact, then there exists a unique decomposition T = Tn + Ts where Tn , Ts : E → X are weakly compact, Tn is normal and Ts is singular. We note that the preceding Proposition 6.4 and Corollary 6.5 apply in the special cases that M is either non-atomic and E oc = {0} or is atomic and all minimal projections have equal trace. This follows from Proposition 5.3 and Proposition 5.4 in the case that M is non-atomic and from Proposition 5.3 and Proposition 5.5 in the case that M is atomic and that all minimal projections have equal trace. Many of the preceding results continue to hold for continuous linear mappings on an arbitrary von Neumann algebra M. We first recall Akemann’s well-known characterisation of weak compactness in the predual M∗ of an arbitrary von Neumann algebra M, which is restated here for the convenience of the reader. Theorem 6.6. (Akemann; see [16]) Let M be an arbitrary von Neumann algebra with predual M∗ . If K ⊆ M∗ is bounded, then the following statements are equivalent. (i) K is relatively σ(M∗ , M)-compact. (ii) There exists 0 ≤ ϕ ∈ M∗ such that given > 0, there exists δ > 0 such that a ∈ M, ϕ(aa∗ + a∗ a) < δ implies sup{|ψ(a)| : ψ ∈ K} < . (iii) For every system {xα } ⊆ M with xα ↓α 0, sup{|φ(xα )| : φ ∈ K} →α 0. The following result has been observed in [2], Proposition 3.2. We present a proof based directly on Akemann’s criterion rather than the results of [18].
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Proposition 6.7. Let M be an arbitrary von Neumann algebra and let X be a Banach space. If T : M → X is weakly normal then T is weakly compact and normal. Proof. Since T : M → X is weakly normal, it follows that T : M → X is σ(M, M∗ ) to σ(X, X ∗ ) continuous. Since the unit ball of M is σ(M,M∗ )-compact, it follows immediately that T is weakly compact. That T is normal follows directly from the implication (i)=⇒(iii) of Akemann’s theorem. It is also shown in [2] that if T : M → X is weakly compact and weakly singular, then there exists a net 0 ≤ eα ↑α 1 such that T (eα xeα ) →α 0 for every x ∈ M. We show that such mappings are singular. Denote by M∗s the space of singular linear functionals on M∗ . Note that it is still the case that, if ψ ∈ M∗ , then ψ ∈ M∗s if and only if ψ vanishes on some order dense ideal in M. See, for example, [16] Theorem III.3.8. Further, the same arguments as earlier in the case of semi-finite algebras show that, if 0 ≤ ϕ is singular, then N (ϕ) = {x ∈ M : ϕ(|x|) = 0 = ϕ(|x∗ )} is an order dense ideal in M. The Yosida-Hewitt projection in M∗ will continue to be denoted by Pn . As noted earlier, in this case the projection Pn is implemented by a central projection in M∗∗ given by the formula Pn ψ = ψ(z·), ψ ∈ M ∗ . Proposition 6.8. Let M be an arbitrary von Neumann algebra. If X is a Banach space and if T : M → X is weakly compact and weakly singular, then T is singular. Proof. Denote by B(X ∗ ) the unit ball in the Banach dual X ∗ . Since T : M → X is weakly compact, it follows by Schauder’s theorem that T ∗ (B(X ∗ )) ⊆ M∗ is relatively σ(M∗ , M∗∗ )-compact. Applying Akemann’s theorem to the von Neumann algebra M∗∗ , there exists 0 ≤ ϕ ∈ M∗ such that given > 0, there exists δ > 0 such that a ∈ M∗∗ , ϕ(aa∗ + a∗ a) < δ implies sup{| a, T ∗ x∗ | : x∗ ∈ B(X ∗ )} < . In particular, if 0 ≤ a ∈ M∗∗ , ϕ(a) = 0 implies sup{| a, T ∗ x∗ | : x∗ ∈ B(X ∗ )} = 0. Noting that 0 ≤ (I − Pn )ϕ is singular, suppose that 0 ≤ x ∈ M satisfies x ∈ N ((I − Pn )ϕ) and set 0 ≤ y = (I − Pn )∗ π(x) ∈ M∗∗ , where π : M → M∗∗ denotes the natural embedding. Note that ϕ(y) = (I − Pn )∗ π(x), ϕ = π(x), (I − Pn )ϕ = 0. By the above remarks,
y, T ∗ x∗ = 0,
x∗ ∈ B(X ∗ ).
Consequently,
T x, x∗ = T ∗∗ (I − Pn )∗ π(x), x∗ = (I − Pn )∗ π(x), T ∗ x∗ = y, T ∗ x∗ = 0.
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It follows that T vanishes on the order dense ideal N ((I − Pn )ϕ) and so T is singular. The weak version of the Yosida-Hewitt decomposition given in Proposition 6.1 remains valid, mutatis mutandi if E is replaced by an arbitrary von Neumann algebra M. This observation, combined with the preceding two propositions now yields the following result. Proposition 6.9. Suppose that M is an arbitrary von Neumann algebra and that X is a Banach space. If T : M → X is a weakly compact linear mapping, then there exists a unique decomposition T = Tn + Ts , where Tn , Ts : M → X are weakly compact linear mappings with Tn normal and Ts singular. It has been shown by H. Pfitzner [13] that M is a Grothendieck space, that is, for sequences in the dual space M∗ , the notions of weak∗ and weak convergence coincide. As is well known, this implies that each continuous linear mapping from M to a separable Banach space is necessarily weakly compact. This yields the following refinement of Proposition 6.9 preceding. Corollary 6.10. Suppose that M is an arbitrary von Neumann algebra and that X is a separable Banach space. If T : M → X is a continuous linear mapping, then there exists a unique decomposition T = Tn + Ts , where Tn , Ts : M → X are weakly compact linear mappings with Tn normal and Ts singular. The preceding discussion now raises the interesting question as to when a continuous linear mapping from a symmetrically normed M-bimodule of τ -measurable operators to a Banach space X maps order intervals to relatively weakly compact sets, and whether such operators admit a Yosida-Hewitt decomposition into singular and normal parts. See, for example, [1] and the references contained therein.
References [1] A. Basile and A.V. Bukhvalov, On a unifying approach to decomposition theorems of Yosida-Hewitt type, Annali di Mat. pura ed applicata 173(1997), 107–125. [2] J.K. Brooks and J.D. Maitland-Wright, Representing Yosida-Hewitt decompositions for classical and non-commutative vector measures, Expositiones Math. 19(2001), 273–383. [3] J. Dixmier, von Neumann Algebras, Mathematical Library, vol. 27. Amsterdam: North Holland (1981). [4] P.G. Dodds, T.K. Dodds and B. de Pagter, Non-commutative Banach function spaces, Math. Z. 201(1989), 583–597. [5] P.G. Dodds, T.K. Dodds and B. de Pagter, A general Markus inequality, Proc. Centre Math. A.N.U., 24(1990), 47–57. [6] P.G. Dodds, T.K. Dodds and B. de Pagter, Weakly compact subsets of symmetric operator spaces, Math. Proc. Camb. Phil. Soc. 110(1991), 169–182. [7] P.G. Dodds, T.K. Dodds and B. de Pagter, Non-commutative K¨ othe duality, Trans. Amer. Math. Soc. 339(1993), 717–750.
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[8] P.G. Dodds and B. de Pagter, The non-commutative Yosida-Hewitt decomposition revisited (to appear). [9] P.G. Dodds, T.K. Dodds. F.A. Sukochev and O.E. Tikhonov, A non-commutative Yosida-Hewitt theorem and convex sets of measurable operators closed locally in measure, Positivity 9(2005), 457–484. [10] T. Fack and H. Kosaki, Generalized s-numbers of τ -measurable operators, Pacific J. Math. 123(1986), 269–300. [11] S.G. Krein, Ju.I. Petunin and E.M. Semenov, Interpolation of linear operators, Translations of Mathematical Monographs, Am. Math. Soc. 54(1982). [12] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15(1974), 103–116. [13] H. Pfitzner, Weak compactness in the dual of a C ∗ -algebra is determined commutatively, Math. Ann. 298(1994), 349–371. [14] M. Takesaki, On the conjugate space of operator algebra, Tˆ ohoku Math. J. 10(1958), 194–203. [15] M. Takesaki, On the singularity of a positivity linear functional on operator algebra, Proc. Japan Acad. 35(1959), 365–366. [16] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York-HeidelbergBerlin, 1979. [17] M. Terp, Lp -spaces associated with von Neumann algebras, Notes, Copenhagen University (1981). [18] J.D.M. Wright, Operators from C ∗ -algebras to Banach spaces, Math. Z. 172(1980), 123–129. [19] K. Yosida and E. Hewitt, Finitely additive measures, Trans. Amer. Math. Soc 72(1952), 46–66. [20] A.C. Zaanen, Riesz spaces II, North-Holland, Mathematical Library AmsterdamNew York-Oxford, 1983. Peter G. Dodds School of Computer Science, Mathematics and Engineering Flinders University GPO Box 2100 Adelaide 5001, Australia e-mail: [email protected] Ben de Pagter Department of Mathematics, Faculty ITS Delft University of Technology PO Box 5031 NL-2600 GA Delft, The Netherlands e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 199–204 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Ideals of Subseries Convergence and Copies of c0 in Banach Spaces Lech Drewnowski and Iwo Labuda Abstract. For a sequence x = (xn ) in a Banach space, define C(x) to be the N set ∞of all elements (εn ) of the Cantor cube K = {0, 1} for which the series n=1 εn xn is subseries convergent. The main result of the paper is that a Banach space X contains no isomorphic copy of c0 if and only if, for every sequence x in X, the set C(x) is Fσ in K. A similar equivalence involving ‘ideals of Pettis integrability’ is also shown. Mathematics Subject Classification (2000). Primary 46B15, 46B25, 46G10. Secondary 28A05, 54H05. Keywords. Subseries convergence, unconditional convergence, ideal of sets, Fσ set, Fσδ set, Banach space not containing c0 , Pettis integrability.
It was already known to Orlicz around 1929 that a series n xn in a Banach space isunconditionally convergent iff it is subseries convergent, i.e., in his language, n εn xn converges for every sequence (εn ) of zeros and ones. He knew that such a series is always perfectly bounded, that is, the set of all its finite sums is bounded (cf. [11]). He also proved that the converse holds in a weakly sequentially complete Banach space. A fully satisfactory result, today known as the BessagaPe lczy´ nski c0 -theorem, had to wait until 1957: Every perfectly bounded series in a Banach space is (subseries) convergent iff the space contains no isomorphic copy of c0 ([10, Th. 2], [2], [7, Ch. V, Th. 8]). Our main result, Theorem 5, provides new and somewhat curious characterizations of Banach spaces without copies of c0 . We came upon the concept of ‘ideals of subseries convergence’, together with the equivalence between conditions (a) and (b) in Theorem 5, at an early stage of work on [4] in 1994. The guiding idea was that the ideals in question could possibly be used to distinguish between some classes of Banach or F -spaces (as in Theorem 5 itself). Conditions involving Pettis integrability (including the second part of Proposition 1 and all of Proposition 2) The first author was partially supported by The Ministry of Science and Higher Education, Poland, Grant no. N N 201 2740 33.
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are due to Z. Lipecki; they appeared in a different context in his correspondence with the first author in 2000. During the present conference, he also suggested the inclusion of condition (c) in Theorem 5. We are very grateful to him for the permission to use this material, some other helpful comments, and the reference [9]. More results on ideals of subseries convergence will be presented in [5]. We write F for the family of all finite subsets of N = {1, 2, . . . }, and P for the family of all subsets of N. A family C ⊂ P is an ideal if B ⊂ A ∈ C implies B ∈ C, and A∪B ∈ C whenever A, B ∈ C. We identify P with the Cantor cube K = {0, 1}N, a metrizable compact space, via the correspondence A → (χA (n)), where χA is the characteristic function of A in N. Consequently, any topological statements on subfamilies of P should be understood in the sense of this identification. Let X = (X, ·) be a Banach space. Then with every sequence x = (xn ) in X we associate its ideal of subseries convergence (or unconditional convergence), C(x) = A ∈ P : xn is subseries convergent . n∈A
C(x) is indeed an ideal in P containing F, and its counterpart in K is ∞ C(x) = (εn ) ∈ K : εn xn is subseries convergent . n=1
We note that C(x) = F iff lim inf n xn > 0, and that the series n xn is subseries convergent iff C(x) = P iff C(x) is closed. Along with C(x), we also consider a solid (in the sense of [1]) vector subspace P (x) of ω, the space of all scalar sequences with the product topology, defined by ∞ P (x) = (αn ) ∈ ω : αn xn is subseries convergent . n=1
Proposition 1. For every sequence x in X, C(x) is an Fσδ subset of K, and P (x) is an Fσδ subset of ω. Proof. From the Cauchy condition for the subseries (or unconditional) convergence of a series j εj xj for (εj ) ∈ K it follows that ∞ ) ∞ 1 7 7 , (εj ) ∈ K : εj xj C(x) = k n=1 k=1
F ∈Fn
j∈F
where Fn = {F ∈ F : min F n}. Since, for each finite set F ⊂ N, the mapping (εj ) → j∈F εj xj from K into R is continuous, the sets on the extreme righthand side of the above formula are closed. Hence C(x) is an Fσδ set in K. The proof in the case of P (x) is similar, replacing K with ω. (In fact, the result for P (x) implies that for C(x) = K ∩ P (x) because K is closed in ω.) If (S, Σ, μ) is a finite positive measure space, then Σ(μ) denotes the complete semimetric space (Σ, dμ ), where dμ (A, B) = μ(A ' B). We say that (S, Σ, μ) is nontrivial if S admits an infinite Σ-partition (Sn ) with μ(Sn ) > 0 for all n. It is
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worth noting that P can be treated as a very special case of such measure spaces because its topology can be determined by the metric dν , for any finite strictly positive measure ν on P. Proposition 2. Let (S, Σ, μ) be a finite positive measure space. If a function f : S → X is strongly measurable, then the family P(f ) of all sets A ∈ Σ such that the function f χA is Pettis μ-integrable is an Fσδ ideal in Σ(μ). Proof. It is a known fact (see [3]) that f can be written ∞as g + h (μ-a.e.), where g is a bounded strongly measurable function and h = n=1 xn χAn for a sequence (xn ) in X and a Σ-partition (An ) of S. Clearly, P(f ) = P(h). Moreover, ∞ μ(A ∩ An )xn is subseries convergent . P(h) = A ∈ Σ : n=1
Since the map A → μ(A ∩ An ) from Σ(μ) to ω is continuous, applying Proposition 1 we conclude that P(h) is an Fσδ in Σ(μ). We recall that a set A ⊂ N is said to be of density zero if |A ∩ {1, . . . , n}| = 0. n Clearly, the family Z of all such sets is an ideal in P containing F. Moreover, it follows directly from the definition that Z is Fσδ in P (cf. Proof of [4, Prop. 3.3]). (This is also a byproduct of Proposition 1 and Proposition 4 below.) The fact that Z is not Fσ in P is more subtle, but our feeling is that it must have been known long before 1994, when we came upon it. Despite our efforts, we were unable to find any really early reference or a direct proof comparable to ours. Thus, for instance, in [9, p. 421] there is a suggestion that Z is not Fσ . Also, according to Professor Solecki, Z is a standard example of an Fσδ ideal which is not Fσ and, as he pointed out, this is implicit, e.g., in [12] (proof of Th. 3.4, Case 1). Therein, and especially in [8], references to further works on ideals in P can be found. For methods of proof similar to that below, though in a different context, see [6, Sec. 5]. lim
n→∞
Proposition 3. The ideal Z is not an Fσ in P. Proof. Let Z denote the subset n of K corresponding to Z. Clearly, if (ηn ) ∈ K, then (ηn ) ∈ Z iff limn n−1 j=1 ηj = 0. Define a metric ρ in Z by the equality n 1 ρ (ηn ), (ηn ) = sup |ηj − ηj |. n n j=1
It is known (cf. [11, Sec. 2]), and easily seen, that the metric space (Z, ρ) thus obtained is complete and that it is continuously included in K. ' Suppose that Z is an Fσ in K. Thus Z = n Wn for a sequence (Wn ) of closed subsets of K. Each of these sets is also closed in (Z, ρ) so, by the Baire
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category theorem, one of them, W := Wn0 say, contains a ρ-ball with center (εj ) and radius r. Fix N so large that 1 r εj < n j=1 2 n
for all n N.
If ζN +1 , . . . , ζn ∈ {0, 1} (n > N ) are such that 1 n−N
n
ζj <
j=N +1
r 2
and (ηj ) := (ε1 , . . . , εN , ζN +1 , . . . , ζn , 0, 0, . . . ), then n n n n 1 1 1 1 |εj − ηj | = |εj − ζj | εj + ζj n j=1 n n j=1 n j=N +1
<
1 r + 2 n−N
j=N +1
n
ζj < r.
j=N +1
Therefore, ρ (εj ), (ηj ) < r, and (ηj ) ∈ W . In particular, if m = (2/r) and ζj = 1 only for j = N + m, N + 2m, N + 3m, . . . , then (ε1 , . . . , εN , ζN +1 , . . . , ζn , 0, 0, . . . ) is in W for every n > N . But W is closed in K, so also the sequence (ηj ), where ηj = εj for 1 j N , and ηj = ζj for j > N must be in W ⊂ Z. However, this sequence is not in Z ! A contradiction. The sequence x in c0 below has already been used in the proof of [4, Prop. 5.1]. Proposition 4. Let x = (xn ) be the sequence in c0 defined by xn = 2−k+1 ek
for n ∈ Dk = {n ∈ N : 2k−1 n < 2k }, k = 1, 2, . . . ,
where (ek ) is the standard basis of c0 . Then C(x) = Z, and C(x) is not Fσ in P. Proof. If A ⊂ N, then n∈A∩Dk
|A ∩ Dk | xn = 2−k+1 |A ∩ Dk | = , |Dk |
and the latter quantity tends to 0 iff A ∈ Z (see [4, Prop. 3.2]). It follows that the series n∈A xn converges iff A ∈ Z. Thus C(x) = Z; apply Proposition 3. Theorem 5. Let (S, Σ, μ) be a nontrivial finite positive measure space. Then for a Banach space X the following are equivalent. (a) X contains no isomorphic copy of c0 . (b) For every sequence x in X, the set C(x) is Fσ in K. (c) For every strongly measurable function f : S → X, the ideal P(f ) is Fσ in Σ(μ).
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Proof. (a) implies (b) and (c): By the Bessaga-Pe lczy´ nski c0 -theorem, since X contains no copy of c0 , a series in X is subseries convergent iff it is perfectly bounded. Therefore, ∞ 7 ) C(x) = (εj ) ∈ K : εj xj k k=1 F ∈F
j∈F
and, consequently, C(x) is an Fσ subset of K. Likewise, examining the proofs of Propositions 1 and 2, we see that P(f ) is Fσ in Σ(μ). (b) implies (a): Apply Proposition 4. (c) implies (a): Suppose it is not so; then we may assume that X = c0 . Fix an infinite partition (Sn ) of S with σn := μ(Sn ) > 0 for all n, and let x = (x n ) be the sequence from Proposition 4. Next, define a function h : S → c0 by h = n yn χSn (pointwise sum), where yn = xn /σn for each n. Recall that then P(h) consists of all those sets A ∈ Σ for which the series n μ(A ∩ Sn )yn is subseries convergent. By (c), P(h) is Fσ in Σ(μ). Let A stand for the σ-algebra in S generated by (Sn ). Then A ∩ P(h) is Fσ in the (complete) metric space (A, dμ ). Moreover, the map ϕ : (A, dμ ) → K defined by ϕ(A) = (μ(A ∩ Sn )/σn ) is a homeomorphism onto. It follows that ϕ(A ∩ P(h)) = C(x) is Fσ in K; a contradiction. Remark 6. Let x = (xn ) be a sequence in X such that C(x) is not Fσ . Obviously, C(x) is contained in the ideal B(x) consisting of the sets A ⊂ N for which the series n∈A xn is perfectly bounded. Since B(x) or, more precisely, its counterpart in K, can be expressed in the same manner as C(x) in the first part of the proof of Theorem 5, it is Fσ , and so C(x) = B(x). Thus there is a subsequence (yn ) of (xn ) such that the series n yn is perfectly bounded and not subseries convergent. In consequence (by [2] or [7, Ch. V, Th. 8]), there are disjoint Fn ’s in F such that the sequence bn = j∈Fn xj (n ∈ N) is equivalent to the unit vector basis of c0 . Remark 7. The equivalence of (a) and (b) in Theorem 5 holds also for p-Banach spaces, while in the case of ‘true’ F -spaces some new phenomena arise [5].
References [1] C. Aliprantis and O. Burkinshaw, Locally Solid Riesz Spaces with Applications to Economics, 2nd ed., Math. Surveys and Monographs, vol. 105, Amer. Math. Soc., 2003. [2] C. Bessaga and A. Pelczy´ nski, On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151–164. [3] J.K. Brooks, Representations of weak and strong integrals in Banach spaces, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 266–270. [4] L. Drewnowski and I. Labuda, Vector series whose lacunary subseries converge, Studia Math. 138 (2000), 53–80.
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[5] L. Drewnowski and I. Labuda, Ideals of subseries convergence and F -spaces, in preparation. [6] L. Drewnowski and Z. Lipecki, On vector measures which have everywhere infinite variation or noncompact range, Dissert. Math. 339 (1995), 1–39. [7] J. Diestel, Sequences and Series in Banach Spaces, Springer, 1984. [8] I. Farah, Analytic quotients: theory of liftings for quotients over analytic ideals on the integers, Mem. Amer. Math. Soc. 148 (2000), no. 702, xvi+177. [9] W. Just and A. Krawczyk, On certain Boolean algebras P(ω)/I, Trans. Amer. Math. Soc. 285 (1984), 411–429. [10] A. Pelczy´ nski, On B-spaces containing subspaces isomorphic to the space c0 . Bull. Acad. Polon. Sci. Cl. III 5 (1957), 797–798. [11] W. Orlicz, On perfectly convergent series in certain function spaces (in Polish), Comment. Math. Prace Mat. 1 (1955), 393–414. English translation in: W. Orlicz, Collected Papers, Part I, Polish Scientific Publishers, Warsaw, 1988, pp. 830–850. [12] S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), 51–72. Lech Drewnowski Faculty of Mathematics and Computer Science A. Mickiewicz University Umultowska 87 PL-61–614 Pozna´ n, Poland e-mail: [email protected] Iwo Labuda Department of Mathematics University of Mississippi University, MS 38677, USA e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 205–214 c 2009 Birkh¨ auser Verlag Basel/Switzerland
On Operator-valued Measurable Functions Jan H. Fourie Abstract. In this paper we discuss several results about classes of vectorvalued (more specifically, operator-valued) measurable functions. The results we discuss are mostly applications of a useful lemma (Lemma 2.1 in this paper) about measurable operator-valued functions. The lemma and its elementary proof, as well as some special versions thereof, are also discussed in this note. Mathematics Subject Classification (2000). Primary 46B28; 46B45; Secondary 47B10. Keywords. Vector-valued measurable function, strongly p-integrable function, integral multiplier function.
1. Introduction Throughout this paper (Ω, Σ, μ) is a finite measure space and X, Y denote Banach spaces. The closed unit ball of a Banach space X is denoted by BX and the space of bounded linear operators from X to Y is denoted by L(X, Y ); X ∗ is the (continuous) dual space of X and for x ∈ X, x∗ ∈ X ∗ the value x∗ (x) is sometimes, when convenient, denoted by x, x∗ . The reader is referred to the book of Diestel and Uhl (cf. [4]) for the standard definitions and results in connection with measurable X-valued functions and Bochner integrable functions f : Ω → X. Our approach in this note is to first state and prove an important lemma (Lemma 2.1) in Section 2 and then to discuss results about strongly p-integrable functions (in Section 2.1), integral multiplier functions (in Section 2.2), (p, q)-integral functions (in Section 2.3) and the space of strongly measurable functions h : Ω → L(X, Y ) such that r h(t)(x) dμ(t) < ∞, ∀x ∈ X (in Section 2.4), in which Lemma 2.1 (or speΩ cial versions thereof) plays a fundamental role. The ideas in this paper originated from the work in (for instance) the papers [1], [5] and other related articles, where sequences of bounded linear operators (so-called summing multipliers) on Banach spaces and Banach space-valued strongly integrable functions were discussed. The research was partially supported by the North-West University (Potchefstroom Campus), South Africa and the South African NRF-grant, GUN 2053733.
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2. Measurable operator-valued functions The following lemma, with elementary proof, plays important role in the study of operator-valued measurable functions: Lemma 2.1. Let h : Ω → L(X, Y ) be measurable. Then for each > 0, there exists a measurable g : Ω → X such that g (t) ∈ BX , μ-a.e. and h(t) < h(t)(g (t)) + , μ-a.e. Proof. Since h : Ω → L(X, Y ) is measurable, there is a sequence (hn ) of simple functions hn : Ω → L(X, Y ) that converges almost everywhere to h; say E ∈ Σ such mn that μ(E) = 0 and h(t) − hn (t) → 0 for all t ∈ Ω\E. If wemnlet hn (t) = i=1 χAi,n (t)Si,n where Ai,n ∈ Σ, Ai,n ∩ Aj,n = ∅ for i = j and ∪i=1 Ai,n = Ω, then clearly # # mn # # # # Si,n χAi,n (t)# → 0 if n → ∞, for all t ∈ Ω\E. #h(t) − # # i=1
We may thus assume that the scalar function h · (t) = h(t) is measurable. Let a fixed > 0 be given. The fact that h : Ω → L(X, Y ) is measurable, implies by the Pettis Measurability Theorem that there exists A ∈ Σ such that μ(A) = 0 and h(Ω\A) is a (norm) separable subset of L(X, Y ). Let (Tn ) ⊂ L(X, Y ) be a dense countable subset of h(Ω\A). The sets An = h−1 · ((Tn − /3, Tn + /3)) are measurable and thus the sets Bn = An ∩ (Ω\A) are measurable, too. For each t ∈ Ω\A, there is n ∈ N such that h(t) − Tn < /3, i.e., t ∈ Bn . Thus we see that Ω\A = ∪n Bn . For each n ∈ N, take any (one) xn ∈ X with xn = 1, such that Tn < Tn (xn ) + /3 and for each n ∈ N fix this (single) 'nvector xn . Now consider a standard disjointification procedure, i.e., let Fn = i=1 Bi and then put G1 = F1 , G2 = F2 \F1 , G3 = F3 \F2 , etc. Then {Gn }∞ 1 is a disjoint partition of Ω\A. The function g : Ω → X such that xn χGn (t), g (t) = n
is measurable. Clearly, g (t) ≤ 1, μ-a.e. Also, if t ∈ Ω\A, then t ∈ Gn for a unique Gn and then t ∈ Bn . Therefore, we have h(t) − h(t)(g (t))
< ≤
h(t) − Tn + Tn (xn ) + /3 − h(t)(xn ) /3 + 2Tn − h(t) < .
This shows that h(t) < h(t)(g (t)) + , μ-a.e.
In the following subsections we discuss some results about classes of vectorvalued functions in which Lemma 2.1 (or special versions thereof) plays important role:
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2.1. Strongly p-integrable functions We recall the usual notation for vector-valued Lp -spaces. Throughout the paper, for 1 ≤ p < ∞, p will be the conjugate number of p, i.e., p1 + p1 = 1. If 1 ≤ p < ∞, Lp (μ, X) denotes the Banach space of X-valued Bochner integrable functions f : Ω → X; we call f ∈ Lp (μ, X) absolutely p-integrable. Also, L∞ (μ, X) denotes the Banach space of all essentially bounded μ-Bochner integrable functions f : Ω → X. For x ∈ X and f : Ω → X ∗ we let xf : Ω → K be the function t → f (t)(x). The space of equivalence classes of weak* p-integrable functions is defined by Lpweak∗ (μ, X ∗ ) = {f : Ω → X ∗ : f is measurable, xf ∈ Lp (μ), ∀ x ∈ X}. The norm is given by ∗ gweak p
1 := sup ( | x, g(t) |p dμ(t)) p . x ≤1
Ω
Definition 2.2. Let 1 ≤ p ≤ ∞. We call a μ-measurable function h : Ω → X a strongly p-integrable function if for each weak* p -integrable function g : Ω → X ∗ , the function Ω → K :: t → h(t), g(t) is in L1 (μ). Let Lp μ, X be the vector space of (equivalence classes of) strongly p-integrable functions h : Ω → X such that hLpμ,X := sup | h(t), g(t) | dμ(t) < ∞. ∗ ≤1 g weak p
Ω
For 1 ≤ p < ∞, (Lp μ, X , hLpμ,X ) is a Banach space. An important special case of Lemma 2.1 was considered independently by Bu and Lin in the paper [2]. The description thereof in our setting is as follows: Let h : Ω → X be a measurable function. Considering X as a normed subspace of X ∗∗ , the function h defines a measurable h : Ω → L(X ∗ , K). Applying our Lemma 2.1 to this situation, we conclude that the following lemma, formally stated (and proved) in the paper [2] by Bu and Lin, is true: Lemma 2.3. ([2], Lemma 1) Let h : Ω → X be μ-measurable. Then for each > 0, there exists a μ-measurable g : Ω → X ∗ such that g (t) ∈ BX ∗ , μ-a.e. and h(t) < | h(t), g (t) | + , μ-a.e. Applying Lemma 2.3, we get Proposition 2.4. Let 1 ≤ p < ∞. Then Lp μ, X ⊆ Lp (μ, X) with hLp(μ,X) ≤ hLpμ,X for all h ∈ Lp μ, X . Moreover, in case of p = 1 we have L1 μ, X = L1 (μ, X), isometrically.
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Proof. Let 1 < p < ∞ and let h ∈ Lp μ, X . For > 0, let g : Ω → X ∗ be the measurable function given by Lemma 2.3. Take f ∈ Lp (μ) and observe that t → f (t)g (t) is a measurable function and for every x ∈ X,
p
1/p
| x, f (t)g (t) |
≤ f Lp (μ) x .
Ω
Therefore,
h(t)|f (t)| dμ(t)
Ω
≤
| h(t), f (t)g (t) | dμ(t) + f L1(μ) Ω
≤ hLp μ,X f Lp (μ) + f L1(μ) . This shows that Lp μ, X ⊆ Lp (μ, X) by a norm ≤ 1 inclusion. The inclusion L1 μ, X ⊆ L1 (μ, X) follows by a similar argument, while the ∗ ∞ ∗ inverse inclusion follows by using that L∞ w (μ, X ) = L (μ, X ). Recall that a Banach space operator u : X → Y is called integral (in the sense of Grothendieck) if the induced bilinear form βu : X × Y ∗ → K : (x, y ∗ ) → y ∗ (ux) is integral, i.e., if there exists a measure μ ∈ C(BX ∗ × BY ∗∗ )∗ such that ∗ ∗ y (ux) = βu (x, y ) = x∗ (x)y ∗∗ (y ∗ ) dμ(x∗ , y ∗∗ ). BX ∗ ×BY ∗∗
The integral norm of u is given by u = βu . The Banach space of integral operators from X to Y is denoted by I(X, Y ). In [5], using Proposition 2.4 (and known results of Grothendieck on integral operators), it is proved that for 1 < q < ∞ the (Banach) space Lq μ, X can be identified with a space of integral operators as follows:
Theorem 2.5. (JHF, [5]) Let 1 < q < ∞. The Banach space Lq μ, X is isometrically isomorphic to the Banach space (I(Lq (μ), X), · ) of integral operators.
Knowing that Lq (μ) is reflexive and Lq (μ) has the metric approximation property, it then follows from the work of Grothendieck on projective tensor products that:
Corollary 2.6. (JHF, [5]) Let 1 < q < ∞. Then the space Lq μ, X is isometrically
∧
isomorphic to the projective tensor product Lq (μ) ⊗ X, the isometry being given n n ∧ gi ⊗ xi → gi (·)xi to Lq (μ) ⊗ X. by the unique extension of i=1
i=1
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2.2. Classes of (operator-valued) integral multiplier functions To begin, we mention the following measurability result: Lemma 2.7. If g : Ω → X and h : Ω → L(X, Y ) are μ-measurable functions, then the function Fh,g : Ω → Y such that Fh,g (t) = h(t)(g(t)) is μ-measurable too. For 1 ≤ p < ∞, the space of (equivalence classes of) weakly p-integrable functions, is given by Lpw (μ, X) = {f : Ω → X : f is measurable, x∗ f ∈ Lp (μ), ∀ x∗ ∈ X ∗ }. The quantity
1 gweak := sup ( |x∗ g(t)|p dμ(t)) p p x∗ ≤1
defines a norm on
Ω
Lpw (μ, X).
Definition 2.8. (1) A measurable function h : Ω → L(X, Y ) is called a (p, q)-integral multiplier function (for the pair (X, Y )) if for each g ∈ Lqw (μ, X), the function Fh,g : Ω → Y :: t → h(t)(g(t)) is in Lp (μ, Y ) and the linear operator ˆh : Lqw (μ, X) → ˆ = Fh,g , is bounded. In this case, we let Lp (μ, Y ), given by h(g) ˆ = sup ( |Fh,g (t)|p dμ(t)) 1p . πp,q (h) := h g weak ≤1 q
Ω
(2) In general, if E(Ω, X) and F (Ω, Y ) are normed spaces of μ-measurable functions from Ω into X and from Ω into Y respectively – both spaces containing the simple functions – then a (E(Ω, X), F (Ω, Y ))-integral multiplier function (for the pair (X, Y )) is a measurable function h : Ω → L(X, Y ) such that for each g ∈ E(Ω, X), the function Fh,g : Ω → Y : t → h(t)(g(t)) is in F (Ω, Y ) ˆ : E(Ω, X) → F (Ω, Y ) is bounded. In this case we and the corresponding h let ˆ = h(E,F ) := h sup Fh,g F (Ω,Y ). g E(Ω,X) ≤1
The vector space of equivalence classes of (E(Ω, X), F (Ω, Y ))-integral multiplier functions is simply denoted by (E(Ω, X), F (Ω, Y )). In the special case of the vector space of equivalence classes of (p, q)-integral multiplier functions, we use Lπp,q (X, Y ) rather than (Lqw (μ, X), Lp (μ, Y )), for historical reasons only. Proposition 2.9. The vector space (E(Ω, X), F (Ω, Y )) is a normed space with norm given by · (E,F ) . Let L∞ (μ,X) ⊆ E(Ω,X) and F (Ω,Y ) ⊆ L1 (μ,Y ) and h ∈ (E(Ω,X),F (Ω,Y )) be given. The function h : Ω → L(X, Y ) being measurable, for given > 0 we let g : Ω → X be the measurable function given by Lemma 2.1. Clearly, because of L∞ (μ, X) ⊆ E(Ω, X), also g ∈ E(Ω, X); thus t → h(t)(g (t)) is in F (Ω, Y ) ⊆ L1 (μ, Y ). It is therefore clear that:
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Corollary 2.10. If L∞ (μ, X) ⊆ E(Ω, X) and F (Ω, Y ) ⊆ L1 (μ, Y ), then (E(Ω, X), F (Ω, Y )) ⊂ L1 (μ, L(X, Y )). Moreover, if the embeddings L∞ (μ, X) ⊆ E(Ω, X) and F (Ω, Y ) ⊆ L1 (μ, Y ) are continuous (in particular, norm ≤ 1), then so is the embedding of the function space (E(Ω, X), F (Ω, Y )) into L1 (μ, L(X, Y )). In particular, we thus conclude that Lπp,q (X, Y ) ⊂ L1 (μ, L(X, Y )) by a norm ≤ 1 embedding. 2.3. (p, q)-integral functions Definition 2.11. For any Banach space X, we introduce the following X-valued function spaces: (i) The normed space of (p, q)-integral functions in X, is given by
=
Lπp,q (μ, X) := Lπp,q (X ∗ , K) ∩ L1 (μ, X) p1 1 p | h(t), g(t) | dμ(t) < ∞} , {h ∈ L (μ, X) : sup g weak ≤1 q
with norm
Ω
πp,q (h) :=
p1 | h(t), g(t) | dμ(t) . p
sup g weak ≤1 q
Ω
(ii) The normed space of (p, q)∗ -integral functions in X, is given by
=
Lπp,q μ, X := (Lqw∗ (μ, X ∗ ), Lp (μ, K)) ∩ L1 (μ, X) p1 sup | h(t), g(t) |p dμ(t) < ∞} , {h ∈ L1 (μ, X) : ∗ g weak ≤1 q
with norm ∗ πp,q (h)
Note that
Ω
:=
sup
∗ g weak ≤1 q
p1 | h(t), g(t) | dμ(t) . p
Ω
Lq μ, X = Lπ1,q μ, X . Using the description in Corollary 2.6 of Lπ1,q μ, X , we obtain the following Proposition 2.12. Let 1 ≤ p < q < ∞. For 1/r = 1/p − 1/q we have that (Lπ1,q μ, X , Lpw (μ, Y )) = Lrw (L(X, Y )). If we observe that for 1p = 1q + 1r ; k ∈ Lr (μ) and x ∈ X, the function ∗ (kx) ≤ kLr (μ) x, then kx : Ω → X : (kx)(t) = k(t)x is in Lπp,q μ, X with πp,q a standard Hahn-Banach argument yields the following Proposition 2.13. Let 1 ≤ p ≤ q. The space (Lπp,q μ, X , Lp (μ, Y )) embeds into L(X, Lq (μ, Y )) by a norm ≤ 1 embedding.
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We consider more inclusion results in the following: Proposition 2.14. (i) Let 1 ≤ p < q and put
1 r
=
1 p
− 1q . Then,
(Lq (μ, X), Lp (μ, Y )) = Lr (μ, L(X, Y )). In particular, L∞ (μ, L(X, Y )) ⊆ (Lq (μ, X), Lp (μ, Y )). (ii) L∞ (μ, L(X, Y )) ⊆ (Lp (μ, X), Lp (μ, Y )) for all 1 ≤ p ≤ ∞. (iii) Let 1 ≤ q < p. Then (Lq (μ, X), Lp (μ, Y )) ⊆ L∞ (μ, L(X, Y )). Proof. We only consider the proof of (Lq (μ, X), Lp (μ, Y )) ⊆ Lr (μ, L(X, Y )) in (i), illustrating the application of Lemma 2.1: Let h ∈ (Lq (μ, X), Lp (μ, Y )). Since L∞ (μ, X) ⊆ Lq (μ, X) and Lp (μ, Y ) ⊆ L1 (μ, Y ), it follows from Corollary 2.10 that h ∈ L1 (μ, L(X, Y )). We show that h ∈ Lr (μ, L(X, Y )) : r
r
Suppose f ∈ Lr (μ). Put f1 (t) = f (t) q and f2 (t) = f (t) p . Then f (t) = f1 (t)f2 (t) and f1 ∈ Lq (μ), f2 ∈ Lp (μ). For > 0 let g : Ω → X be the function obtained in Lemma 2.1. Then t → f1 (t)g (t) is measurable and moreover, it is in Lq (μ, X). Thus, t → h(t)(f1 (t)g (t)) is in Lp (μ, Y ). Since f2 ∈ Lp (μ), it follows that t → h(t)(f1 (t)g (t))|f2 (t)| is in L1 (μ). Also, h(t)|f (t)| ≤ h(t)(g (t))|f (t)| + |f (t)| = h(t)(f1 (t)g (t))|f2 (t)| + |f (t)| μ-a.e. Therefore, the function t → h(t)|f (t)| is in L1 (μ), with h(t)|f (t)| dμ(t) ≤ h(t)[f1 (t)g (t)]|f2 (t)| dμ(t) + |f (t)| dμ(t). Ω
Ω
Ω
r Since f ∈ Lr (μ) was arbitrary, this shows that h ∈ L (μ, L(X, Y )).q Moreover, since > 0 is arbitrary and since f Lr (μ) ≤ 1 implies Ω f1 (t)g (t) dμ(t) ≤ 1, the above inequality also indicates that 1r r h(t) dμ(t) ≤ h(Lq (μ,X),Lp (μ,Y )) . Ω
Remark 2.15. It follows from Proposition 2.14 that (Lq (μ, K), Lp (μ, X)) = Lr (μ, L(K, X)) when 1 ≤ p < q and 1r = 1p − 1q . In other words, identifying X with L(K, X), this gives Lπp,q (K, X) = Lr (μ, X) for these choices of p, q, r. 2.4. A new class of operator-valued functions Definition 2.16. For two Banach spaces X and Y we let Lrs (L(X, Y )) denote the space of all μ-measurable functions h : Ω → L(X, Y ) such that h(t)(x)r dμ(t) < ∞, ∀x ∈ X . Ω
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On this vector space we define
hLrs = sup
1/r h(t)(x) dμ(t) . r
x ≤1
Ω
If h : Ω → L(X, Y ) is measurable, then there exists a sequence of simple functions hn : Ω → L(X, Y ) that converges to h. Let h∗ : Ω → L(Y ∗ , X ∗ ) : h∗ (t) = (h(t))∗ .
kn ∗ For each hn (t) = j=1 χEj,n (t)uj,n , we consider the simple function hn (t) = kn ∗ ∗ ∗ j=1 χEj,n (t)uj,n . It can then easily be verified that hn (t) = (hn (t)) and that ∗ ∗ ∗ hn (t) − h (t) → 0 if n → ∞. So h is measurable too. In the next result, an interesting “duality relationship” between the new class of operator-valued functions introduced in Definition 2.16 and the space of multiplier functions (Lq (μ, X), Lpw (μ, Y )) is discussed. Again, Lemma 2.1 comes into play. Suppose a measurable h : Ω → L(X, Y ) is given and the associated “dual” measurable function h∗ : Ω → L(Y ∗ , X ∗ ) is defined as before. Fix a y ∗ ∈ Y ∗ . By Lemma 2.7, the function h∗ (·)y ∗ is measurable, too. Therefore, we may consider the following special case of Lemma 2.1: Consider X ∗ = L(X, K), then it follows from Lemma 2.1 that there exists for any given > 0 a measurable function g : Ω → X such that g (t) ∈ BX , μ-a.e. and h∗ (t)y ∗ ≤ |(h∗ (t)y ∗ )[g (t)]| + , μ-a.e. Proposition 2.17. Let 1 ≤ p < q, 1r = 1p − 1q and let h : Ω → L(X, Y ) be measurable. Then h ∈ (Lq (μ, X), Lpw (μ, Y )) ⇐⇒ h∗ ∈ Lrs (L(Y ∗ , X ∗ )). Proof. We prove h ∈ (Lq (μ, X), Lpw (μ, Y )) =⇒ h∗ ∈ Lrs (L(Y ∗ , X ∗ )): Let h ∈ (Lq (μ, X), Lpw (μ, Y )) and fix any y ∗ ∈ Y ∗ . Let > 0 be given; then let g : Ω → X be the measurable function for which g (t) ∈ BX , μ-a.e. and h∗ (t)y ∗ ≤ |(h∗ (t)y ∗ )[g (t)]| + , μ-a.e. Then we have: 1/r ∗ ∗ r h (t)(y ) dμ(t) Ω
1/r |(h (t)y )[g (t)]| dμ(t) + μ(Ω)1/r ∗
≤ Ω
=
∗
sup f Lq (μ) =1
≤
r
1/p | h(t)[f (t)g (t)], y ∗ |p dμ(t) + μ(Ω)1/r
Ω
sup g Lq (μ,X) ≤1
1/p | h(t)[g(t)], y | dμ(t) + μ(Ω)1/r . ∗
p
Ω
Since the above inequality holds for any y ∗ ∈ Y ∗ and arbitrary > 0, it is clear that h∗ ∈ Lrs (L(Y ∗ , X ∗ )) and that h∗ Lrs ≤ h(Lq (μ,X),Lpw (μ,Y )) . The converse statement is easy to verify.
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Again, Lemma 2.3 plays an important role in proving that Proposition 2.18. Let 1 ≤ p ≤ q. Then (Lπp,q μ, X , Lp (μ, Y )) = Lqs (L(X, Y )). In Proposition 2.13 we show that for 1 ≤ p ≤ q, the multiplier function space (Lπp,q μ, X , Lp (μ, Y )) embeds into the space L(X, Lq (μ, Y )) of bounded linear operators. We are now ready to see what the range space of this embedding is. It is clear that if for a μ-measurable h : Ω → L(X, Y ) we consider the linear operator uh : X → L0 (μ, Y ) from X to the space L0 (μ, Y ) of μ-measurable Y -valued functions defined by uh x(·) = h(·)x for all x ∈ X, then uh maps into Lq (μ, Y ) if and only if h ∈ Lqs (L(X, Y )). In this case, it is clear that uh = hLqs . Therefore, we conclude from Proposition 2.13 that Corollary 2.19. (Lπp,q μ, X , Lp (μ, Y )) is isometrically isomorphic to the subspace of L(X, Lq (μ, Y )) consisting of operators of the form ux(·) = h(·)x, where h ∈ Lqs (L(X, Y )). Using Corollary 2.6, one verifies the following Proposition 2.20. Let 1 ≤ p < q < ∞ and 1/r = 1/p − 1/q. Then (Lπ1,q μ, X , Lp (μ, Y )) = Lrs (L(X, Y )) . Next we discuss some elementary examples of integral multipliers and integral functions. Examples. (1) Let 1 ≤ p ≤ q < ∞. For fi ∈ L∞ (μ, Y ) and x∗i ∈ X ∗ , i = 1, . . . , n, consider n h : Ω → L(X, Y ) : h(t) = x∗i ⊗ fi (t). i=1
For each g ∈ Lqw (μ, X) and Fh,g (t) = h(t)[g(t)] we have Fh,g ∈ Lq (μ, Y ) ⊆ Lp (μ, Y ). Thus h ∈ Lπp,q (X, Y ). (2) Let 1 ≤ p ≤ q, f ∈ L∞ (μ, X) and x ∈ X, x = 1. Consider hf,x : Ω → L(X ∗ , X) : hf,x (t) = x ⊗ f (t). Then for Fh,g (t) = hf,x (t)[g(t)], where g ∈ Lqw∗ (μ, X ∗ ), we have p1 ∗ Fh,g (t)pX dμ(t) ≤ f L∞(μ,X) gweak . q Ω
∗ Thus, hf,x ∈ (Lqw∗ (μ, X ∗ ), Lp (μ, X)) and πp,q (h) ≤ f L∞(μ,X) . q 1 1 1 r ∗ (3) Let p = q + r , f ∈ L (μ, X); g ∈ Lw∗ (μ, X ) and x ∈ X, x = 1. Define hf,x : Ω → L(X ∗ , X) as in Example 2 by hf,x (t) = x ⊗ f (t) and let Fh,g (t) = hf,x (t)[g(t)] = g(t), x f (t). Then p1 1q 1r p q Fh,g (t)X dμ(t) ≤ | g(t), x | dμ(t) f (t)r dμ(t) , Ω
i.e., h ∈ (Lqw∗ (μ, X ∗ ), Lp μ, X)).
Ω
Ω
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(4) T : X → Y is a q-summing operator if and only if given any probability space (Ω, Σ, μ) and any strongly measurable function g : Ω → X such that g ∈ Lqw (μ, X), then T ◦ g ∈ Lq (μ, Y ) (cf. [3] (p. 56)). Now, for T ∈ Πq (X, Y ) given, let hT : Ω → L(X, Y ) be the constant function hT (t) = T for all t ∈ Ω. For any g ∈ Lqw (μ, X) it follows that t → hT (t)[g(t)] = T (g(t)) = (T ◦ g)(t) is a function in Lq (μ, Y ). Therefore, hT ∈ Lπq,q (X, Y ). Thus, with each T ∈ Πq (X, Y ) we associate hT ∈ Lπq,q (X, Y ). n ∗ (5) Let h : Ω → L(X, Y ) have the form h = i=1 (xi ⊗ yi )χEi , with Ei ∈ q Σ, Ei ∩ Ej = ∅, ∀ i = j. For g ∈ Lw (μ, X), we have Fh,g (t) = h(t)(g(t)) = n ∗ f ∈ Lp (μ), 1 ≤ p < ∞ and 1p + 1q = 1, i=1 xi (g(t))yi χEi (t) and for each n ∗ i χEi (t). Therefore, i=1 i (g(t))|y it follows that f (t)Fh,g (t) ≤ |fn(t)x ∗ f (t)F (t) dμ(t) ≤ ( max y ) |f (t)x g(t)| dμ(t), for all f ∈ h,g i i i=1 Ω Ω 1≤i≤n
Lp (μ), showing that Fh,g ∈ Lq (μ, Y ) for all g ∈ Lpw (μ, X). Thus, h ∈ (Lpw (μ, X), Lq (μ, Y )). (7) Note that Lp (μ, X) = (Lp (μ, X), L1 (μ, K)) ⊆ Lp (μ, X)∗ , isometrically. Equality holds once X ∗ has the Radon-Nikodym property. Acknowledgment The author thanks the referee for valuable remarks and for pointing out a mistake in the proof of Lemma 2.1 in the first version of the paper.
References [1] J.L. Arregui and O. Blasco, (p, q)-Summing sequences of operators. Quaestiones Math.26(4) (2003), 441–452. [2] Q. Bu and P.K. Lin, Radon-Nikodym property for the projective tensor product of K¨ othe function spaces. J. Math. Anal. Appl. 293 (2004), 149–159. [3] J. Diestel, H. Jarchow and A.Tonge, Absolutely Summing Operators. Cambridge studies in advanced mathematics. 43, Cambridge Univ. Press, 1995. [4] J. Diestel and J.J. Uhl, Vector measures. Mathematical Surveys. 15, Amer. Math. Soc. Providence, 1977. ˜ π X by vector-valued function [5] J.H. Fourie, Representation of (projective) Lq (μ)⊗ spaces. J. of Math. Anal. and Appl. 332 (2007), 753–766. Jan H. Fourie School of Computer, Statistical and Mathematical Sciences North-West University (Potchefstroom Campus) Private Bag X6001 Potchefstroom 2520, South Africa e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 215–229 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Logarithms of Invertible Isometries, Spectral Decompositions and Ergodic Multipliers T. Alastair Gillespie Abstract. Firstly, we consider when certain invertible isometries on Banach spaces have (bounded linear) logarithms and when they are trigonometrically well bounded, i.e., have spectral decompositions similar to that of a unitary operator. We then survey aspects of the theory of trigonometrically wellbounded operators, including an outline of some recent results. Mathematics Subject Classification (2000). Primary 47B40, 47A60, 47B38; Secondary 42B15, 43A20. Keywords. Spectral decompositions, trigonometrically well-bounded operators, functional calculus, invertible isometries.
1. Introduction The aim of this paper is to survey some features of a strand of Banach space operator theory that has been developed over the last thirty years. The original motivation of this lay in a problem concerning the existence of a (bounded linear) logarithm for an invertible operator that generates in the weak operator topology a weakly compact group. Specific examples of such operators are translations on Lp (G), where 1 ≤ p < ∞ and G is a compact abelian group and, when p = 1, such a logarithm may not exist. However, when 1 < p < ∞, logarithms always exist; indeed, such translation operators have a spectral diagonalisation of the form 2π eiλ dE(λ) (1.1) 0−
akin to, but weaker than, that for a unitary operator on Hilbert space and a 2π logarithm is given by i 0− λ dE(λ). This led some time later to a more systematic investigation of operators of the form (1.1), now called trigonometrically wellbounded operators. This background is described in more detail in §2 of the paper and is followed in §3 by some observations concerning the existence of logarithms for more general
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invertible isometries. Some salient features of the theory of trigonometrically wellbounded operators are discussed in §4, whilst in the final section of the paper more recent developments in this theory are outlined. The notation in the paper is for the most part standard. Given a Banach space X, B(X) denotes the algebra of bounded linear operators on X and, for 1 ≤ p < ∞, pX is the space of two-sided p-summable sequences having entries in X with its natural norm (p when X = C). The spectrum of T ∈ B(X) is denoted by σ(T ). Given a compact interval J in R and 1 ≤ q < ∞, BVq (J) is the Banach algebra of functions f : J → C of bounded q-variation with norm given by f BVq (J) = sup |f | + varq (f, J). J
When q = 1, this reduces to the usual algebra BV (J) of functions of bounded variation on J. The algebras BV (T) and BVq (T) are defined in the natural way by identifying a function f on T with the corresponding functions f˜ on [0, 2π] given by f˜(t) = f (eit ). The subalgebras of absolutely continuous functions in BV (J) and BV (T) are denoted, respectively, by AC(J) and AC(T). In a similar way, the Marcinkiewicz class Mq (T) is defined as the Banach algebra of functions f : T → C such that f Mq (T) ≡ sup |f | + sup varq (f, Δk ) < ∞, T
k∈Z
where Δk is the dyadic arc in T corresponding to [sk , sk+1 ] in [0, 2π] with sk = 2k−1 π (k ≤ 0) and sk = 2π − 2−k π (k > 0) via the map t → eit and varq (f, Δk ) = varq (f˜, [sk , sk+1 ]). We also recall the class of UMD Banach spaces. Although originally defined in terms of martingale difference sequences, the characterisation relevant for present purposes is that due to Burkholder and Bourgain. More precisely, a Banach space X is a UMD space if, for some (equivalently all) p in the range 1 < p < ∞, the discrete Hilbert transform Hd (that is convolution by the sequence h = {hn } where hn = n−1 for n = 0 and h0 = 0) is bounded on pX (see, for instance, the discussion in [9, §2]). Notice that, when 1 < p < ∞, Hd x = h∗x is defined co-ordinatewise for every x ∈ pX ; it follows from this that, if Hd is pX -bounded on any dense subspace of pX , then X is a UMD space. is to be interpreted as lim . Finally, a sum of the form 0<|n|<∞
N →∞ 0<|n|≤N
2. Logarithms of measures and translations The results discussed in the present paper have their origin in work of the author and T.T. West dating back to the 1970’s. This concerned an invertible operator T on a Banach space with the property that the closure in the weak operator topology of the semigroup {T n : n ∈ N} generated by T is in fact a group that is compact in the weak operator topology.
Logarithms of Invertible Isometries
217
Such operators are characterized by the following two properties: (a) supn∈N T n < ∞; (b) the closed linear span of the eigenvectors of T (corresponding to necessarily unimodular eigenvalues) equals the underlying Banach space. (See [19],[20].) Such operators are called G-operators and canonical examples are the translation operators on Lp (G), where G is a compact abelian group and 1 ≤ p < ∞; here, the characters on G are eigenvectors for translation operators. One question that arose at that time was: if T is a G-operator on a Banach space X, does there exist A ∈ B(X) such that eiA = T ? (The reason for writing such a logarithm as iA rather than A is by analogy with the fact that every unitary operator on a Hilbert space can be written as eiA with A self-adjoint.) This was answered in the negative in [21] and relied on the fact that, if G is a locally compact abelian group, then the Dirac measure δx at a point x ∈ G has a logarithm in the measure algebra M (G) if and only if x has finite order. It follows from this that, if ω ∈ T has infinite order, then translation by ω on L1 (T) does not have a logarithm in B(L1 (T)). For such a logarithm would have to commute with all translation operators and hence would be given by convolution by a measure μ that would be a logarithm of δω in M (T). This raises the following question: if 0 = s ∈ R, does translation by s on L1 (R) have a logarithm in B(L1 (R))? It seems likely that the answer is again no; a partial result along these lines, in which a restriction on the spectrum of the logarithm is prescribed, will be discussed in the next section. The situation when 1 < p < ∞ is entirely different. Here, translation operators on Lp (G) for G a locally compact abelian group have a rich spectral structure akin to that of a unitary operator and this ensures that they do have a logarithm. The key notion here is that of a spectral family on a Banach space. Definition 2.1. A spectral family on a Banach space X is a projection-valued function E(·) : R → B(X) with the following properties. (i) E(λ)E(μ) = E(λ ∧ μ) for λ, μ ∈ R; (ii) limμ→λ− E(μ) exists and limμ→λ+ E(μ) = E(λ) in the strong operator topology for λ ∈ R; (iii) E(λ) → 0 as λ → −∞ and E(λ) → I as λ → ∞ in the strong operator topology; (iv) E∞ ≡ sup E(λ) < ∞. λ∈R
Also, E(·) is said to be concentrated on [a, b], where −∞ < a ≤ b < ∞, if E(λ) = 0 for λ < a and E(λ) = I for λ ≥ b. It follows from the principle of uniform boundedness that (iv) in fact follows from (ii) and (iii) whilst, when X is reflexive, (iv) will imply the existence in the strong operator topology of limμ→λ− E(μ) and limμ→λ+ E(μ). Suppose that E(·) is a spectral family on X with E(·) concentrated on [a, b]. b Then it can be shown that, given f ∈ AC([a, b]), the integral a− f (λ) dE(λ) exists
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T.A. Gillespie
in B(X) in the strong operator topology and the mapping b f (λ) dE(λ) f→ a−
defines an identity preserving algebra homomorphism of AC([a, b]) into B(X) which is compact relative to the strong operator topology. In particular, b A= λ dE(λ) (2.1) a−
belongs to B(X) and has an AC([a, b]) functional calculus with the above compactness property. Conversely, if A ∈ B(X) has an AC([a, b]) functional calculus which is compact when B(X) is given the strong (or weak) operator topology, then A has the form given in (2.1) for some uniquely determined spectral family E(·), necessarily concentrated on [a, b]. Such operators A are called well-bounded operators of type (B); when X is reflexive, one only requires a norm continuous AC([a, b]) functional calculus and in this situation they are referred to as being well bounded. For a full account of the basic theory of these operators, see [12]. It can be shown that, if U = eiA for some well-bounded operator of type (B), then U can be written as 2π eiλ dE(λ) (2.2) U= 0−
for a uniquely determined spectral family E(·) concentrated on [0, 2π] and satisfying limλ→2π− E(λ) = I. Operators U of the form (2.2) are characterized as those invertible operators having an AC(T) functional calculus that is compact relative to the strong (or weak) operator topology. Such operators U are called trigonometrically well bounded since, in the reflexive case, they are characterized by the existence of a constant K such that f (U ) ≤ Kf BV (T) for all trigonometric polynomials f (see [2]). Returning to translation operators, it was shown in [16], [17] that, if G is a locally compact abelian group, 1 < p < ∞ and Us is translation by s ∈ G on Lp (G) (that is, (Us f )(t) = f (t − s) t-a.e.), then Us can be written as (2.2) and 2π so has a logarithm in B(Lp (G)) given by i 0− λ dE(λ). This result was in part the motivation for the introduction of the class of trigonometrically well-bounded operators in [2]. More recent developments in their theory will be discussed later. For the moment, though, it should be mentioned that G in fact plays no particular role here. It is the underlying Banach space structure of Lp (G) that is significant. More precisely, let X be a UMD space (in particular some Lp space or a closed subspace thereof, where 1 < p < ∞) and let U ∈ B(X) be an invertible operator such that supn∈Z U n < ∞. Then U is trigonometrically well bounded and so has a representation of the form (2.2) (see [9, Theorem 4.5]).
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3. Logarithms of invertible isometries We now turn to the more general problem of examining when an invertible isometry U on a Banach space X has a logarithm. Firstly, if σ(U ) = T, then there is a branch of log z defined on a neighbourhood of σ(U ) and the analytic functional calculus can then be used to define a logarithm of U , that is A ∈ B(X) such that eiA = U . However, a more explicit description is as follows. For simplicity, we consider the case when 1 ∈ / σ(U ); the general case of σ(U ) = T reduces to this by rotation. Proposition 3.1. Let U be an invertible isometry on a Banach space X with 1 ∈ / σ(U ). Then (i) the series ∞ ∞ n−1 U n and n−1 U −n n=1
n=1
converge in norm; (ii) if ∞ ∞ A0 = πI + i n−1 U n − i n−1 U −n = πI + i n=1
n=1
n−1 U n ,
0<|n|<∞
= U and σ(A0 ) ⊆ (0, 2π); then e (iii) if A ∈ B(X) satisfies eiA = U and σ(A) ⊆ (0, 2π), then A = A0 . iA0
Proof. We have N
n−1 U n
=
(I − U )−1 {U − N −1 U N +1 −
→
(I − U )−1 {U −
n=1
in norm as N → ∞. Hence
∞
N
Un } n(n − 1) n=2
∞
Un } n(n − 1) n=2
n−1 U n converges in norm. Similarly,
n=1
∞
n−1 U −n
n=1
converges in norm. Let 0 < r < 1. We have log(1 − rz) = −
∞
rn z n /n
n=1
for |z| < r−1 , where log is the branch of the logarithm on (say) the right half-plane with log 1 = 0, and so ∞
n n exp r z /n = (1 − rz)−1 (3.1) n=1
for |z| < r
−1
. By the analytic functional calculus for U , it follows that ∞
n n exp r U /n = (1 − rU )−1 n=1
(3.2)
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T.A. Gillespie
for 0 < r < 1. Now ∞ n−1 rn U n
=
(I − rU )−1 {rU −
→
(I − U )−1 {U −
n=1
∞ rn U n } n(n − 1) n=2
∞
∞ Un }= n−1 U n n(n − 1) n=2 n=1
as r → 1− in norm. Letting r → 1− in (3.2) now gives
∞ −1 n = (I − U )−1 . n U exp n=1
Similarly,
exp
∞
n−1 U −n
= (I − U −1 )−1 .
n=1
With A0 as in (ii), we now have eiA0 = −(I − U )(I − U −1 )−1 = U. By the spectral mapping theorem, σ(A0 ) = {π + i
ω n /n : ω ∈ σ(U )} ⊆ (0, 2π)
0<|n|<∞
from the Fourier expansion of eit → t for 0 < t < 2π. This proves (ii). Let A be as in (iii). Then A commutes with U and hence with A0 . If φ is a character on a maximal abelian subalgebra of B(X) containing A0 and A, then φ(A0 ) = π + i n−1 φ(U )n = π + i n−1 einφ(A) = φ(A) 0<|n|<∞
0<|n|<∞
since φ(A) ∈ (0, 2π). Hence Q = A0 − A is quasinilpotent and satisfies eiQ = I, from which it follows that Q = 0. This establishes (iii). An invertible isometry U on a Banach space X has a natural functional calculus associated with the Wiener algebras W (T) of complex-valued functions on T with absolutely convergent Fourier series given by f (U ) =
∞
fˆ(n)U n .
n=−∞
Since W (T) is an admissible algebra in the sense of [11, p. 59], it follows that U is a decomposable operator. In particular, associated with each closed subset F of T is the spectral maximal subspace XU (F ) consisting of those elements of X with local U -spectrum contained in F . The subspace XU (F ) is closed and invariant under the commutant of U . Further, σ(U |XU (F )) ⊆ F and XU (F ) contains every closed U -invariant subspace M for which σ(U |M ) ⊆ F . (See [11] or the more recent [22] for a full discussion of these matters.)
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Given ε with 0 < ε < π, let Iε denote the closed arc {eiλ : ε ≤ λ ≤ 2π − ε} of T. Also, let ∨ denote ‘closed linear span’. Lemma 3.2. With the above notation, ∨{XU (Iε ) : 0 < ε < π} = (I − U )X = ∨{f (U )X : f ∈ W (T) and f (1) = 0}. Proof. The inclusions ∨{XU (Iε ) : 0 < ε < π} ⊆ (I − U )X ⊆ ∨{f (U )X : f ∈ W (T) and f (1) = 0} are clear since I − U is invertible on each XU (Iε ) and (I − U )X = f0 (U )X, where f0 (eit ) = 1 − eit . Let f ∈ W (T) satisfy f (1) = 0. Then there is a sequence {fn } in W (T) converging to f with each fn vanishing on a neighbourhood of 1 (see [23, 2.6.4]). It follows easily, by [11, 3.1.14], that fn (U )X ⊆ XU (Iε ) for small enough ε and, since fn (U ) → f (U ) in norm, the inclusion ∨{f (U )X : f ∈ W (T) and f (1) = 0} ⊆ ∨{XU (Iε ) : 0 < ε < π} is now clear. Note that
n
−1
U x converges in norm for each x ∈ (I − U )X. We n
0<|n|<∞
have the following corollary. Corollary 3.3. Suppose that there exists A ∈ B(X) with σ(A) ⊆ [0, 2π] such that n−1 U n on (I − U )X. eiA = U . Then A = 0<|n|<∞
Proof. For 0 < ε < π, A leaves invariant XU (Iε ) and Hence
σ(A|XU (Iε )) ⊆ σ(A) ∩ {λ : eiλ ∈ Iε } ⊆ (0, 2π).
n−1 U n = A on X0 = span {XU (Iε ) : 0 < ε < π} by Proposition
0<|n|<∞
3.1. Let z ∈ (I − U )X and let {zk } be a sequence in X0 with zk → z, such a sequence existing by Lemma 3.2. It is easily checked, by rearranging the sums as the proof of Proposition 3.1, that n−1 U n (I − U )zk → n−1 U n (I − U )z 0<|n|<∞
as k → ∞. However,
0<|n|<∞
n−1 U n (I − U )zk = A(I − U )zk → A(I − U )z
0<|n|<∞
and so
n−1 U n (I − U )z = A(I − U )z = (I − U )Az.
0<|n|<∞
Since z ∈ (I − U )X,
n−1 U n z converges and so (I − U )w = 0, where
0<|n|<∞
w=
0<|n|<∞
n−1 U n z − Az.
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Noting that w ∈ (I − U )X, we thus have w ∈ ker (I − U ) ∩ (I − U )X = {0} since U is an isometry (cf. [13, VIII.5.2]). Thus w = 0 as required. We can apply these results to the bilateral shift on a vector-valued p space as follows. Theorem 3.4. Let X be a Banach space, let 1 ≤ p < ∞, and let U denote the bilateral shift on pX . Then the following are equivalent. (i) There exists A ∈ B(pX ) with σ(A) ⊆ [0, 2π] such that eiA = U . (ii) U is trigonometrically well bounded; that is, there is a spectral family E(·) concentrated on [0, 2π] such that 2π U= eiλ dE(λ). 0−
(iii) 1 < p < ∞ and X is a UMD space. Proof. The implications (iii)⇒(ii) and (ii)⇒(i) are easy. For (iii)⇒(ii), note that, if 1 < p < ∞ and X is a UMD space, then so is pX by Fubini’s theorem. Thus U , being invertible and power-bounded, is trigonometrically well bounded by [9, 2π Theorem 4.5]. If (ii) holds, then U = eiA where A = 0− λdE(λ) ∈ B(X) with σ(A) ⊆ [0, 2π]. Finally, suppose that (i) holds. Note that in this setting n−1 U n x = Hd x 0<|n|<∞
for those x ∈ pX for which this makes sense, where Hd is the discrete Hilbert transform. Thus, by Corollary 3.3, A = Hd on (I − U )pX . Let δk ⊗ u denote the sequence in pX with u ∈ X as the kth entry and 0 elsewhere. Then δ0 ⊗u−δk ⊗u ∈ (I − U )pX and so Hd (δ0 ⊗ u − δk ⊗ u)pX ≤ 21/p Au. A simple calculation shows that n−1 U n (δ0 ⊗ u − δk ⊗ u)1X → ∞ as 0<|n|<∞
|k| → ∞ when u = 0. Hence 1 < p < ∞. Now (I − U )pX = pX when 1 < p < ∞; thus Hd = A is bounded on pX and X is a UMD space. Corollary 3.5. Let 0 = s ∈ R and let Us denote translation by s on L1 (R). Then there does not exist A ∈ B(L1 (R)) with σ(A) ⊆ [0, 2π] such that eiA = Us . Proof. By dilation invariance, we can take s = 1. Considering L1 (R) as 1X , where X = L1 [0, 1], U1 becomes the bilateral shift and the result follows immediately from Theorem 3.4.
4. Trigonometrically well-bounded operators Let U be a trigonometrically well-bounded operator on a Banach space X; that is, U can be written as (2.2) for some spectral family E(·) on X concentrated on
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223
[0, 2π]. It is then possible to define a BV (T)-functional calculus for U by setting 2π f (eiλ ) dE(λ) (4.1) f (U ) = 0−
for f ∈ BV (T), the integral existing as a Riemann-Stieltjes integral in the strong operator topology. (For a discussion of the existence of such integrals, see [12, Chapter 17] or the shorter account in [3, §2].) It has already been mentioned that every invertible power-bounded operator on a UMD space is trigonometrically well bounded. Another similar result, in which power-boundedness is relaxed but the UMD space is an Lp space, is the following. Recall that a linear operator T on Lp (μ), where 1 ≤ p < ∞, is separation-preserving if (T f )(T g) = 0a.e. whenever f g = 0a.e. In this case, there is a uniquely defined positive operator |T |, the linear modulus of T , on Lp (μ) such that |T ||f | = |T f | for all f ∈ Lp (μ). Further, if T is invertible, then so is |T | and the norms of T n and |T |n are equal for all n ∈ Z. Theorem 4.1. ([7], Theorem 4.2) Let U be an invertible separation-preserving operator on Lp (μ), where μ is σ-finite and 1 < p < ∞. Suppose that n=N 1 |U |n < ∞. sup 2N + 1 N ≥0 n=−N
Then U is trigonometrically well bounded. Given a trigonometrically well-bounded operator U , it is natural to ask whether it is possible to extend the BV (T) functional calculus given by (4.1) to a larger Banach algebra A of functions on T by setting 2π f (U ) = f (eiλ ) dE(λ) (4.2) 0−
for f ∈ A. In the case when U is the bilateral shift on p (1 < p < ∞), which is certainly trigonometrically well bounded since it is a translation operator on Lp (Z), f (U ) in (4.1) can be shown to be the operator corresponding to the pmultiplier f on T. Consequently, for a general U , (4.2) can be thought of as an ‘ergodic multiplier’ associated with the group {U n }n∈Z . Such extensions have been established in a number of situations if restrictions are placed both on U and on the underlying space X. The following give illustrations of when this happens. (a) X a UMD space and sup{U n : n ∈ Z} < ∞ ; A = M1 (T) (see [5, Theorem 1.1]). (b) X a subspace of an Lp -space, 1 < p < ∞, and sup{U n : n ∈ Z} < ∞ ; A = Mq (T) with 1 ≤ q < ∞ and |2−1 − p−1 | < q −1 (see [6, Theorem 4.10]). (c) X = Lp (μ), where μ is σ-finite and 1 < p < ∞; U invertible and separationpreserving, with mean-bounded linear modulus as in Theorem 4.1; A = Mq (T) for some q ∈ (1, ∞) (see [8, Theorem 10]).
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These results are proved using transference techniques, exploiting know multiplier results in a scalar- or vector-valued setting. Notice that in each instance X is a UMD space. In the final section of the present paper, some recent results for trigonometrically well-bounded operators on super-reflexive, and hence arbitrary UMD, spaces will show inter alia that it is always possible to extend the BV (T)functional calculus on such spaces to BVq (T) for some q ∈ (1, ∞). These results are proved using very different ideas.
5. Trigonometrically well-bounded operators on super-reflexive spaces and norm growth of iterates Let U be a trigonometrically well-bounded operator. Then U n is controlled by it the variational norm of the function e → e on T and so U n = O(|n|) as |n| → ∞. In general, it is not possible to improve on this rate of growth. To see this, let Xk denote Ck with norm (z1 , . . . , zk )k = |z1 | +
k
|zj − zj−1 |,
j=2
where k ≥ 2. Let 1 = λ1 < λ2 < · · · < λk < 2π and define Uk on Xk by Uk z = (eiλ1 z1 , . . . , eiλk zk ). It is easy to check that |f (1)| +
k j=2
|f (λj ) − f (λj−1 )| ≤ f (Uk ) ≤ |f (1)| + 2
k
|f (λj ) − f (λj−1 )| (5.1)
j=2
8∞ for all trigonometric polynomials f . Taking X to be the 2 -direct sum k=2 Xk 8∞ and U = k=2 Uk on X, with the union of the partitioning points associated with the Uk ’s forming a dense subset of [0, 2π], we obtain from (5.1) an invertible operator U on a reflexive space such that 2−1 f BV (T) ≤ f (U ) ≤ 2f BV (T)
(5.2)
for all trigonometric polynomials f . Thus U is trigonometrically well bounded and, by (5.2), π|n| ≤ U n ≤ 2(1 + 2π)|n| for n ∈ Z. The question arises as to whether the growth of the iterates of a trigonometrically well-bounded operator can be improved if some geometric condition is imposed on the underlying Banach space. A motivation for this comes from consideration of the bilateral shift on a weighted sequence space. More precisely, let w = {wk } be a sequence of positive weights on Z and consider the bilateral shift U defined on the associated weighted sequence space pw , where 1 < p < ∞. It can be shown that U is trigonometrically well bounded if and only if w is an Ap weight (see [4]). (This was a precursor of the mean-boundedness result for separation-preserving mappings discussed in the previous section.)
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The weight w is, by definition, an Ap weight if there exists a constant C such that
$
%$ wn
k∈I
%p−1 −1/(p−1) wk
≤ C|I|p
(5.3)
k∈I
for all finite intervals I in Z. It is easy to check that U n = supk∈Z (wk+n /wn )1/p and then (5.3) simply yields (considering intervals I with end-points k and k + n) the growth rate U n = O(|n|) when w is an Ap weight. However, it is known that, given an Ap weight w, there exists ε > 0 such that w1+ε is also an Ap weight (see [15, IV.2.7] in the continuous setting) and then (5.3) with w replaced by w1+ε gives the growth estimate U n = O(|n|1/(1+ε) ) for U acting on pw . As a specific example of this phenomenon, consider the weight w given by w0 = 1 and wn = |n|α for n = 0. Then w is an Ap weight if and only if −1 < α < p − 1 (cf. [15, p. 407]) and, when this condition is satisfied, U n = O(|n||α|/p ). Hence there is a growth of U n of at worst |n|β , where β = min{1/p, 1/p} and p is the index conjugate to p. In particular, taking p = 2, we obtain growth of order at worst |n|1/2 . This example raises the question: does such growth always hold for trigonometrically well-bounded operators on Lp spaces when 1 < p < ∞ and, in particular (taking p = 2), do we have growth of order |n|1/2 for U n as |n| → ∞ when U is a trigonometrically well-bounded operator on a Hilbert space? A preliminary question would be: can the order of growth O(|n|) at least be improved to O(|n|γ ) for some γ with 0 < γ < 1 when the underlying space is an Lp space and 1 < p < ∞. This preliminary question was answered in the affirmative in [10] and, in the process, a number of interesting results concerning trigonometrically well-bounded operators on super-reflexive, as opposed to reflexive, Banach spaces were obtained. We conclude this paper with an outline of these results. Complete details are in preparation and will appear elsewhere. The key new idea introduced in [10] is that of the q-variation of a spectral family, defined as follows. Definition 5.1. Let E(·) be a spectral family on a Banach space X and let 1 ≤ q < ∞. Then the q-variation varq (E) of E is defined as ⎧ ⎫1/q N ⎨ ⎬ (E(λj ) − E(λj−1 ))xq , varq (E) = sup ⎩ ⎭ j=1
where the supremum extends over all finite partitions −∞ < λ0 < λ1 < · · · < λN < ∞ of R and all x ∈ X with x ≤ 1. In general, all that can be said about varq (E) is that 1 ≤ varq (E) ≤ ∞. However, when X is super-reflexive (this means that every Banach space that is finitely representable in X is automatically reflexive; see for instance [1, p. 225])
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T.A. Gillespie
spectral families do have finite q-variation for some q with 1 < q < ∞. In fact, a somewhat more precise result is true. To state this, recall the notation E∞ for supλ∈R E(λ), where E(·) is a spectral family. We always have 1 ≤ E∞ < ∞. Theorem 5.2. ([10], Theorem 2.1) Let E(·) be a spectral family of a super-reflexive Banach space X and let 4E∞ < κ < ∞. Then there exists q ∈ (1, ∞) depending only on κ and X, such that varq (E) ≤ κ. The proof of this result is technical and too long to give here. At its heart is the characterisation due to R.C. James of super-reflexivity in terms of failure of the finite tree property and the constructs used to prove this result (see [1, 4.I.3] for an account of these matters). As an immediate Corollary, we have the following less quantitative result. Corollary 5.3. ([10], Corollary 2.2) If E(·) is a spectral family on a super-reflexive Banach space X, then there exists q ∈ (1, ∞) depending on E∞ and X such that varq (E) < ∞. The fact that a spectral family on a super-reflexive Banach space has finite q-variation for some q ∈ (1, ∞) has the satisfactory consequence that the growth of U n as |n| → ∞ for a trigonometrically well-bounded operator U on such a space can be improved from O(|n|) to O(|n|α ) for some α ∈ (0, 1). The extra ingredient here is the following classical result of L.C. Young ([24]). Lemma 5.4. Let J be a compact interval and let 1 < p, q < ∞ satisfy p−1 +q −1 > 1. If f ∈ BVp (J) and g ∈ BVq (J) have no common discontinuities, then # # # # # f (t) dg(t)# ≤ Kp,q f BV (J) varq (g, J) p # # J
where Kp,q = 1 + ζ(p−1 + q −1 ) and ζ is the Riemann ζ-function. The precise growth result is as follows. Theorem 5.5. ([10], Theorem 2.2) Let U be a trigonometrically well-bounded operator on a super-reflexive Banach space X with spectral family E(·). Then there exist constants C > 0 and α ∈ (0, 1), both depending on E∞ and X, such that U n ≤ C|n|α for all n ∈ Z\{0}. Proof. Firstly fix q ∈ (1, ∞) as in Corollary 5.3 so that varq (E) < ∞ and then fix p ∈ (1, ∞) such that p−1 + q −1 > 1. We have 2π n einλ (dE(λ)x, ϕ) (U x, ϕ) = (E(0)x, ϕ) + 0
for x ∈ X and ϕ ∈ X ∗ . By Lemma (5.4), we obtain |(U n x, ϕ)| ≤ {E∞ + Kp,q n BVp [0,2π] varq (E)}xϕ,
(5.4)
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227
where n denotes the function λ → einλ on [0, 2π]. An elementary calculation (see [10, Lemma 2.7]) gives the existence of an absolute constant C0 such that n BVp [0,2π] ≤ C0 |n|1/p . Now (5.4) gives the required inequality U n ≤ C|n|α with α = 1/p and C = E∞ + C0 Kp,q . A further application of Lemma 5.4 yields the following ‘ergodic multiplier’ result for trigonometrically well-bounded operators on super-reflexive spaces. Its proof hinges on estimation of the approximating Riemann-Stieltjes sums involved in the integral appearing in the result. Theorem 5.6. ([10], Theorem 4.1) Let U be a trigonometrically well-bounded operator on a super-reflexive Banach space X with spectral family E(·). Then there exists q ∈ (1, ∞), depending on E∞ and X, such that the map 2π ϕ(eiλ ) dE(λ) ϕ → ϕ(U ) ≡ 0−
defines a continuous algebra homomorphism from BVq (T) to B(X). Specializing to the case of a Hilbert space H, it might be hoped that var2 (E) would be finite for every spectral family E(·) on H, but such a hope is easily dispelled. If this were the case, we would have var2 (E) < ∞ and var2 (E ∗ ) < ∞
(5.5) ∗
for every spectral family E(·) on a Hilbert space since E(·) is also a spectral family. However, we have the following result. Theorem 5.7. ([10], Theorem 3.1) Let E(·) be a spectral family on a Hilbert space H. Then there exists a spectral measure E on H, defined on the Borel subsets of R, such that E(λ) = E((−∞, λ]) for λ ∈ R if and only if (5.5) holds. Proof. (Sketch) The necessity of (5.5) follows from the fact that a spectral measure on a Hilbert space is similar to a selfadjoint spectral measure (cf. [12, Proposition 8.2] or [14, XV.6]). For sufficiency, suppose that (5.5) holds. Then the CauchySchwarz inequality gives n |({E(λj ) − E(λj−1 )}x, y)| ≤ var2 (E)var2 (E ∗ )xy j=1
for x, y ∈ H and ∞ < λ0 < · · · < λn < ∞. It follows that the Boolean algebra generated by {E(λ) : λ ∈ R} is bounded and hence, again by [12, Proposition 8.2], is similar to a Boolean algebra of selfadjoint projections. Standard arguments now give the existence of a spectral measure E defined on the Borel subsets of R such that E((−∞, λ]) = E(λ) for all λ. Since it is easy to construct from a conditional basis a spectral family on a Hilbert space that does not generate a spectral measure (see [18]), it follows from Theorem 5.7 that there are spectral families on Hilbert space with infinite 2-variation. In fact, using specific examples of conditional bases, it is possible to
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T.A. Gillespie
construct, for arbitrary s ∈ [2, ∞), a trigonometrically well-bounded operator U with spectral family E(·) on a Hilbert space such that vars (E) = ∞ (see [10, Theorem 3.4]) but, in this example, U n ∼ log |n|. These observations mean that the method of proof of Theorem 5.5 cannot be used to show that the iterates U n of a trigonometrically well-bounded operator U on a Hilbert space have norm growth of order |n|1/2 . Nevertheless, it remains open whether this is indeed the case.
References [1] B. Beauzamy, Introduction to Banach Spaces and their Geometry, North-Holland, 1982. [2] E. Berkson and T.A. Gillespie, AC functions on the circle and spectral families J. Operator Theory 13 (1985), 33–47. [3] E. Berkson and T.A. Gillespie, Steˇckin’s theorem, transference, and spectral decompositions J. Functional Anal. 70 (1987), 140–170. [4] E. Berkson and T.A. Gillespie, The spectral decomposition of weighted shifts and the Ap condition Coll. Math. 60/61 (1990), 507–518. [5] E. Berkson and T.A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces Studia Math. 112 (1994), 13–49. [6] E. Berkson and T.A. Gillespie, The q-variation of functions and spectral integration of Fourier multipliers Duke Math. J. 88 (1997), 103–132. [7] E. Berkson and T.A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators Trans. Amer. Math. Soc. 349 (1997), 1169–1189. [8] E. Berkson and T.A. Gillespie, The q-variation of functions and spectral integration from dominated ergodic estimates J. Fourier Anal. and Appl. 10 (2004), 149–177. [9] E. Berkson, T.A. Gillespie and P.S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform Proc. London Math. Soc. (3) 53 (1986), 489–517. [10] D. Blagojevic, Spectral Families and Geometry of Banach Spaces, PhD Thesis, University of Edinburgh, 2007. [11] I. Colojoar˘ a and C. Foia¸s, Theory of Generalized Spectral Operators, Gordon and Breach, 1968. [12] H.R. Dowson, Spectral Theory of Linear Operators, London Math. Soc. Monographs, Academic Press, 1978. [13] N. Dunford and J.T. Schwartz, Linear Operators, Part I, Wiley-Interscience, 1957. [14] N. Dunford and J.T. Schwartz, Linear Operators, Part III, Wiley-Interscience, 1971. [15] J. Garc´ıa-Cuerva and J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, 1985. [16] T.A. Gillespie, Logarithms of Lp translations Indiana Univ. Math. J. 24 (1975), 1037–1045. [17] T.A. Gillespie, A spectral theorem for Lp translations J. London Math. Soc. (2) 11 (1975), 499–508. [18] T.A. Gillespie, Commuting well-bounded operators on Hilbert spaces Proc. Edin. Math. Soc. (Series II) 20 (1976), 167–172.
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[19] T.A. Gillespie and T.T. West, Operators generating weakly compact groups Indiana Univ. Math. J. 21 (1972), 671–688. [20] T.A. Gillespie and T.T. West, Operators generating weakly compact groups II Proc. Royal Irish Acad. 73A (1973), 309–326. [21] T.A. Gillespie and T.T. West, Weakly compact groups of operators Proc. Amer. Math. Soc. 49 (1975), 78–82. [22] K.B. Laursen and M.M. Neumann, An Introduction to Local Spectral Theory, London Math. Soc. Monographs, New Series, Oxford University Press, 2000. [23] W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1967. [24] L.C. Young, An inequality of H¨ older type, connected with Stieltjes integration Acta Math. 67 (1936), 251–282. T. Alastair Gillespie School of Mathematics and Maxwell Institute for Mathematical Sciences University of Edinburgh James Clerk Maxwell Building Mayfield Road Edinburgh EH9 3JZ, Scotland e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 231–243 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Norms Related to Binomial Series Herbert Hunziker and Hans Jarchow Abstract. We investigate norms related to the vectors hn = (hn (k))k∈N in RN (or CN ) (n ∈ N) where n! 1 hn (k) = = n+k . k(k + 1)···(k + n) k n We estimate, and in a few cases even calculate, the norms of the hn ’s as elements of the usual sequence spaces r . We also show that for ‘almost all’ p, q the matrix H with entries hn (k) defines a bounded linear operator from p into q , with rather strong compactness properties. Mathematics Subject Classification (2000). Primary 26D15, 47A30; Secondary 11M06, 47B10. Keywords. Binomial series, norms of related vectors and matrices.
0. Introduction We consider norms associated with the sequence of vectors hn = (hn (k))k∈N in RN (or CN ), each hn (k) being given by hn (k) =
1 n! = n+k . k(k + 1)···(k + n) k n
Occasionally, we will extend this by setting h0 (k) := 1/k (k ∈ N). Obviously, the hn are members of the standard Banach sequence spaces r , 1 ≤ r ≤ ∞. The rth powers of their r -norms, ∞ 1/r hn r = hn (k)r , k=1
belong to the family of ‘binomial series’ referred to in the title. These norms are unknown for ‘almost all’ r ≥ 1, in particular for odd integers, a fact which is connected with classical unsolved questions concerning ζ(·), the Riemann zeta function.
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To compute hn r is easy only for r = 1, ∞. Appropriate recurrence identities will enable us to settle in addition the cases r = 2 and r = 3. Among others, our investigations will lead us to new, even though involved, series expansions of ζ(2) and ζ(3), with fairly good convergence properties. Of particular interest is the diagonal case k = n. For example, the value of ∞
(1)
hn (n) =
∞
1 2n
n n n=1 n=1 √ is known to be π/(3 3) (cf. J. Borwein, D. Bailey, R. Girgensohn [3], p. 26). Relatives of (1) appear in several identities involving values of ζ(·). special 2 (1/n ) = π 2 /6): Recall L. Euler’s rapidly convergent series for ζ(2) (= ∞ n=1 (2)
ζ(2) = 3
∞
1 ,
n2 2n n n=1
cf. K. Knopp [5], p. 275. Another familiar identity, ∞ 5 (−1)n+1 , ζ(3) = 2 n=1 n3 2n n
(3)
has been used by R. Ap´ery [1] in his proof of ζ(3)’s irrationality. For more on this and on related topics see Section 1.7 of [3]. The present paper is divided into five sections. The results just alluded to are to be found in the first three sections. We stress that their derivation requires elementary tools only. In the last two sections we are going to look at the matrix H = hn (k) n,k∈N as a Banach space operator p → q , for 1 ≤ p, q ≤ ∞ (k counts the columns and n the rows). We estimate the corresponding operator norm and we show that, for many values of p and q, the operator actually belongs to some well-known small ideals of Banach space operators (nuclear operators and variants thereof,. . . ), cf. A. Pietsch [6]. We employ standard and/or self-explaining terminology and notation. We are indebted to the referee for several valuable comments and for having discovered quite a few annoying misprints.
1. Generalities Our first result is quite easy: Proposition 1.1. For each n ∈ N: 1 (a) hn ∞ = , n+1
(b)
hn 1 =
1 . n
For the sake of completeness, we include a proof.
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Proof. (a) Just observe that hn (·) is decreasing, so that hn ∞ = hn (1). (b) By partial summation, ∞ ∞ < = n! 1 1 = n! − hn 1 = k k···(k + n) k···(k + n) (k + 1)···(k + n + 1) k=1
∞ = (n + 1)! k=1
k=1
1 1 − = (n + 1)hn 1 − 1. k···(k + n) (n + 1)!
It follows readily that, for 1 < p < r < ∞, 1 1 < hn r < hn p < . n+1 n Hence ∞ ∞ hn pr < hn pp n=1
n=1
and: Corollary 1.2. Regardless of p, r > 1, we have (hn r )n ∈ p \ 1 . Just observe that, if 1 < p < r, then ∞ ∞ p hn r < hn pp < ∞. n=1
n=1
Moreover, ∞ ∗ 1/r hn r = ( hn (k)hn (k)r−1 )1/r ≤ hn 1 hn 1/r ∞
if r ≥ 1,
k=1
so that:
1 1/r . n n+1 Here r∗ is the usual conjugate exponent of r. We will see [2.2.(a) and 3.5] that, at least for r = 2, 3, the inequality in 1.3 is strict. – We are going to look at these two special cases in the next two sections.
Corollary 1.3. hn r ≤
1 1/r
∗
2. 2 -norm First of all, h0 22 = ζ(2) (= π 2 /6) since h0 (k) = 1/k. We prove: Theorem 2.1. For each n ∈ N0 : 2(2n + 1) 3 hn 22 − (a) hn+1 22 = . n+1 (n + 1)2 ∞ ∞ < 3k!2 1 = 2n 2n 2 =3 (b) hn 2 = 2k . (2k + 2)! n n k2 k k=n k=n+1 Note that Euler’s rapidly convergent expansion (2) is the case n = 0 of (b).
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Proof. (b) follows from (a) by induction, and (a) is obtained by twicefold partial summation. First: ∞ ∞ n!2 1 1 2 hn 22 = = n! k − k 2 ···(k + n)2 k 2 ···(k + n)2 (k + 1)2 ···(k + n + 1)2 k=1
k=1
= 2n!2 (n + 1)
∞ k=1
+ (n + 1)!2
1 (k + 1)2 ···(k + n + 1)2
∞
1 k(k +
k=1 ∞ < 2
= 2(n + 1)n!
k=1
+ (n + 1)!2
∞ k=1
1)2 ···(k
+ n + 1)2
= 1 1 − k 2 ···(k + n)2 (n + 1)!2
1 k(k + 1)2 ···(k + n + 1)2 ∞
= 2(n + 1)hn 22 −
1 2 + (n + 1)!2 , 2 n+1 k(k + 1) ···(k + n + 1)2 k=1
whence hn 22 =
( )
∞ 1 (n + 1)!2 2 − . (n + 1)(2n + 1) 2n + 1 k(k + 1)2 ···(k + n + 1)2 k=1
The other application of partial summation allows to rewrite the series in ( ) in a useful fashion: ∞ 1 k=1
k(k + 1)2 ···(k + n + 1)2
=
∞ k k=1
1 1 − k(k + 1)2 ···(k + n + 1)2 (k + 1)(k + 2)2 ···(k + n + 2)2
= (2n + 3)
∞ k=1
Thus ∞ k=1
∞
1 1 1 − + (n + 1)2 . 2 2 2 k(k + 1) ···(k + n + 1) (n + 1)! k 2 ···(k + n + 1)2 k=1
1 k(k +
1)2 ···(k
+n+
1)2
=
1 1 − hn+1 22 2 2(n + 1)(n + 1)! 2(n + 1)n!2
which, in combination with ( ), gives n+1 3 + hn+1 22 . hn 22 = 2(n + 1)(2n + 1) 2(2n + 1)
Remark 2.2. By 2.1.(b) the numbers hn 2 are transcendental so that, at least when p is 2, there is no hope for equality in 1.3 (see 3.5 for p = 3).
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3. 3 -norm For small integers n, and possibly up to terms containing ζ(3), the 3 -norms of the hn ’s are known. In fact, h0 33 = ζ(3), and by [5], p. 281, Ex. 127.b), ∞ 1 2 . = 10 − 6ζ(2) = 10 − π (A) h1 33 = k 3 (k + 1)3 k=1
Moreover, by direct verification, h2 33
(B)
=
∞ k=1
29 8 = − 6ζ(3). k 3 (k + 1)3 (k + 2)3 4
Also [see again [5], p. 281, Ex. 128.b); take note of a leaflet with corrections], ∞
1 3 (n + 1)3 (n + 2)3 (n + 3)3 n n=1
h3 33 = 216 (C) =8
∞ 6 217 (n + 3)3 − n3 − 9n(n + 3) . = 70ζ(2) − n3 (n + 1)3 (n + 2)3 (n + 3)3 54 n=1
By similar reasoning, 10 385 , 96 etc. All these results are covered by what will be proved next. We start with h4 33 = 90ζ(3) −
(D)
Theorem 3.1. For each n ≥ 1, hn+1 33 =
28n2 + 24n + 6 3(9n2 − 1) − hn−1 33 . n2 (n + 1)3 (n + 1)2
To ease our notational burden, let us agree that for any integer k ≥ 2 a term “ ma ..k ..nb ” is shorthand for “ ma ·(m + 1)k ···(n − 1)k ·nb ” (for m, n, a, b ∈ N, m < n). For example, k 2 ..3 ..(k + n + 1) stands for k 2·(k + 1)3···(k + n)3·(k + n + 1). Proof. ∞
hn+1 33 1 k+n+1−k = 3 (n + 1)! n+1 k 3 ..3 ..(k + n + 1)3 k=1
< k +n+1−k k+n+1−k = 1 − 2 3 2 (n + 1) k ..3 ..(k + n + 1) k 2 ..3 ..(k + n + 1)3 ∞
=
∞
k=1
=
1 (n + 1)3 + 3
∞ k=1
∞ < k=1
k=1
1 −3 k 3 ..3 ..(k + n)3
∞ k=1
1 k 2 ..3 ..(k + n + 1)
= 1 1 1 − + k..3 ..(k + n + 1)2 k 3 ..3 ..(k + n)3 (n + 1)!3 ∞
k=1
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=
∞ 3 4 k+n−k − (n + 1)3 (n + 1)!3 n(n + 1)4 k 2 ..3 ..(k + n)3 k=1
∞ ∞ k+n+1−k k+n−k 3 6 − + (n + 1)5 k..3 ..(k + n + 1) n(n + 1)4 k 3 ..3 ..(k + n)2 k=1
k=1
19n2 + 12n + 3 3n + 1 hn−1 33 = 2 − 6 n (n + 1)3 (n + 1)!3 n2 (n + 1)5 (n − 1)!3 ∞ ∞ < k+n−k k+n−k = 3n + 1 + 3 3 + n (n + 1)5 k..3 ..(k + n)2 k 2 ..3 ..(k + n) k=1
k=1
19n2 + 12n + 3 3n + 1 hn−1 33 = 2 − 6 n (n + 1)3 (n + 1)!3 n2 (n + 1)5 (n − 1)!3 ∞ = n−1 3n + 1 < 1 − 3 3 − 2 5 3 3 3 n (n + 1) k ..3 ..(k + n − 1) n (n − 1)! k=1
28n + 24n + 6 9n2 − 1 hn−1 33 −3 3 · 3 3 + 1) (n + 1)! n (n + 1)5 (n − 1)!3 2
=
n2 (n
We rewrite 3.1 so as to have (n + 1)2 28n2 + 24n + 6 − hn+1 33 , (∗) hn−1 33 = 3n2 (n + 1)(9n2 − 1) 3(9n2 − 1) Theorem 3.2. For each N ∈ N: (a) h2N −2 33 =
∞ (3N − 3)! k!3 (56k 2 + 80k + 29) (−1)N +k−1 · 3 (N − 1)! 4(2k + 1)2 (3k + 3)! k=N −1
(b) h2N −1 33 =
(6N − 5)(2N − 1)(N − 1)!3 (2N − 1)!3 (3N − 3)! ∞ 56k 2 + 24k + 3 (−1)k−N (2k − 1)!3 (3k − 3)! · × . (2k − 1)(6k − 5)!(k − 1)!3 6k 2 (2k + 1)(36k 2 − 1) k=N
Proof. (a) If n is odd, n = 2N − 1, then (∗) becomes h2N −2 33 = ϕ(N ) − ψ(N )h2N 33 where ϕ(N ) = and
56N 2 − 32N + 5 12N (4N 2 − 4N + 1)(9N 2 − 9N + 2)
N3 . (3N − 2)(3N − 1)(3N ) Put χ(N ) = ψ(1)···ψ(N ) and check that χ(N ) = N !3 /(3N )!. Hence ψ(N ) =
ψ(N )ψ(N + 1)···ψ(N + ) =
(3N − 3)! (N + )!3 χ(N + ) = · χ(N − 1) (N − 1)!3 (3N + )!
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and so, since limN →∞ χ(N )h2N 33 = 0, h2N −2 33 = ϕ(N ) − ψ(N )h2N 33 = ϕ(N ) − ψ(N )ϕ(N + 1) + ψ(N )ψ(N + 1)ϕ(N + 2) − ψ(N )ψ(N + 1)ψ(N + 2)ϕ(N + 3) ± ··· ∞ (3N − 3)! k!3 (56k 2 + 80k + 29) . (−1)N +k−1 · 3 (N − 1)! 4(2k + 1)2 (3k + 3)!
=
k=N −1
(b) If n is even, n = 2N , then (∗) becomes h2N −1 33 = ϕ(N ) − ψ(N )h2N +1 33 where ϕ(N ) =
56N 2 + 24N + 3 + 1)(36N 2 − 1)
6N 2 (2N
and ψ(N ) =
(2N + 1)2 . 3(36N 2 − 1)
We are going to work with an artificially complicated version of the ψ’s: (2N + 1)2 (6N − 4)(6N − 3)(6N − 2)6N · 3(36N 2 − 1) (6N − 4)(6N − 3)(6N − 2)6N (6N − 5)!(N − 1)!3 2N − 1 (2N + 1)!3 (3n)! · · = (2N − 1)!3 (3N − 3)! 2N + 1 N !3 (6N + 1)!
ψ(N ) =
For k ∈ N0 , we put χ(k) = ψ(N )···ψ(N + k); we also put χ(−1) = 1. We obtain h2N −1 33
= ϕ(N ) +
∞
(−1) χ(k − 1)ϕ(N + k) = k
k=1
∞
(−1)k χ(k − 1)ϕ(N + k)
k=0
(6N − 5)!(2N − 1)(N − 1)!3 = (2N − 1)!3 (3N − 3)! ∞ (2k − 1)!3 (3k − 3)! 56k 2 + 24k + 3 . (−1)k−N · · × (2k − 1)(6k − 5)!(k − 1)!3 6k 2 (2k + 1)(36k 2 − 1) k=N Both results are of particular interest in case N = 1: Corollary 3.3. (a) (b)
h0 33 = h1 33 =
∞ k=0 ∞ k=1
(−1)k ·
k!3 (56k 2 + 80k + 29) . 4(3k + 3)!(2k + 1)2
(−1)k−1 ·
56k 2 + 24k + 3 (2k − 1)!3 (3k − 3)! · . (2k − 1)(k − 1)!3 (6k − 5)! 6k 2 (2k + 1)(36k 2 − 1)
Recall that h0 33 = ζ(3) and h1 33 = 10 − π 2 . Therefore 3.3 can be used to modify 3.2:
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Corollary 3.4. For each integer N ≥ 2: N −1 = < 3 2 3 N −1 (3N )! k k! 56k + 80k + 29) ζ(3) − · (−1) · (a) h2N 3 = (−1) N !3 4(2k + 1)2 (3k + 3)! k=0
(b)
(6N − 5)!(2N − 1)(N − 1)!3 = (−1) · (2N − 1)!3 (3N − 3)! N −1 = < (−1)k−1 (2k − 1)!3 (3k − 3)! 56k 2 + 24k + 3 . × 10 − π 2 − · (2k − 1)(6k − 5)!(k − 1)!3 6k 2 (2k + 1)(36k 2 − 1) N −1
h2N −1 33
k=1
We claim that convergence of both series in 3.3 is more rapid than of other known relevant series. Let’s first look at 3.3.(a). Let sn be the nth partial sum of the latter series in (a). Recall from (3) ∞ ]}. Let σ(n) be the nth partial sum of that ζ(3) = (5/2) k=1 {(−1)k+1 /[k 3 2k k ∞ this series and αn the nth partial sum of k=1 1/k 3 . The sequence (sn ) converges better than (σn ) which in turn converges better than (αn ). This is apparent from: n
αn
σn
sn
5 10 20 50
1.185662 037 037 . . . 1.197531 985 674 . . . 1.200867 841 958 . . . 1.201860 863 164 . . .
1.302 067 625661 . . . 1.302 056 900941 . . . 1.202 056 903159 . . . 1.202 056903 159 . . .
1.202056 904 152 . . . 1.202056 903 159 . . . 1.202056 903 159 . . . 1.202056 903 159 . . .
Now we turn to the other case. Denote by tn the nth partial ∞sum of the series in 3.3.(b), and by βn and τn the nth partial sums of π 2 = 6 k=1 1/k 2 and ∞ π 2 = 18 k=1 (k − 1)!2 /(2k)!, respectively (as for the latter, see [5], p. 275). The sequence (tn ) [resp. (τn )] converges better than (τn ) [resp. (βn )]: n 5 10 20 50
βn
10 − tn
τn
8.781 666 666 666 . . . 9.298606 386 999 . . . 9.576979 463 478 . . . 9.750796 401 729 . . .
9.868 928 571 428 . . . 9.869 604 130788 . . . 9.869 604 401089 . . . 9.869 604 401089 . . .
9.869 604 400 781 . . . 9.869604 401 089 . . . 9.869604 401 089 . . . 9.869604 401 089 . . .
Note that 3.3 also yields: Corollary 3.5. All hN 33 are irrational; each h2N −1 33 is even transcendental. In fact, π is transcendental and ζ(3) is irrational [1]. Hence, as in 3.2.(a), there is no hope for equality in 1.3 for h2N −1 33 either. Remarks 3.6. (a) It should be clear, in principle, how to continue in order to determine the values hn N N for N, n ∈ N, N ≥ 4. But this leads to increasingly complicated and clumsy expressions. We decided not to pursue the general case.
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(b) The Bernoulli numbers Bj , however, allow to present h1 nn in an appealing manner. We mention without proof: n 2 = <2n − 1 2n − 1 − 2k n n+1 h1 n = (−1) ζ(2k) −2 n − 2k n−1 k=1 n 2 < 2n − 1 2n − 1 − 2k B2k π 2k = . − · = (−1)n+1 (−1)k+1 (2k)! n−1 n − 2k k=1
4. H as an operator Let 1 ≤ p, q ≤ ∞. Whenever the matrix H induces a bounded operator p → q , we will denote it by Hp,q . Of course, in such a case, we wish to obtain information, e.g., on the corresponding operator norm Hp,q and on further ideal (quasi-)norms of Hp,q , if available. First a ‘negative’ result: no matter which 1 ≤ p ≤ ∞ we choose, H doesn’t induce a map p → 1 . Indeed, He1 is the sequence (hn (1))n = (1/(n + 1))n . Here and in what follows, en denotes the nth coordinate unit vector (δn,k )k . In contrast, by straightforward application of H¨ older’s inequality: Proposition 4.1. If q > 1 and p ≥ 1 then Hp,q exists, with ∞ 1/q Hp,q ≤ hn qp∗ . n=1
In two cases, we can be precise: Proposition 4.2. For any q > 1, ∞ 1 1/q H∞,q = nq n=1
and
H1,q =
1/q 1 . (n + 1)q n=1
∞
In other words, ζ(q) = H∞,q q = H1,q q + 1. Proof. (a) For every unit vector ξ ∈ ∞ we get from 1.1.(a) ∞ ∞ # ∞ #q 1/q # # H∞,q ξq = hn (k)ξk = hn (k)ξk # # ≤
k=1 ∞ ∞ n=1
k=1
q
n=1 k=1
∞ ∞ 1/q q 1/q 1 1/q hn (k) = hn q1 = , nq n=1 n=1
so that H∞,q ≤
∞
1/nq
1/q
n=1
As for equality, look at Hξ( where ξ( = (1, 1, 1, . . . ).
.
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(b) Again by 1.1.(a) we have, for unit vectors ξ ∈ 1 , ∞ # ∞ ∞ ∞ #q q # # hn ∞ ξ1 H1,q ξqq = hn (k)ξk # ≤ = # so that H1,q ≤
n=1 k=1 ∞ ( n=1 1/(n
n=1 q 1/q
+ 1) )
1 , (n + 1)q n=1
. Look at H1,q e1 for equality.
By similar arguments, H doesn’t map 1 into any of the classical Lorentz spaces 1,q , q < ∞. But it maps ∞ into 1,∞ . In what follows, we use ‘ →’ is to signalize that we are dealing with a formal identity (canonical bounded inclusion map between appropriate Banach spaces). Natural factorization Hp,q
H1,q : 1 → p −→ q
H∞,q
and Hp,q : p → ∞ −→ q
leads immediately to the next statement: ∞
∞ 1 1 q ≤ Hp,q ≤ for all p ≥ 1. Corollary 4.3. If q > 1 then q q (n + 1) n n=1 n=1
In other words, Hp,q q belongs to the interval [ζ(q) − 1, ζ(q)]. Problems. (a) Let 1 < p < ∞. By definition, Hp,p∗ equals the weak p∗ -norm ([4]) of (hn )n as a sequence in ∗p . Can this be expressed in terms of the hn ’s only? (b) Calculate Hp,q for further values of p and q. To conclude this section, we note: Corollary 4.4.
∞ k=2
H1,k k =
∞
H∞,k k − 1 = 1 .
k=2
In fact, as is well known, ∞ ∞ < ∞ ∞ 1= 1 1 = = 1. − 1 − [ζ(k) − 1] = k n n n(n − 1) n=2 n=2 k=2 k=0 k As a consequence, ∞ k=2 (Hp,k − 1) ≤ 1 for all p ≥ 1.
5. Further ideal properties By a classical result of A. Grothendieck, every operator 1 → 2 is (absolutely) 1-summing. Also, every operator ∞ → q is 2-summing when q ≤ 2, resp. r-summing when 2 < q < r < ∞ (see [6] or [4]). Natural factorization reveals that if q ≥ 2, then H1,q is 1-summing, whereas Hp,q is 2-summing for 1 ≤ p ≤ ∞ and 1 < q ≤ 2, resp. r-summing for 1 ≤ p ≤ ∞ and 2 < q < r < ∞. As we shall see, the Hp,q enjoy even stronger ideal theoretic properties. We confine ourselves to just a few cases.
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Fix 1 ≤ p ≤ ∞ and 1 < q ≤ ∞. Then Hp,q is defined and, e.g., in the sense of pointwise convergence, we can write Hp,q =
∞
hn ⊗ e n .
n=1
Taking 1.3 into account we see: Proposition 5.1. Hp,q is q-nuclear whenever 1 ≤ p < ∞ and q ≥ p∗ . We refer to Ch. 18 of [6] for the concept of q-nuclearity and related properties of Banach space operators. Here is a possibility to improve upon 5.1. Given N ∈ N, think of HN =
N −1
hn ⊗ e n
n=1
as an operator p → q of rank < N . The N th approximation number, sN (Hp,q ), of Hp,q (cf. [6]) then satisfies sN (Hp,q ) ≤ Hp,q − HN = sup
ξ∈Sp
∞ # ∞ ∞ #q 1/q 1/q # # hn (k)ξk # ≤ hn qp∗ . # n=N
k=1
n=N
Here Sp denotes the unit sphere of p . In particular, (sN (Hp,q ))N is a null sequence, so that Hp,q is compact. More is true. Given 0 < q < ∞, let
Sq be the qth “approximation number ideal” which is made up of all general Banach space operators u : X → Y satisfying (sN (u))N ∈ q . This is a quasi-Banach ideal, ∞ with σq (u) = ( N =1 sN (u)q )1/q as its canonical quasi-norm; see again [6]. If H and K are Hilbert spaces, then Sq (H, K) is the corresponding Schatten q-class of operators H → K. In particular, S2 (H, K) is the collection of all Hilbert-Schmidt operators H → K, and S1 (H, K) consists of all nuclear (or trace class) operators H → K. See [6] or [4] for details. Proposition 5.2. H2,2 is a Hilbert-Schmidt operator, the Hilbert-Schmidt norm σ2 (H2,2 ) being given by ∞ ∞ n 1 2n ζ(2) − σ2 (H2,2 )2 = hn 22 = 3 2k . n k2 k n=1 n=1 k=1
Proof. The first statement follows from what has been mentioned at the beginning of this section. Modulo 2.1, the assertion on the Hilbert-Schmidt norm is also easy: ∞ k=1
H2,2 ek 22 =
∞ ∞ k=1 n=1
hn (k)2 =
∞ n=1
hn 22 .
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By natural factorization, if 1 ≤ p ≤ 2 ≤ q, then Hp,q belongs to every ideal of Banach space operators whose Hilbert space components coincide with the corresponding Hilbert-Schmidt classes. In particular (see, e.g., [4], 5.30), Proposition 5.3. If 1 ≤ p ≤ 2 ≤ q, then Hp,q is in S2 , and it is r-nuclear for every r > 1. We add two further partial improvements of 5.1: Proposition 5.4. For any choice of p ≥ 1 and q > 2, the operator Hp,q is a member of Sq . Proof. We apply 1.3. Accordingly, since q > 2, σq (Hp,q )q = ≤ ≤ ≤
∞ N =1 ∞
sN (Hp,q )q ≤ 1
N =1 n=N ∞ ∞
n=N
hn qp∗
N =1 n=N ∞
nq/p
N =1 ∞
∞ ∞
n=N
1 ≤ nq
·
1 (n + 1)q/p∗
∞ 1/p∗ 1 1/p 1 nq (n + 1)q n=N
∞ N =1
1 < ∞. N q−1
For the last inequality we refer to G. Bennett [2], Lemma 4.7. Recall (cf. [4], [6]) that 2 -valued operators with domain 1 or ∞ are 2-summing and that any composition of two 2-summing operators is nuclear. Hence: Proposition 5.5. For any 1 < q < ∞ and 1 ≤ p ≤ 2, the operators H1,q and Hp,∞ are nuclear. ∞ Consider hn = (hn (k))k as an element of ∞ and think of n=1 hn ⊗ en as a representation of the nuclear operator H1,2 . It is tempting to believe that this is a nuclear representation. But this would mean that h n en < ∞ holds n ∞ which contradicts hn ∞ = 1/(n + 1) (1.1.(a)). Similarly, n=1 hn ⊗ en cannot be a nuclear representation of the nuclear operator H2,∞ . Problems. (a) Find nuclear representations for H1,2 and H2,∞ . (b) Using 1.3 it can be shown that H2,2 belongs to Sr even for all r > 1. Hence, e.g., in 5.3, S2 can be replaced with Sr . What’s about the case r = 1? Suppose that H2,2 is trace class (the trace is given by formula (1)). For 1 ≤ p ∞ ≤ 2 ≤ q, Hp,q is then not only nuclear but even a member of S1 . But again, n=1 hn ⊗ en fails to be a nuclear representation of H2,2 .
Norms Related to Binomial Series
243
References [1] R. Ap´ery, Irrationalit´e de ζ(2) et ζ(3). Ast´erisque 61 (1979) 11–13. [2] G. Bennett, Factorizing the Classical Inequalities. Memoirs AMS 120 1996. [3] J. Borwein, D. Bailey, R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery. AK Peters 2004. [4] J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators. Cambridge Univ. Press 1995. [5] K. Knopp, Theorie und Anwendung der unendlichen Reihen (4th ed.) Springer-Verlag 1947. [6] A. Pietsch, Operator Ideals. VEB Deutscher Verlag der Wissenschaften 1978, NorthHolland 1980. Herbert Hunziker Abteilung Mathematik Alte Kantonsschule Aarau CH-5001 Aarau, Switzerland e-mail: [email protected] Hans Jarchow Institut f¨ ur Mathematik Universit¨ at Z¨ urich CH-8057 Z¨ urich, Switzerland e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 245–254 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Vector-valued Extension of Linear Operators, and T b Theorems Tuomas P. Hyt¨onen Abstract. I recently proved that a large class of singular integral operators T ∈ L (Lp (μ)) have bounded tensor extensions to T ∈ L (Lp (μ; X)) provided that X is a UMD space with a certain additional property. Here I eliminate this extra condition on X by assuming a bit more about T , and sketch a program towards its full elimination. Mathematics Subject Classification (2000). Primary 42B20; Secondary 60G46. Keywords. Probabilistic method, paraproduct, non-doubling measure.
1. Introduction Given an operator T ∈ L (Lp (μ)) and a Banach space X, one asks whether (the tensor extension of) T , first defined on the algebraic tensor product Lp (μ) ⊗ X by T
K k=1
K φk ⊗ ξk := T φk ⊗ ξk ,
φk ∈ Lp (μ),
ξk ∈ X,
k=1
also extends to a bounded operator on Lp (μ; X). It is well known that the nicest operators T (those which are pointwise dominated by a positive operator S ∈ L (Lp (μ)) in the sense that |T φ| ≤ S|φ|) allow such an extension for all Banach spaces X, and all operators T allow the extension for the nicest spaces X (the Hilbert spaces, of course). But if neither the space nor the operator is so nice, things become more tricky. About 25 year ago, Bourgain [1] and Burkholder [3] proved that the classical Hilbert transform, the mother of all singular integrals so to say, extends to Lp (R; X) if and only if X is a UMD space, i.e., the martingale differences are unconditional in Lp (0, 1; X). (This condition is actually independent of p ∈ (1, ∞).) The author is supported by the Academy of Finland, project 114374.
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T.P. Hyt¨ onen
Since then, the extension property in UMD spaces has been proven for ever bigger classes of singular integral operators, and the farthest reaches of the present technology are as follows: Let μ be a Borel measure on RN which satisfies the upper bound μ(B(x, r)) ≤ Crd ,
d ∈ (0, N ],
(1.1)
for any ball B(x, r) of centre x ∈ R and radius r > 0. A d-dimensional Calder´ on– Zygmund kernel is a function K(x, y) of variables x, y ∈ RN , x = y, such that C , (1.2) |K(x, y)| ≤ |x − y|d N
1 |x − x |α |x − x | < , (1.3) , d+α |x − y| |x − y| 2 for some α > 0. An operator T acting on some functions is called a Calder´ on– Zygmund operator with kernel K if T f (x) = K(x, y)f (y) dμ(y), x∈ / supp f. (1.4) |K(x, y) − K(x , y)| + |K(y, x) − K(y, x )| ≤ C
RN
A measure μ is said to satisfy the doubling property if μ(B(x, 2r)) ≤ Cμ(B(x, r))
∀ balls B(x, r) ⊂ RN .
The following result was obtained by Bourgain [2] for the Lebesgue measure dμ = dx (thus d = N in (1.1)–(1.3)) and convolution kernels K(x, y) = k(x−y), by Figiel [5] for dμ = dx but dropping the convolution assumption, and in the stated generality by myself. (Because of space limitations, I need to skip the discussion of various related results on Fourier multipliers, pseudodifferential operators etc.) Theorem 1.1 ([6]). For μ and K as in (1.1)–(1.3), where μ also satisfies the doubling property, an operator T ∈ L (L2 (μ)) as in (1.4) extends to T ∈ L (Lp (μ; X)) for all p ∈ (1, ∞) and all UMD spaces X. How is something like this proven? There seems to be no way of directly exploiting the assumption T ∈ L (L2 (μ)). Instead, one relies on a characterization of this boundedness from harmonic analysis. The following generalization of the famous T b theorem of David, Journ´e and Semmes [4] is due to Nazarov, Treil and Volberg [10]. Recall that a function b ∈ L1loc (μ) is called weakly accretive if # 1 ## # b(y) dμ(y)# ≥ δ > 0 ∀ cubes Q = x + [0, )N ⊂ RN . # μ(Q) Q The definition of the BMO space associated to a general μ as in (1.1) requires some care (see [10]), but I omit the details, as they are not explicitly needed below. Theorem 1.2 ([10]). For μ and K as in (1.1)–(1.3), an operator T as in (1.4), first defined on some test functions, extends to T ∈ L (L2 (μ)) if and only if, for two bounded weakly accretive functions b1 and b2 there hold 1. the weak boundedness property: | 1Q b2 , T (b1 1Q ) | ≤ Cμ(Q) for all cubes Q, 2. the T b conditions: T b1 ∈ BMO(μ), T ∗ b2 ∈ BMO(μ).
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247
So to prove Theorem 1.1, one observes that the assumption T ∈ L (L2 (μ)) is equivalent to the above weak boundedness and T b conditions. And then one adapts the argument of Theorem 1.2 that weak boundedness and T b1 , T ∗ b2 ∈ BMO(μ) imply T ∈ L (L2 (μ)) to show that the same conditions even imply that T ∈ L (Lp (μ; X)). Obviously this adaptation is non-trivial; what was said is just the general underlying philosophy. The reader may have noticed that doubling is not assumed in Theorem 1.2, so why does it appear in Theorem 1.1? It turned out that only with this extra assumption I was able to carry out the proof in the generality of all UMD spaces. The lack of doubling could be compensated by assuming a bit more about the Banach space X, namely the Rademacher maximal function (RMF) property introduced in [7]. As the present theme is avoiding this property, I also avoid the definition due to limited space; see [6] or [7]. Theorem 1.3 ([6]). For μ and K as in (1.1)–(1.3), an operator T ∈ L (L2 (μ)) as in (1.4) extends to T ∈ L (Lp (μ; X)) for all p ∈ (1, ∞) and all UMD spaces X for which both X and X ∗ have the RMF property. The theorem covers, by results in [7], the standard examples of UMD spaces (UMD function lattices, and the reflexive non-commutative Lp spaces) but it is unknown if it covers all UMD spaces. As the RMF property is not well understood, it would be useful to eliminate this assumption from Theorem 1.3. One way would be to prove that UMD already implies RMF, but in the lack of an idea how to do it, one is lead to look at Theorem 1.3 per se. And in this paper I prove: Theorem 1.4. For μ and K as in (1.1)–(1.3), an operator T ∈ L (L2 (μ)) as in (1.4) extends to T ∈ L (Lp (μ; X)) for all p ∈ (1, ∞) and all UMD spaces X if for two bounded weakly accretive functions b1 and b2 there holds T b1 ∈ L∞ (μ),
T ∗ b2 ∈ L∞ (μ).
This does not yet allow the elimination of RMF from Theorem 1.3, since these T b conditions, unlike those of Theorem 1.2, need not hold for all T ∈ L (L2 (μ)). However, there is some hope, based on the following “local” T b theorem of Nazarov, Treil and Volberg [9], that the ideas in the proof of Theorem 1.4 could lead to the desired elimination. Theorem 1.5 ([9]). For μ and K as in (1.1)–(1.3), an operator T as in (1.4), first defined on some test functions, extends to T ∈ L (L2 (μ)) if and only if there exist, for all cubes Q, functions b1Q and b2Q such that: # 1 ## # biQ ∞ ≤ 1, biQ (y) dμ(y)# ≥ δ > 0, supp biQ ⊆ Q, # μ(Q) Q T b1Q∞ ≤ C,
T ∗ b2Q ∞ ≤ C.
Theorems 1.4 and 1.5 motivate the following: Conjecture. For μ and K as in (1.1)–(1.3), an operator T ∈ L (L2 (μ)) as in (1.4) extends to T ∈ L (Lp (μ; X)) for all p ∈ (1, ∞) and all UMD spaces X.
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T.P. Hyt¨ onen
The idea of proof would be to adapt the argument of Theorem 1.5 to show that the same conditions even imply that T ∈ L (Lp (μ; X)), in a similar way as the argument of Theorem 1.2 was adapted for Theorem 1.3, but hopefully avoiding RMF by the ideas of Theorem 1.4. This seems like a reasonable program, but to carry it out in detail is beyond the present contribution. The rest of this paper is concerned with the proof of Theorem 1.4. In Section 2, I review the argument for Theorem 1.3 and point out the part which needs to be modified, and Section 3 then concentrates on the required modification.
2. Review of the proof of the T b theorem To bound the operator norm T L (Lp (μ;X)) , one needs to estimate the pairing
g, T f ,
gLp (μ;X ∗ ) = f Lp(μ;X) = 1.
It is possible to reduce to the a priori bounded situation T L (Lp (μ;X)) < ∞, but looking for a new quantitative bound for the norm which only depends on the constants appearing in the assumptions. Then one can even fix f and g as above in such a way that T L (Lp (μ;X)) ≤ 2| g, T f |. The analysis of the pairing is based on the following conditional expectations and their adapted versions: 1
f Q f dμ, EQ f := 1Q f Q , EbQ1 f := 1Q b1 ,
f Q := μ(Q) Q
b1 Q Ek f Ek f := EQ f Ebk1 f := EbQ1 f = b1 , Ek b1 Q∈Dk
Q∈Dk
where the Dk are refining partitions of RN into dyadic cubes of side-length (Q) = 2k . One also needs the corresponding differences ) 1 1 . f − Ebk1 , DbQ1 f := 1Q Dblog f Q ∈ D := D Dbk1 f := Ebk−1 k (Q) 2
k∈Z
Similarly one defines Ek , and using another dyadic system ' D = k∈Z Dk in place of D and another function b2 in place of b1 . The reason for two different dyadic systems D and D is that one eventually wants to choose them randomly and independently of each other. This is because there are certain b pairings D R2 g, T DbQ1 f for which it seems impossible to get good estimates from the assumptions, and one wants to show that these situations are in a sense rare. So one partitions D into two subcollections Dgood and Dbad , and similarly D = Dgood ∪Dbad . How this is done precisely is not important for the present discussion; see [6], Section 5. In the estimation of g, T f , one splits the functions f = fgood + fbad and g = ggood + gbad , where, e.g., fgood := DbQ1 f . It is a consequence of the
b E k2 ,
b D k2
b D R2 ,
Q∈Dgood
T b Theorems UMD property that f =
249
DbQ1 f with unconditional convergence, and hence
Q∈D
fgood p f p . For b1 ≡ 1, this is essentially the definition of UMD; otherwise, it requires an argument, which is given in [6], Section 4. Also the operator T is split into several parts. Without going into the lengthy details in [6], Sections 7–12, one could just call them the unconditional part T uc, the bad part T bad , and the paraproduct parts ΠT and Π∗T ∗ , which will be considered more carefully below. Then T = T uc + T bad + ΠT + Π∗T ∗ , and
g, T f = ggood , T ucfgood + ggood , ΠT fgood + ΠT ∗ ggood , fgood + ggood , T bad fgood + gbad , T fgood + g, T fbad .
(2.1)
In [6] (as in [10] for the scalar case), the first three terms were estimated directly for any D and D (which are implicit behind the splittings of f , g and T ), with desired bounds of the form Cgp f p = C, where C depends on the constants in the assumptions. The unconditional part heavily relies on the UMD assumption, whereas in the paraproduct parts one resorts to the RMF conditions: X ∈ RM F for ΠT and X ∗ ∈ RM F for ΠT ∗ . The final three terms were handled by a probabilistic argument: taking the averages ED and ED over the choices of the dyadic systems (with respect to an appropriate probability defined in [6], Sect. 5, or [10], Sect. 9.1), one finds that ED ED | ggood , T bad fgood | + | gbad , T fgood | + | g, T fbad | ≤ T L (Lp (μ;X)) , where can be made (by adjusting the definition of “bad”) as small as desired at the cost of increasing the above-mentioned bound C. Taking = 13 , it follows that 1 T L (Lp (μ;X)) ≤ 2| g, T f | = 2ED ED | g, T f | ≤ 2 C + T L (Lp (μ;X)) , 3 and moving over some terms completes the argument. The plan towards eliminating RMF is to apply a probabilistic argument also to the estimation of the paraproduct parts, which were the only places were RMF was needed. The argument will be different from the one above in that the operator norm T L (Lp (μ;X)) will not appear, but one gets an “honest” bound C as for the unconditional part, although only after taking the expectations ED ED .
3. Probabilistic approach to the paraproduct A few words about the notation: As usual, εk will be independent Rademacher random variables (P(εk = +1) = P(εk = −1) = 12 ) on a probability space (Ω, A , P). The Lp norm is often denoted simply be · p , with the underlying measure space understood from the expression inside the norm, in such a way that all the variables are “integrated out”. So if f : RN → X, then ε k Dk f = ε k Dk f p . f p = f Lp(μ;X) , k
p
k
L (μ⊗P;X)
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T.P. Hyt¨ onen
I consider only the paraproduct ΠT ∗ , the treatment of ΠT being completely analogous. The definition ([6], Section 8) is
g R b2 1 ΠT ∗ g := (DbQ1 )∗ T ∗ b2 = (Dbk−r )∗ vk−r b−1 2 E k g , (3.1)
b2 R good R∈D Q∈D ;Q⊂R (Q)=2−r (R)
k∈Z
where
vk−r :=
(DbQ1 )∗ T ∗ b2
R∈Dk Q∈D good ;Q⊂R (Q)=2−r (R) 1 and one observes that DbQ1 and Dbk−r , and hence their adjoints, are projections. The difficulty with the paraproduct (3.1) compared to the one in Figiel’s vectorvalued T b theorem [5], and some subsequent results preceding [6], is the presence b 1 related to the of the conditional expectations and their differences E k2 and Dbk−r different dyadic systems D and D. This destroys the martingale structure, which was successfully exploited in the earlier arguments. On the other hand, having two different systems appears to be essential in order to handle the last three terms in the expansion (2.1) by the probabilistic method. Under the assumption that T ∗ b2 ∈ L∞ (μ), the same is true uniformly for the functions vk−r . Indeed, one readily checks that the projections (Ebk1 )∗ , and hence −1 their differences (Dbk1 )∗ , are uniformly bounded on L∞ (μ). Also |b−1 by 2 | ≤ δ weak accretivity and Lebesgue’s differentiation theorem. I turn to the analysis of the paraproduct term in (2.1), substituting (3.1): b b1
ΠT ∗ ggood , fgood =
vk−r E k2 ggood , b−1 2 Dk−r fgood = I + II + III,
k
where the three terms correspond to the splitting 2 2 2 2 ) + (E k−r−1 − Ebk−r−1 ) + Ebk−r−1 E k2 = (E k2 − E k−r−1
b
b
b
b
The I term is easiest to estimate: # # # # b b2 b1
vk−r (E k2 − E k−r−1 )ggood , b−1 D f Lemma 3.1. # # gp f p . good 2 k−r k
Proof. By the usual randomization trick, # , - # # # b b2 b1 dP# εk vk−r (E k2 − E k−r−1 )ggood , ε b−1 D f LHS = # good 2 −r Ω
k b1 b2 b2 εk vk−r (E k − E k−r−1 )ggood ε b−1 D f ≤ good 2 −r
k r
j=0
k
p
p
b2 1 εk D k−j ggood ε Db−r fgood , p
p
where the bounded factors vk−r and b−1 were “pulled out” from the random2 ized sums by the contraction principle. The remaining norms are dominated by
T b Theorems
251
ggood p gp and fgood p f p , respectively, by the unconditionality of b2 1 the adapted martingale differences D k−j and Db−r . In the III term, one gets back to the martingale situation where both involved conditional expectation are related to D. # # # # b1 2 Lemma 3.2. #
vk−r Ebk−r−1 ggood , b−1 2 Dk−r fgood # gp f p . k
Proof. One starts just like in the previous proof, and estimates the factor with f in a similar way, to the result that vk−r b2 LHS εk Ek−r−1 ggood f p, Ek−r−1 b2 p k
and the factor |b2 /Ek−r−1 b2 | 1 can be pulled out by the contraction principle. One also checks the measurability condition vk−r = Ek−r−1 vk−r . This is needed to apply a UMD space-valued “Carleson embedding” (Theorem 3.4 of [6]): εk vk−r Ek−r−1 ggood sup μ(S)−1/2 1S εk vk−r ggood p . p
k
S∈D
Written out, the L2 norm above reads εk 1S k≤log2 (S)+r+1
1S
2
k−r−1≤log2 (S)
(DbQ1 )∗ T ∗ b2
R∈Dk Q∈Dgood ;Q⊂R (Q)=2−r (R)
2
εQ (DbQ1 )∗ T ∗ b2
2
Q∈D (Q)≤2(S)
≤ 1S (DbS1(1) )∗ T ∗ b2 + 2
εQ (DbQ1 )∗ T ∗ b2 ,
Q∈D;Q⊆S
2
where the first step was based on removing the restriction to good Q’s by the contraction principle, and reindexing the random signs by the cubes instead of the size of the cubes, which can be done since each x ∈ RN has only one dyadic cube of a given side-length containing it. In the second step, the only non-zero summand with (Q) = 2(S) was separated; this corresponds to Q = S (1) , the dyadic “father” of S. The first term above is immediately estimated by μ(S)1/2 (DbS1(1) )∗ T ∗ b2 ∞ μ(S)1/2 T ∗b2 ∞ , while for the second, one first applies the unconditionality of the projections DbQ1 to obtain εQ (DbQ1 )∗ T ∗ b2 (DbQ1 )∗ T ∗ b2 = 1S [I − (EbS1 )∗ ]T ∗ b2 , Q∈D Q⊆S
2
Q∈D Q⊆S
2
2
and then one again estimates by μ(S)1/2 T ∗b2 ∞ , as (EbS1 )∗ is bounded on L∞ (μ). So the desired conclusion is reached under the assumption that T ∗ b2 ∈ L∞ (μ).
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T.P. Hyt¨ onen
The probabilistic method is used for the II term, reindexed as follows: # # # # b b1
vk+1 (Ebk2 − E k2 )ggood , b−1 Lemma 3.3. ED ED # 2 Dk+1 fgood # gp f p . k
Proof. After expanding Ebk2 − E k2 = Ebk2 (I − E k2 ) + (Ebk2 − I)E k2 = b
b
b
2 Ebk2 D k−m −
b
m≥0
2 Dbk−m E k2 ,
b
m≥0
one is lead to estimate the following, hoping for summable decay in m: b2 b 2 1
vk+1 (Ebk2 D k−m − Dbk−m E k2 )ggood , Dbk+1 fgood , k
2 = Now look at the operator composition Ebk2 D k−m
b
(3.2)
Ebk2 D R2 . The b
R∈Dk−m
b D R2
range of consists of functions with a vanishing integral and supported on R. If R is a subset of some Q ∈ Dk , such functions are annihilated by Ebk2 . Hence 2 2 2 = Ebk2 χk,m D k−m = Ebk2 D k−m χk,m Ebk2 D k−m
b
b
b
where χk,m is the indicator (identified with the corresponding pointwise multi plication operator) of the union of cubes R ∈ Dk−m which intersect with the boundary of some Q ∈ Dk . Observe that χk,m ≤ χ ˜k,m , where χ ˜k,m is the indicator of {x : dist(x, ∂Q) ≤ 2k−m for some Q ∈ Dk }, which depends only on D b and not on D . Similarly, observing that D R2 annihilates functions which coincide b b 2 2 on R with a constant times b2 , it follows that Dbk−m E k2 = χk,m Dbk−m E k2 , where ˜k,m are defined analogously by interchanging the roles of D and D above. χk,m ≤ χ By the same randomization trick which started the proof of Lemma 3.1, # # # # b2 b1
vk+1 Ebk2 χk,m D k−m ggood , b−1 D f # # good 2 k+1 k
v b2 k+1 b2 ≤ εk Ek χk,m D k−m ggood f p . Ek b2 p k
The bounded multiplying functions vk+1 b2 /Ek b2 are extracted by the contraction principle, and the conditional expectations Ek by the vector-valued Stein inequality due to Bourgain ([2], Lemma 8). Using the contraction principle again, χk,m is b2 b2 replaced by χ ˜k,m and D k−m ggood by D k−m g, which depends only on D . Then for fixed D , b2 εk χ ˜k,m D k−m g ED k
≤
p
RN
RN
#p < # =s/p p /s 1/p # # b2 ED εk χ ˜k,m (x)D k−m g(x)# ∗ dP(ε) dμ(x) # Ω
X
# #p 1/p p /s b2 # # sup ED [χ ˜k,m (x)]s εk D k−m g(x)# dP(ε) dμ(x) , # ∗ k
k
Ω
k
X
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253
where s ∈ (max{p , q}, ∞) and the space X ∗ has cotype q ∈ [2, ∞) (which every UMD space does by well-known results). The last estimate above is Lemma 3.1 of [8], which permits pulling out from a randomized sum an independent multiplying sequence of Ls -integrable random variables under the stated conditions. One readily checks that for fixed x, k and m, the probability (when the dyadic system D is chosen randomly) that x is within the distance 2k−m from the boundary of some Q ∈ Dk is of the order 2−m as m → ∞. Hence it follows that b2 b2 εk χ ˜k,m D k−m g 2−m/s εk D k−m g 2−m/s gp , ED p
k
p
k
which is summable in m ∈ N as desired. Turn then to the second part in (3.2). Moving some factors to the dual side, randomizing and applying Young, it is estimated by # # # # b b1 2
Dbk−m E k2 ggood , χk,m vk+1 b−1 D f # 2 k+1 good # k
p p b b1 2 εk Dbk−m E k2 ggood + λp εk χk,m vk+1 b−1 D f ≤ λ−p 2 k+1 good , p
k
(3.3)
p
k
where λ > 0 is yet to be chosen. For the second term, one applies the contraction 1 1 by 1, χk,m by χ ˜k,m , and Dbk+1 fgood by Dbk+1 f , as principle to replace vk+1 b−1 2 before. Then, by a similar reasoning as a above, p p 1 1 ED εk χ ˜k,m Dbk+1 f 2−mp/s εk Dbk+1 f 2−mp/s f pp , p
k
p
k
where one should take s ∈ (max{p, r}, ∞) so that X has cotype r ∈ [2, ∞). 1/p −1/p , it suffices to bound (the expectation over Taking λ = 2m/(sp ) gp f p the dyadic systems of) the first term on the right of (3.3) (without the factor λ−p ) by Cm gpp , with Cm subexponential in m. To prove the mentioned bound, write b b 2 εk Dbk−m ggood p ggood p gp E k2 = I − (I − E k2 ), and observe that 2 2 2 by unconditionality. Splitting Dbk−m = Ebk−m−1 − Ebk−m , it remains to consider b 2 b2 εk Ek−m (I − E k )g as the other term is similar with m + 1 in place of m. Expand ∞ m−1 ∞ b2 b b2 b2 I − E k2 = D k− = D k− + D k−m−
=0
=0
=0
For the first m terms, one uses the contraction principle, Stein’s inequality and unconditionality, b2 b2 b2 εk Ek−m D k− ggood εk D k− ggood gp ; Ek−m b2 p p k
k
so the total contribution of these terms is dominated by mgp .
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2 2 2 2 D k−m− = Ebk−m χk−m, D k−m− , which was already hanAnd finally, Ebk−m dled earlier in the proof by the probabilistic method to the result that b2 2 ED εk Ebk−m χk−m, D k−m− ggood 2−/s gp ;
b
k
b
p
so the expectations of the remaining terms are summable to Cgp . Thus the final claim, which was left, has been verified with the subexponential constant Cm ≤ C(m + 1), and the proof is complete. Remark 3.4. If μ satisfies the doubling property, the results of this section could be extended to T ∗ b2 ∈ BMO(μ). But, as Theorem 1.1 states, the doubling situation was already handled in a different way in [6] in any case.
References [1] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21 (1983), no. 2, 163–168. [2] , Vector-valued singular integrals and the H 1 -BMO duality. Probability theory and harmonic analysis (Cleveland, Ohio, 1983), 1–19, Dekker, New York, 1986. [3] D. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions. Conf. on harmonic analysis in honor of A. Zygmund, Vol. I, II (Chicago, 1981), 270–286, Wadsworth, Belmont, CA, 1983. [4] G. David, J.-L. Journ´e, S. Semmes, Op´erateurs de Calder´ on–Zygmund, fonctions para-accr´ etives et interpolation. Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56. [5] T. Figiel, Singular integral operators: a martingale approach. Geometry of Banach spaces (Strobl, 1989), 95–110, Cambridge Univ. Press, Cambridge, 1990. [6] T. Hyt¨ onen, The vector-valued non-homogeneous T b theorem. Preprint, arXiv:0809. 3097, 2008. [7] , A. McIntosh, P. Portal, Kato’s square root problem in Banach spaces. J. Funct. Anal. 254 (2008), no. 3, 675–726. [8] , M. Veraar, R-boundedness of smooth operator-valued functions, Integral Equations Operator Theory 63 (2009), no. 3, 373–402. [9] F. Nazarov, S. Treil, A. Volberg, Accretive system T b-theorems on nonhomogeneous spaces. Duke Math. J. 113 (2002), no. 2, 259–312. [10] , , , The T b-theorem on non-homogeneous spaces. Acta Math. 190 (2003), no. 2, 151–239. Tuomas P. Hyt¨ onen Department of Mathematics and Statistics University of Helsinki Gustaf H¨ allstr¨ omin katu 2b FI-00014 Helsinki, Finland e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 255–269 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Some Recent Applications of Bilinear Integration Brian Jefferies Abstract. Three topics featuring bilinear integration are described: the noncommutative Feynman-Kac formula, the connection between stationary state and time-dependent scattering theory and the stochastic integration of vectorvalued processes. Mathematics Subject Classification (2000). Primary 28B05; Secondary 46B28, 46G10, 46N50, 60H05. Keywords. Bilinear integration, semivariation, random evolution, scattering theory, stochastic integral.
1. Introduction In the survey article [9], published nearly thirty years ago now, I. Kluv´ anek outlined a number of themes of vector measure theory that have applications to other areas of mathematics and the physical sciences. The present article deals with three of these themes, all related to bilinear integration, that is, the integration of a vector or operator-valued function with respect to a vector or operator-valued measure. The earliest account of bilinear integration with respect to a vector measure m was given by R. Bartle [2]. There convergence in semivariation of m with respect to a given bilinear map is used to approximate integrable functions. Later, I. Dobrakov [3], [4] considered the integration of vector-valued functions with respect to operator-valued measures and considerably relaxed the approximation of integrable functions to convergence almost everywhere by simple functions based on sets with finite bilinear semivariation of m. For the three applications of bilinear integration outlined below, it turns out that there may be too few sets of finite bilinear semivariation or the bilinear semivariation is not defined. However, an appeal to an underlying tensor product can be exploited. For the first application, the perturbation of semigroups, see The support of an FRG grant from UNSW is gratefully acknowledged.
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[9, Example 4, p. 125], we need to integrate an operator-valued function F : Ω → L(X) with respect to an operator-valued measure M : S → L(Lp (μ)). Here X is a Banach space and Lp (μ) is the space of pth-integrable functions (1 ≤ p < ∞) with respect to a σ-finite measure μ. The indefinite integral takes values in the space of continuous linear operators on the Bochner space Lp (μ, X) of X-valued pth-integrable functions. In this context, the bilinear map (A, B) −→ A ⊗ B, A ∈ L(X), B ∈ L(Lp (μ)), need not have values in the space of continuous linear operators on Lp (μ, X) unless, say, X is Hilbert space and p = 2. Theme 8 of Kluv´ anek’s survey refers directly to bilinear integration and its application to the connection between stationary and time-dependent scattering theory in quantum mechanics, as described in the paper [1] of Amrein, Georgescu and Jauch. A closer examination of [1] reveals that the theory requires the integration of an operator-valued function with respect to a vector measure formed by letting a spectral measure act on a fixed vector, all taking place in Hilbert space. Such a vector measure will typically have infinite bilinear semivariation. Although Bartle’s integral is mentioned in [1], a type of Riemann approximation to the integrals in question is used there instead. The approach advocated here is to “decouple” the bilinear integral into a tensor product integral to which the operator-vector evaluation map may be applied. The application of Fubini’s Theorem in the tensor product integral reduces to the scalar case and this is the essence of the connection between stationary and time-dependent scattering theory. Stochastic integration was mentioned in Theme 5 of Kluv´ anek’s survey. Recently a theory of stochastic integration of vector-valued processes with respect to Brownian motion was developed by van Neerven, Veraar and Weis [13]. With the assumption that the underlying Banach space E has the UMD property, a key feature of the theory of [13] is that the stochastic integral for suitable E-valued processes can be “decoupled” into a tensor product integral to which the pointwise multiplication map on the sample space may be applied. A common theme of these examples is the crucial role played by the underlying tensor product in the integration process in the absence of sufficiently many sets with finite bilinear semivariation. In Section 2 below, the formal definition of bilinear integration of vector- and operator-valued functions with respect to vector- and operator-valued measures in tensor products is set out. A simple example shows that the bilinear semivariation of a vector measure may have only the values zero and infinity. A random evolution described in Section 3 is a family of operator-valued random variables modelling an evolution influenced by some random changes in the environment. Integrals with respect to certain operator-valued measures represent the superposition of two evolutions along the lines of [9, Example 4, p. 125], where only scalar integrands were considered. Because the component evolutions act on different spaces, bilinear integration in tensor products arises naturally in this context. The use of bilinear integration in scattering theory is outlined in Section 4. This is forthcoming joint work of the author with L. Garcia-Raffi and S. Okada, so the reduction to the case where we can apply the scalar version of Fubini’s Theorem is considered
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only for bounded selfadjoint operators. The full force of the argument requires the development of an underlying tensor product topology suited to unbounded operators. The decoupled nature of a bilinear integral in a tensor product is an explicit feature of the approach to stochastic integration of vector-valued functions of [13], which is outlined in Section 5. However, there is no space to mention γradonifying maps which is also an essential feature of the work [13].
2. Bilinear integration in tensor products We now turn to the definition of bilinear integration in tensor products. Let X and Y be Banach spaces. We shall be looking at the integration of an X-valued function f : Ω → X with respect to a vector measure m : S → Y , that is, a σ-additive, Y -valued set function defined on the measurable space (Ω, S). The ˆ τ Y takes values in some complete normed indefinite integral f ⊗ m : S → X ⊗ ˆ τ Y of X and Y . Related to this context, we also integrate tensor product X ⊗ an operator-valued function F : Ω → L(X) with respect to an operator-valued measure M : S → L(Y ), σ-additive for the strong operator topology on the space L(Y ) of continuous linear operators on Y . We then seek conditions for which the ˆ τ Y ) has values in the space of continuous indefinite integral F ⊗ M : S → L(X ⊗ ˆ τ Y and the identity linear operators on X ⊗ (F ⊗ M )(S) (x ⊗ y) = (F x) ⊗ (M y) (S) holds for every x ∈ X, y ∈ Y and S ∈ S. Suppose that τ is the topology defined on X ⊗ Y by a norm · τ with the property that there exists C > 0 such that (T1) x ⊗ yτ ≤ Cx y for all x ∈ X and y ∈ Y , and (T2) X ⊗ Y may be identified with a linear subspace of the continuous dual ˆ τ Y ) of X ⊗τ Y and x ⊗ y ≤ Cx y for all x ∈ X (X ⊗τ Y ) = (X ⊗ and y ∈ Y . If conditions (T1), (T2) hold, then τ is said to be a norm tensor product ˆ π Y of the topology on X ⊗ Y . For example, the projective tensor product X ⊗ Banach spaces X and Y is obtained by taking the completion of the linear space X ⊗ Y with respect to the norm ⎧ ⎫ ⎨ ⎬ uπ = inf xj .yj : u = xj ⊗ yj , u ∈ X ⊗ Y. ⎩ ⎭ j
j
The infimum is over all possible representations of u in the linear space X ⊗Y . It is ˆ π Y can be represented as u = ∞ well known that any element u of X ⊗ j=1 xj ⊗ yj with ∞ x .y < ∞ [10, 41.2]. j j j=1 We shall also suppose that the norm tensor product topology τ on X ⊗ Y is completely separated in the sense that the subspace X ⊗ Y of (X ⊗τ Y ) separates ˆ τ Y of the normed space X ⊗τ Y , that is, if u ∈ X ⊗ ˆ τ Y and the completion X ⊗
u, x ⊗ y = 0 for all x ∈ X and y ∈ Y , then u = 0.
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The injective tensor product topology is always completely separated, [10, 45.4]. If one of the Banach spaces X, Y has the approximation property, then the projective tensor product topology on X ⊗Y is completely separated, [10, 43.2.(7)]. Let (Σ, E, μ) be a σ-finite measure space and 1 ≤ p < ∞. Another important example is provided by the space Lp (μ, X) of all strongly μ-measurable X-valued functions that are pth integrable with respect to μ. Then Lp (μ, X) can be identified with the completion of X ⊗ Lp (μ) with respect to the relative norm of Lp (μ, X), which defines a completely separated norm tensor product topology Δp on X ⊗ Lp (μ). Definition 2.1. Let (Ω, S) be a measurable space, and let X and Y be Banach spaces. Suppose that τ is a completely separated, norm tensor product topology on X ⊗ Y . Let m : S → Y be a vector measure. A function f : Ω → X is said to be ˆ τ Y if there exist X-valued S-simple functions φj , j = 1, 2, . . . , m-integrable in X ⊗ ˆ such that φj → f (m-a.e.) as j → ∞, and {(φj ⊗ m)(A)}∞ j=1 converges in X ⊗τ Y for each A ∈ S. Let (f ⊗ m)(A) = A f (ω) ⊗ dm(ω) denote this limit. Sometimes, we write m(f ) for the definite integral (f ⊗ m)(Ω). An operator-valued function F : Ω → L(X) is said to be integrable in ˆ τ Y ) with respect to an operator-valued measure M : S → L(Y ), σ-additive L(X ⊗ for the strong operator topology on L(Y ), if for every x ∈ X and y ∈ Y the Xˆ τ Y with respect to valued function F x : ω −→ F (ω)x, ω ∈ Ω, is integrable in X ⊗ the Y -valued measure M y : S −→ M (S)y, S ∈ S, and for each S ∈ S there exists ˆ τ Y ) such that an operator (F ⊗ M )(S) ∈ L(X ⊗ (F ⊗ M )(S) (x ⊗ y) = (F x) ⊗ (M y) (S) (2.1) holds for every x ∈ X and y ∈ Y . ˆ τ Y → X⊗ ˆ τ Y is uniquely The continuous linear operator (F ⊗ M )(S) : X ⊗ ˆτY . determined by equation (2.1) for each S ∈ S, because X ⊗ Y is dense in X ⊗ ˆ The set function F ⊗ M : S → L(X ⊗τ Y ) is σ-additive for the strong operator topology by the Vitali-Hahn-Saks theorem [6, Lemma 4.3.3]. The size of a vector measure m : S → Y is estimated by its scalar semivariation m(S) = sup j cj m(Sj )Y . The supremum is over scalars cj with |cj | ≤ 1 for j = 1, . . . , n and partitions {S1 , . . . , Sn } of S by elements of S for n = 1, 2, . . . . Two notable approaches to bilinear integration [2],[3] use the concept of vector semivariation, defined in the tensor setting as follows. Let m : S → Y be a vector measure and τ a tensor product topology on X ⊗Y as above. The X-semivariation ˆ τ Y is the set function βX (m) : S → [0, ∞] defined by of m in X ⊗ ⎧ ⎫ ⎬ k ⎨ xj ⊗ m(Sj ) βX (m)(S) = sup (2.2) ⎭ ⎩ τ j=1
for every S ∈ S; the supremum is taken over all pairwise disjoint sets S1 , . . . , Sk from S ∩ S and vectors x1 , . . . , xk from X, such that xj X ≤ 1 for all j = 1, . . . , k and k = 1, 2, . . . .
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We note here a simple example taken from [7, Example 4.1] of a vector measure m whose X-semivariation takes only the values zero and infinity. Example. Let m : B([0, 1]) → L2 ([0, 1]) be the vector measure defined by m(B) = χB for every B ∈ B([0, 1]), the Borel σ-algebra of the interval [0, 1]. Then the L2 ([0, 1])-semivariation of m in the projective tensor product L2 ([0, 1])⊗π L2 ([0, 1]) (see [10, 41.2]) is infinite on any Borel set A with positive Lebesgue measure |A|. To see this, let n be any positive integer and suppose that Aj , j = 1, . . . , n, are pairwise disjoint subsets of A, with Lebesgue measure |A|/n. For each j = 1, . . . , n, let φj = (n/|A|)1/2 χAj . Then the map Φ : L2 ([0, 1]) ⊗π L2 ([0, 1]) → C, defined 1 by Φ(f ⊗ g) = 0 f (t)g(t) dt for every f ∈ L2 ([0, 1]) and every g ∈ L2 ([0, 1]), is continuous by the Cauchy-Schwarz inequality but n n 1 Φ φj ⊗ m(Aj ) = φj .m(Aj ) dλ = |A|1/2 n1/2 . j=1
j=1
0
Because φj 2 = 1 for j = 1, . . . , n and n is any positive integer, the L2 ([0, 1])semivariation of m in L2 ([0, 1]) ⊗π L2 ([0, 1]) is infinite on A. For the vector measure m of the example above, the only L2 -valued functions ˆ π L2 ([0, 1]) in the sense of [2] or [3] are the null functions. m-integrable in L2 ([0, 1])⊗ Nevertheless, using Definition 2.1, the class of L2 -valued functions m-integrable ˆ π L2 ([0, 1]) can be put into one-to-one correspondence almost everyin L2 ([0, 1])⊗ where with the collection of kernels of trace class operators on L2 ([0, 1]) in an obvious way, see [7, Proposition 4.2].
3. Random evolutions The Feynman-Kac formula is used in the theory of stochastic processes to represent solutions of the heat equation with a source or sink term and finds widespread applications in mathematical finance and mathematical physics. I. Kluv´anek pointed out a long time ago [9] that the Feynman-Kac formula may also be viewed as a means to represent the superposition of two general evolutions and is readily interpreted in terms of integration with respect to operator-valued measures. A random evolution Ft : Ω → L(X ), t ≥ 0, is a family of operator-valued random variables acting on some Banach space X of states, whose evolution is influenced by some random changes in the environment. Random events are measured by a family of operator-valued measures Mt : σ(St ) → L(Lp (μ)), t ≥ 0. In order to represent the superposition of the associated evolutions, we need to integrate the L(X )-valued random variable Ft : Ω → L(X ) with respect to the L(Lp (μ))-valued measure Mt : σ(St ) → L(Lp (μ)) to obtain an integral Ft (ω) ⊗ dMt (ω), R(t) = Ω
for each t ≥ 0. Then R is a semigroup of continuous linear operators acting on the function space Lp (μ, X ). The integral of the operator-valued function with respect
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to the operator-valued measure is in the sense of Definition 2.1. The operatorvalued function Ft takes values in the space L(X ), while the values of the operatorvalued measure Mt act on the space Lp (μ), so we are dealing with a type of “decoupled” integral in that the values of Mt do not act on the values of Ft by operator composition. Let Y be a Banach space. An L(X )-valued spectral measure Q on E is a map Q : E → L(Y) that is σ-additive in the strong operator topology and satisfies Q(Σ) = IdY and Q(A ∩ B) = Q(A)Q(B) for all A, B ∈ E. In quantum theory, Q is typically multiplication by characteristic functions associated with the position observables, but the spectral measures associated with momentum operators also appear. Let (Σ, E) be an measurable space. For each s ≥ 0, suppose that Ss is a σ-algebra of subsets of a nonempty set Ω such that Ss ⊆ St for every 0 ≤ s < t. For every s ≥ 0, there are given functions Xs : Ω → Σ with the property that Xs−1 (B) ∈ St for all 0 ≤ s ≤ t and B ∈ E. It follows that the cylinder sets E = {Xt1 ∈ B1 , . . . , Xtn ∈ Bn } : = {ω ∈ Ω : Xt1 (ω) ∈ B1 , . . . , Xtn (ω) ∈ Bn } = Xt−1 (B1 ) ∩ · · · ∩ Xt−1 (Bn ) 1 n
(3.1)
belong to St for all 0 ≤ t1 < · · · < tn ≤ t and B1 , . . . , Bn ∈ E. A semigroup S of operators acting on Y is a map S : [0, ∞) → L(Y) such that S(0) = IdY , the identity map on Y and S(t + s) = S(t)S(s) for all s, t ≥ 0. The semigroup S represents the evolution of state vectors belonging to Y. Now suppose that Mt : St → L(Y) is a σ-additive operator-valued set function. Adopting the terminology of [6], the system (Ω, St t≥0 , Mt t≥0 ; Xt t≥0 ) is called a time homogeneous Markov evolution process if there exists a L(Y)-valued spectral measure Q on E and a semigroup S of operators acting on Y such that for each t ≥ 0, the operator Mt (E) ∈ L(Y) is given by Mt (E) = S(t − tn )Q(Bn )S(tn − tn−1 ) · · · Q(B1 )S(t1 )
(3.2)
for every cylinder set E ∈ St of the form (3.1) and the process is called an (S, Q)process. The basic ingredients are the semigroup S describing the evolution of states and the spectral measure Q describing observation of states represented by vectors in Y. For our purposes, attention is restricted to the Banach space Y = Lp (μ) with 1 ≤ p < ∞ and μ a σ-finite measure. Now let X be a Banach space. A multiplicative operator functional is a measurable mapping Ft : Ω → L(X ), t ≥ 0, such that a.e., (i) t → Ft (ω) is continuous for the weak operator topology (ii) Fs+t (ω) = Fs (θt ω)Ft (ω), F0 (ω) = IdX Here θt : Ω → Ω is a shift map: Xs+t (ω) = Xs (θt ω). If X = R and V : Σ → R t is a suitable measurable function, Ft = e− 0 V ◦Xs ds is an example. For a random evolution, the operators in (ii) are usually written in the opposite order.
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With the right idea of bilinear integration, it is straightforward that for an integrable multiplicative operator functional Ft , t ≥ 0, the formula R(t) = F (ω) ⊗ dM t t (ω), t ≥ 0, gives a semigroup R of continuous linear operators Ω acting on Lp (μ, X ). Proving that Ris a C0 -semigroup requires additional assumptions [6, Chapter 5]. For example, j Aj ⊗ Mt (Ej ) should form a bounded subset of L(Lp (μ, X )) if Aj L(X ) ≤ 1 and (Ej ) are pairwise disjoint, that is, Mt has finite L(X )semivariation in L(Lp (μ, X )). The boundedness properties are satisfied if Mt comes from a Markov process or is otherwise dominated by a positive L(Lp (μ))-valued measure. In the case that Q is the spectral measure of multiplication by characteristic functions acting on Lp (μ), a necessary and sufficient condition that Mt is dominated by a positive L(Lp (μ))-valued measure is that S is a dominated semigroup of operators on Lp (μ) [8, Theorem 3.1]. The following example is from [6, Theorem 5.3.3]. Example. Let cj : R → (0, ∞), j = 1, 2, . . . , be continuous functions such that supx,j cj (x) < ∞ and inf x cj (x) > 0. Suppose that A = {ajk } is an infinite matrix such that it and its transpose define bounded linear operators on ∞ . Then the solution to the equations ∞
∂uj ∂uj (t, x) = cj (x) (t, x) + ajk uk (t, x) ∂t ∂x k=1
uj (0, x) = fj (x),
j = 1, 2, . . .
fj 22 + fj 22 < ∞ can be written as uj (t, x) = ([ Ω Ft ⊗ dMt ]f )j (x). Here Σ = N, X = L2 (R), the multiplicative operator functional Ft is constructed from the operators Vj : f → cj f and Mt is the (S, Q, t)-measure constructed from S(t) = eAt on 2 and Q multiplication by characteristic functions. The sample space Ω consists of piecewise constant paths. Now {ajk } need not be the generator of a Markov process.
with
∞
j=1
4. Scattering theory The connection between the two main approaches to the mathematical formulation of quantum mechanical scattering theory is considered in an old paper by Amrein, Georgescu and Jauch [1]. In the time-dependent scattering theory, the actual time evolution of a wave packet for an interacting particle is studied. In stationary state scattering theory, the behaviour of the resolvent of the Hamiltonian operator is analysed at points of the continuous spectrum. In [1], the authors establish the connection between the two approaches to scattering theory by reducing the problem to a type of Fubini theorem for operatorvalued Riemann integrals. Without setting down all the details, the aim of this section is to describe how the Amrein-Georgescu-Jauch analysis can be viewed
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in terms of a simple Fubini theorem for bilinear integrals of the type given in Definiton 2.1. This is joint work of the author with L. Garcia-Raffi and S. Okada. Let H be a Hilbert space. The free Hamiltonian H0 : D(H0 ) → H is a selfadjoint operator with absolutely continuous spectrum. The interaction of the system is given by a symmetric operator V : D(V ) → H such that D(H0 ) ⊂ D(V ) and the Hamiltonian operator H = H0 + V is assumed to be selfadjoint with domain D(H0 ). The operator V is usually associated with multiplication by a potential function but its exact form will not concern us. We would like to be able to integrate the function f defined by f (t, λ) = Vt∗ V e−i(λ−i)t ∈ L(D(H0 ), H) for t ≥ 0 and λ ∈ R, where Vt = e−it(H0 +V ) for t ∈ R. The proof of Equation (34) of [1] involves interchanging integrals with respect to t and λ variables. Other identities proved in [1] may be obtained by varying the arguments used to treat the operator-valued function f . First, if F is the spectral measure associated with H0 and ψ is a vector belonging to the domain D(H0 ) of H0 , then we expect that f (t, λ) d(F ψ)(λ) = Vt∗ V e−i(λ−i)t d(F ψ)(λ) R R = Vt∗ V e−i(λ−i)t d(F ψ)(λ) (4.1) =e
−t
Integrating with respect to t, we have ∞ f (t, λ) d(F ψ)(λ) dt = R
0
R ∗ Vt V e−itH0 ψ.
∞
e−t Vt∗ V e−itH0 ψ dt
0
for each ψ ∈ D(H0 ). On reversing the order of integration, we need to integrate the ∞ L(D(H0 ), H)-valued function λ −→ 0 Vt∗ V e−i(λ−i)t dt, λ ∈ R with respect to the D(H0 )-valued measure F ψ, so that there is a genuine problem of interpretation. Because F is a spectral measure, the traditional approaches to bilinear integration [2],[3] are not well adapted to integrating an operator-valued function with respect to the vector measure F ψ. Typically, the H-valued measure A −→ (F ψ)(A), A ∈ B(R), will have infinite variation on every set of positive measure, so the theory of [2],[3] is inapplicable. One approach is suggested by equation (4.1), where for each t ∈ R+ , the operator Vt∗ V is taken outside the integral, that is, the integrals with respect to t and λ are “decoupled” when the iterated integral is written in this order. Operator integrals are also “decoupled” in the random evolutions considered previously in Section 3. If we use tensor product notation, then we have ∞ ∞ f (t, λ) ⊗ d(F ψ)(λ)dt = e−t Vt∗ V ⊗ e−itH0 ψ dt. 0
R
0
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Applying the evaluation map J : T ⊗ x −→ T x, informally we have ∞ ∞ f (t, λ) d(F ψ)(λ) dt := J f (t, λ) ⊗ d(F ψ)(λ)dt R 0 0 ∞ R =J e−t Vt∗ V ⊗ (e−itH0 ψ) dt 0 ∞ = e−t J Vt∗ V ⊗ (e−itH0 ψ) dt 0 ∞ = e−t Vt∗ V e−itH0 ψ dt. 0
Writing the iterated integrals in the opposite order: ∞ ∞ f (t, λ) dt d(F ψ)(λ) := J f (t, λ)dt ⊗ d(F ψ)(λ) R 0 R 0 ∞ −t ∗ −itλ =J e Vt V e dt ⊗ d(F ψ)(λ) (4.2) R 0 ∞ = J e−t Vt∗ V e−itλ dt ⊗ d(F ψ)(λ) R ∞ 0 −t ∗ −itλ = e Vt V e dt d(F ψ)(λ). (4.3) R
0
Making sense of the last two equations poses a problem.One solution is to define ∞ the integral (4.3) of the operator-valued function λ −→ 0 Vt∗ V e−i(λ−i)t dt, λ ∈ R with respect to the vector measure F ψ to be equal to the element of H on the right-hand side of equation (4.2) and then appeal to a vector version of Fubini’s theorem for tensor product-valued integrals – we are “decoupling” the integrals and then applying the evaluation map J of operators acting on vectors. The map J : T ⊗ x −→ T x, x ∈ D(H0 ), T ∈ L(D(H0 ), H) extends linearly to the vector space L(D(H0 ), H) ⊗ D(H0 ) of all finite linear combinations j cj (Tj ⊗ the limit xj ) of tensor products Tj ⊗ xj . Integrating a function involves taking ∞ −t of integrals of a sequence of elementary functions, so the integral 0 e Vt∗ V ⊗ ˜ (e−itH0 ψ) dt will belong to some suitable completion L(D(H0 ), H)⊗D(H 0 ) of the linear space L(D(H0 ), H) ⊗ D(H0 ). The choice of a suitable complete linear tensor ˜ product space L(D(H0 ), H)⊗D(H 0 ) is crucial Let us look at a simple (non-physical) example. Suppose that H0 and V are bounded selfadjoint operators acting on the Hilbert space H. Let Vt = e−it(H0 +V ) ˜ in which the tensor product as above. We are now seeking a linear space L(H)⊗H integral ∞ e−t Vt∗ V ⊗ (e−itH0 ψ) dt (4.4) 0
belongs, and for which the evaluation map J : L(H) ⊗ H → H has a continuous ˜ linear extension from L(H) ⊗ H to L(H)⊗H. An obvious candidate is obtained by
264
B. Jefferies
noting that for Tj ∈ L(H) and xj ∈ H, j = 1, . . . , n, we have n n n J Tj ⊗ xj ≤ Tj xj ≤ Tj .xj , j=1
j=1
j=1
for all n = 1, 2, . . . . Then the evaluation map J : T ⊗ x −→T x has a unique ˜ = ∞ Tj xj for any ˆ π H → H given by Ju continuous linear extension J˜ : L(H)⊗ j=1 ∞ ∞ ˆ π H with j=1 Tj .xj < ∞. representation u = j=1 Tj ⊗ xj of u ∈ L(H)⊗ We now see that under the assumptions that H0 and V are bounded selfadjoint operators, the integral (4.4) is actually the Bochner integral in the projective ˆ π H of the L(H) ⊗ H-valued function tensor product L(H)⊗ t −→ e−t Vt∗ V ⊗ (e−itH0 ψ), t > 0.
(4.5)
Because H0 and V are bounded linear operators, the L(H)-valued functions t −→ Vt∗ V , t > 0, and t −→ e−itH0 , t > 0, are continuous for the uniform operator topology. It follows that the L(H) ⊗ H-valued function t −→ Vt∗ V ⊗ (e−itH0 ψ), t > 0, is continuous for the projective tensor product norm · π and ∞ e−t Vt∗ V .(e−itH0 ψ) dt ≤ V ψ/. 0
Consequently, the function (4.5) is Bochner integrable in the projective tensor ˆ π H. Moreover, the continuous H-valued function product L(H)⊗ t −→ e−t Vt∗ V (e−itH0 ψ), t > 0, is Bochner integrable in H and the equalities ∞ ∞ −t ∗ −itH0 J e Vt V ⊗ e ψ dt = e−t J Vt∗ V ⊗ e−itH0 ψ dt 0 0 ∞ = e−t Vt∗ V e−itH0 ψ dt 0
hold. Next, we see that the integral ∞ −t ∗ −itλ e Vt V e dt ⊗ d(F ψ)(λ) R
(4.6)
0
ˆ π H and that the exists as an element of the projective tensor product L(H)⊗ equality ∞ ∞ −t ∗ −itλ e Vt V e dt ⊗ d(F ψ)(λ) = e−t Vt∗ V ⊗ e−itH0 ψ dt R
0
0
is valid, so that the equality ∞ −t ∗ −itλ J e Vt V e dt ⊗ d(F ψ)(λ) = R
0
0
∞
e−t Vt∗ V (e−itH0 ψ) dt.
(4.7)
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also holds. In fact, for x, h, ξ ∈ H, the scalar version of Fubini’s Theorem implies that ∞ −t ∗ −itλ e Vt V x, h e dt d F ψ, ξ (λ) S 0 ∞ = e−t Vt∗ V x, h e−itH0 F (S)ψ, ξ dt 0 ∞ −t ∗ −itH0 = e Vt V ⊗ (e F (S)ψ) dt, x ⊗ h ⊗ ξ , 0
ˆ π H too and equation (4.7) is valid. so the integral (4.6) must belong to L(H)⊗ The above argument breaks down for unbounded H0 and V because the t −→ Vt∗ V , t > 0, and t −→ e−itH0 , t > 0, are no longer continuous for the uniform operator topology. We can only expect them to be continuous in the strong operator topology on their respective domains. Moreover, strong measurability in the uniform operator norm no longer holds, so applying the Bochner integral in the Banach space L(H) is no longer an option. For unbounded densely defined selfadjoint operators H0 and V , in joint work with L. Garcia-Raffi and S. Okada we aim to modify the preceding argument by replacing the uniform operator topology of L(H) in the projective tensor product ˆ π H by the strong operator topology on the appropriate space of operators. L(H)⊗ ˆ π H is auxiliary to the final definition of The projective tensor product L(H)⊗ the integral (4.3), but it is a feature of the “decoupling” approach to bilinear integration that allows the integration of operator-valued functions with respect to spectral measures.
5. Bilinear integration with respect to white noise In this section, we describe how the approach to the stochastic integration of Banach space-valued processes developed by van Neerven, Veraar and Weis [13] is related to bilinear integration as described in Definition 2.1. Let (Σ, S, μ), (Ω, E, P) be probability measure spaces. Let L0 (Ω, E, P) be the space of real-valued random variables with respect to P equipped with convergence in P-measure. A mapping W : S → L0 (Ω, E, P) is called a Gaussian random measure on (Σ, S, μ) if (a) for all disjoint sets A1 , A2 , . . . from S, we have W
∞ ) n=1
∞ An = W (An ), n=1
where the sum converges with probability one; (b) the random variables W (A1 ), . . . , W (An ) are independent for all disjoint sets A1 , . . . , An belonging to S and all n = 1, 2, . . . ;
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B. Jefferies
(c) for every A ∈ E, the random variable W (A) has a normal distribution with mean zero and variance μ(A). We note the following lemma. Lemma 5.1 ([11], Lemma 2.1). Let Xn be symmetric Gaussian random vectors such that Xn → X in probability P. Then X is a symmetric Gaussian random vector and, for every 0 < p < ∞, EXn − Xp → 0 as n → ∞. Thus, W is a vector measure in any of the spaces Lp (Ω, E, P) for 0 ≤ p < ∞. Let E be a Banach space and 1 ≤ p < ∞. The Banach space of all equivalence classes of strongly P-measurable functions f : Ω → E for which Ω f p dP < ∞ is denoted by Lp (Ω, E, P; E) or just Lp (P; E). The norm is given by 1/p f −→ f p dP , f ∈ Lp (P; E). Ω
The relative topology of Lp (P; E) on Lp (Ω, E, P) ⊗ E is completely separated , so we may consider the integral of an E-valued function f : Σ → E with respect to W in Lp (P; E), in the sense of Definition 2.1. Then for each ξ ∈ E , the scalar function f, ξ : σ → f (σ), ξ , σ ∈ Σ, is W -integrable and by the Itˆ o isometry, the random variable Σ f, ξ dW is Gaussian with variance Σ | f, ξ |2 dμ. By [11, Corollary 4.2], an E-valued function f : Σ → E, integrable with respect to W in Lp (P; E) as in Definition 2.1 is necessarily integrable with respect to the Gaussian random measure W in the sense of [11, Definition 2.1]. The converse statement follows from the proof of [11, Theorem 4.1] and [11, Corollary 2.1]. It follows the lemma above, that if f : Σ → E is integrable with respect to W in Lp (P; E) for some 1 ≤ p < ∞, then it is integrable with respect to W in Lp (P; E) for every 1 ≤ p < ∞. A function f : Σ → E is integrable with respect to W in Lp (P; E) for some 1 ≤ p < ∞, if and only if it is strongly measurable and stochastically integrable with respect to W as in [12, Definition 2.1]. It follows from [11, Proposition 6.1] that W has finite E-semivariation in Lp (P; E) if and only if E is a Banach space of type 2. Let (Ω , E , P ) be another probability measure space. Then strongly measurable functions ϕ : Σ → Lp (P ; E) may also be integrated with respect to W in the space Lp (P ⊗ P; E), in the sense of Definition 2.1. If ϕ : Σ → Lp (P ; E) is strongly measurable, then an appeal to [11, Corollary 4.2] shows that ϕ is W -integrable in Lp (P ⊗ P, E), if and only if the E-valued function t −→ ϕ(t), g , t ∈ Σ, is W -integrable in Lp (P ; E) for every g ∈ Lq (P ) p and for every set A that ∈ S, there exists (ϕ ⊗ Wp)(A) ∈ L (P ⊗ P, E)q such
(ϕ ⊗ W )(A), g = A ϕ(t), g dW (t) P-a.e. in L (P ; E) for each g ∈ L (P ). Here 1/p + 1/q = 1 and · , g denotes integration against g in the P -variable. By applying the pointwise multiplication map to the element (ϕ ⊗ W )(A) of Lp (P ⊗ P, E), we obtain the stochastic integral of a strongly measurable function ϕ : Σ → Lp (P ; E) with respect to the Gaussian random measure W . For example, ˆ π L2 (P) denotes the projective tensor product of L2 (P, E) with L2 (P) if L2 (P, E)⊗ [10, 41.2], then an application of the Cauchy-Schwarz inequality shows that there
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ˆ π L2 (P) → L1 (P, E) such that exists a unique continuous linear map J : L2 (P, E)⊗ 2 for every f, g ∈ L (P) and x ∈ E, the equality J((x.g) ⊗ f )(ω) = x.g(ω)f (ω) holds for almost every ω ∈ Ω, that is, J is the pointwise multiplication map. Requiring that the bilinear integral (ϕ ⊗ W )(A) belongs to the projective ˆ π L2 (P) is a restrictive assumption. It is remarkable that tensor product L2 (P, E)⊗ under mild assumption on the Banach space E, the multiplication map J is continuous for the relative topology of L2 (P ⊗ P, E) into L2 (P, E), provided it is restricted to a certain class of definite integrals (ϕ ⊗ W )(A) with respect to a Wiener process W . Let Σ = R+ with μ Lebesgue measure on R+ . If Wt is Wiener process, then we write W ((s, t]) = Wt − Ws for 0 ≤ s < t. For each t > 0, let Ft be the σ-algebra generated by the random variables {Ws }0≤s≤t . A function φ : R+ → L2 (P) ⊗ E is said to be an elementary progressively measurable function if there exist times 0 < t1 < · · · < tN , vectors xmn ∈ E and sets Amn ∈ Ftn−1 , n = 1, . . . , N , m = 1, . . . , M such that φ(t) =
M N
xmn χ
n=1 m=1
Amn
.χ
(tn−1 ,tn ]
t ∈ R+ .
(t),
Then φ has values in every space Lp (P) ⊗ E for 1 ≤ p ≤ ∞, φ is W -integrable in Lp (P) ⊗ E ⊗ Lp (P) for every 1 ≤ p < ∞ and we have M N φ ⊗ dW = (xmn χ ) ⊗ (Wtn − Wtn−1 ). (5.1) R+
n=1 m=1
Amn
Let X denote the linear subspace of L∞ (P)⊗E⊗Lp(P) consisting of all vectors φ ⊗ dW with φ : R+ → L∞ (P) ⊗ E an elementary progressively measurable R+ function. For each 1 ≤ p < ∞, let J : L∞ (P) ⊗ E ⊗ Lp (P) → Lp (P, E) be the linear map defined by J(g ⊗ x ⊗ f )(ω) = xg(ω).f (ω) for almost all ω ∈ Ω.
Definition 5.2. A Banach space E is called a UMD space (or, E has the unconditional martingale difference property) if for any 1 < p < ∞, there exists Cp > 0 such that for any E-valued martingale difference {ξj }nj=1 and n = 1, 2, . . . , the inequality p n n p E j ξj ξj ≤ Cp E j=1 j=1 E
E
holds for every j ∈ {±1}, j = 1, . . . , n. The following result is from [5, Theorems 2 and 2’]. The simplified proof presented below comes from [13, Lemma 3.4]. It is given here to spell out the connection with bilinear integration in tensor products. Theorem 5.3. Let E be a UMD space and 1 < p < ∞. The multiplication map J is continuous from X into Lp (P, E) for the relative topology of Lp (P ⊗ P, E) on X.
268
B. Jefferies
Proof. Suppose that R+ φ ⊗ dW is given by formula (5.1). For each n = 1, . . . , N , M let ξn = m=1 (xmn χ ) and define Amn
dn = ξn .W (tn ) − ξn .W (tn−1 ) ⊗ 1
(5.2)
en = ξn ⊗ W (tn ) − ξn ⊗ W (tn−1 ). N N Then n=1 en = R+ φ ⊗ dW and n=1 dn = J R+ φ ⊗ dW ⊗ 1.
(5.3)
For n = 1, . . . , N , let r2n−1 := 12 (dn + en ) and r2n := 12 (dn − en ). We now show that {rj }2N j=1 is a martingale difference sequence with respect to the filtration {Gj }2N , where j=1 G2n G2n−1
= =
σ(Ftn ⊗ Ftn ) σ Ftn−1 ⊗ Ftn−1 , (W (tn ) − W (tn−1 )) ⊗ 1 + 1 ⊗ (W (tn ) − W (tn−1 )) .
Clearly rj is G2k -measurable for j = 1, . . . , 2k and k = 1, 2, . . . , N . Denote the expectation with respect to P ⊗ P by EP⊗P . Since ξn is Ftn−1 -measurable, we have EP⊗P (dn + en |G2n−2 ) = EP⊗P (dn |Ftn−1 ⊗ Ftn−1 ) + EP⊗P (en |Ftn−1 ⊗ Ftn−1 ) = 0, so that EP⊗P (r2n−1 |G2n−2 ) = 0. On the other hand, dn + en = (ξn ⊗ 1). (W (tn ) − W (tn−1 )) ⊗ 1 + 1 ⊗ (W (tn ) − W (tn−1 )) , so r2n−1 is G2n−1 -measurable and we have 2EP⊗P (r2n |G2n−1 ) = EP⊗P (dn − en |G2n−1 ) = (ξn ⊗ 1).EP⊗P ((W (tn ) − W (tn−1 )) ⊗ 1 − 1 ⊗ (W (tn ) − W (tn−1 ))|G2n−1 ) = 0. The last equality follows from the observation that f = (W (tn )−W (tn−1 ))⊗1 and g = 1 ⊗ (W (tn )− variables independent of Ftn−1 ⊗ Ftn−1 W (tn−1 ) are i.i.d. random and G2n−1 = σ Ftn−1 ⊗ Ftn−1 , f + g . Consequently, EP⊗P (f − g|G2n−1 ) = EP⊗P (f |G2n−1 ) − EP⊗P (g|G2n−1 ) 1 1 = (f + g) − (f + g) = 0, 2 2 and it follows that {rj }2N j=1 is a martingale difference sequence. 2N N 2N N Because n=1 dn = j=1 rj and n=1 en = j=1 (−1)j+1 rj , the inequali p p p N N ties Cp−1 EP⊗P N n=1 en ≤ EP⊗P n=1 dn ≤ Cp EP⊗P n=1 en follow by E E E appealing to the UMD property, so that p p
p −1 φ ⊗ dW ≤ E J φ ⊗ dW ≤ Cp EP⊗P φ ⊗ dW . Cp EP⊗P R+ R+ R+ E
E
E
Hence, J : X → L (P, E) is continuous for the relative topology of L (P ⊗ P, E) on X = { R+ φ ⊗ dW : φ elementary progressively measurable }. p
p
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Consequently, if φ : R+ → Lp (P, E) is the limit a.e. of elementary progressively measurable functions φn , n = 1, 2, . . . , such that { A φn ⊗dW }∞ n=1 converges in Lp (P ⊗ P, E) for each A ∈ B(R+ ), then φ is W -integrable in Lp (P ⊗ P, E) in the sense of Definition 2.1. Moreover, if E is a UMD Banach space, then for t > 0, the stochastic integral of φ with respect to W on the interval (0, t] may be represented by J (0,t] φ ⊗ dW and it belongs to the space Lp (P, E).
References [1] W.O. Amrein, V. Georgescu and J.M. Jauch, Stationary state scattering theory, Helv. Phys. Acta 44 (1971) 407–434. [2] R. Bartle, A general bilinear vector integral, Studia Math. 15 (1956), 337–351. [3] I. Dobrakov, On integration in Banach spaces I, Czech. Math. J. 20 (1970), 511–536. [4] , On integration in Banach spaces, II, Czech. Math. J. 20 (1970), 680–695. [5] D.J.H. Garling, Brownian motion and UMD-spaces, in: “Probability and Banach Spaces” (Zaragoza, 1985), 36–49, Lecture Notes in Math. 1221, Springer-Verlag, Berlin, 1986. [6] B. Jefferies, Evolution Processes and the Feynman-Kac Formula, Kluwer Academic Publishers, Dordrecht/Boston/London, 1996. [7] B. Jefferies and S. Okada, Bilinear integration in tensor products, Rocky Mountain J. Math. 28 (1998), 517–545. [8] , Dominated semigroups of operators and evolution processes, Hokkaido Math. J., 33 (2004), 127–151. [9] I. Kluv´ anek, Applications of vector measures. Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979), pp. 101–134, Contemp. Math., 2, Amer. Math. Soc., Providence, R.I., 1980. [10] G. K¨ othe, Topological vector spaces Vol II, Grundlehren der mathematischen Wissenschaften 237, Springer-Verlag, Berlin, New York, 1979. [11] J. Rosi´ nski and Z. Suchanecki, On the space of vector-valued functions integrable with respect to the white noise, Colloq. Math. 43 (1980), 183–201. [12] J.M.A.M. van Neerven and L. Weis, Stochastic integration of functions with values in a Banach space, Studia Math. 166 (2005), 131–170. [13] J.M.A.M. van Neerven, M.C. Veraar and L. Weis, Stochastic integration in UMD Banach spaces, Ann. Probab. 35 (2007), 1438–1478. Brian Jefferies School of Mathematics UNSW NSW 2052, Australia e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 271–283 c 2009 Birkh¨ auser Verlag Basel/Switzerland
A Complete Classification of Short Symmetric-antisymmetric Multiwavelets Greg Knowles Abstract. In this note we give a complete characterisation of the class of symmetric-antisymmetric multiwavelets with three and four coefficients. That is, we find all three tap multiwavelets satisfying the following conditions: (a) orthogonality, (b) one symmetric component and the other antisymmetric, (c) approximation order two and (d) compact support in the interval [0, 2]. We also find all four tap multiwavelets satisfying the conditions (a), (b), (c) and (d) compact support in the interval [0, 3]. Mathematics Subject Classification (2000). Primary 42C40; Secondary 65T60. Keywords. Wavelets, Multiwavelets, Image Processing, Data Analysis.
1. Introduction The initial idea of wavelet theory was to compose a basis of L2 (R) from integer shifts of dyadic dilates of one function, which was called the wavelet. The simplest example is the well-known Haar basis [Haa10] where the wavelet is ψ(t) = χ[0,1/2] − χ[1/2,1] and its associated scaling function φ(t) = χ[0,1] . Many other wavelets have been found [Mey85, D92, VK95, StTr97]. It was shown by Daubechies in the case of scalar wavelets that it is not possible to simultaneously have orthogonality, compact support and high approximation order. However, in the case of multiwavelets this is possible, as was shown by [GHM94]. After this work many new multiwavelets with these properties were found [Str96, CL96, GV94, GV95, STT97, LC04]. One of the classes of multiwavelets which has achieved most study is the family for which one component is symmetric, and the other is antisymmetric [CMP97, CMP98, CL96, STT97, TSLH97]. Various multiwavelets of this type have been discovered including some of the most commonly used multiwavelets in image analysis [Str96, CMP98, TSLH97, LC04], and in data analysis [Str96, CMP97]. Here we give complete classification of multiwavelets of this class with lengths three and four and satisfying conditions (a)–(d) above.
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2. Multiwavelets The theory of wavelets is based on the idea of a multi-resolution analysis (MRA). Namely, let φ and w be real-valued functions for which (i) The translates φ(x − k) are linearly independent and form a basis of the subspace V0 =< φ(x −k), k ∈ Z >. (ii) For j ∈ Z let Vj =< φ 2j x − k , k ∈ Z >. The subspaces Vj satisfy · · · ⊂ V−1 ⊂ V0
⊂ V1 ⊂ · · · ⊂ Vj ⊂ · · · ,
∪ Vj
=
L2 (R),
∩ Vj
=
0.
j∈Z j∈Z
(iii) Let W0 be such that V1 = V0 ⊕ W0 , and the translates w(x − k), k ∈ Z, of w are linearly independent and form a basis for W0 . The function φ is called the scaling function and w the wavelet. From the above ∞ properties, we can deduce L2 (R) = ⊕ Wj , where Wj =< w 2j x − k , k ∈ Z >. j=−∞
As V0 ⊂ V1 , we have the dilation equation, ck φ (2x − k) . φ(x) = k
On the other hand, we also have the following equation for the wavelet dk φ (2x − k) . w(x) = k
Multiwavelets began with the idea of generalising the two previous equations to n dimensions. Let φ1 (x), . . . , φr (x) be r scaling functions. Then φ = (φ1 , . . . , φr ) is a multi-scaling function if it satisfies the matrix dilation equation, φ(x) = Ck φ (2x − k) , k
for a family Ck of r × r matrices. Also, let w = (w1 , . . . , wr ) be r wavelet functions. Then w is a multi-wavelet if it satisfies the matrix wavelet equation, w(x) = Dk φ (2x − k) , k
for a family Dk of r × r matrices. A multi-scaling function φ has approximation order of m when all polynomials of degree from 0 to m − 1 can be exactly reproduced by a linear combination of integer translates of φk , k = 1, . . . , r. The refinement mask of the multiwavelet is defined as 1 P (ω) = Ck e−iωk 2 k∈Z
when this sum exists.
Short Symmetric-antisymmetric Multiwavelets
273
The matrices Ck and Dk are the taps of the high and low pass multi-filters associated with the multi-scaling function and multiwavelet. In this paper, we find all the multiwavelets with three and four taps satisfying conditions (a), (b), (c) and (d) in the abstract, in two dimensions (r = 2). From now on we reduce the search for multiwavelets and multi-scaling functions to finding their associated matrices Ck and Dk . Before we begin, we have to take into account the following family of orthogonal matrices. Let 0 1 1 0 0 1 U = A= , S= , B= , −S, −A, −B 1 0 0 −1 −1 0 Theorem 2.1. Let w be a multiwavelet with a multi-scaling function φ and associated coefficients Dk and Ck such that w is orthogonal, has one symmetric component and the other antisymmetric, has approximation order two and has compact ˜ be defined by the similarity transform: C˜k = U Ck U and ˜ and φ support. Let w ˜ Dk = U Dk U where U ∈ U . ˜ which has the same ˜ is a multiwavelet (with multi-scaling function φ) Then w ˜ is orthogonal, has one symmetric component properties as w in the sense that w and the other antisymmetric, has approximation order two and has the same support as w. Moreover, the only orthogonal matrices with this property are those given above. Proof. The proof of the last part of the theorem can be found in [STT97]. ˜ is orthogonal, has one symmetric component and the other The proof that w antisymmetric, has approximation order one and that it has compact support can be found in [STT97]. ˜ has approximation order two. By using So we only need to prove that w frequency domain methods (see [Str96]), the multi-scaling function φ provides approximation order m if and only if • The translates φk (t − l), l ∈ Z, k = 1, 2, . . . , r are linearly independent. 2 Ck e−iωk ) satisfies the following • The refinement mask P (ω) (P (ω) = 12 k=1
conditions, there are vectors uk ∈ R2 (u0 = 0) such that for n = 0, . . . , m− 1, n n k−n uk (2i) (Dn−k P )(0) = 2−n un k k=0 (2.1) n n k−n n−k uk (2i) (D P )(π) = 0 k k=0
˜ (w) of w. ˜ Then we have that Let P (w) be the refinement mask of w and P ˜ (π) = U P (π)U . Also, if the non-zero vectors u0 and ˜ (w) = U P (w)U and P P ˜ 0 = u0 U and u ˜ 1 = u1 U are non-zero and satisfy u1 satisfy (2.1) for w then u ˜ the approximation order conditions for the multiwavelet w.
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The theorem implies that multiwavelets and multi-scaling functions (up to a change of sign) occur in families of four, corresponding to the transformations U = A, S, B. Remark 2.2. Suppose we have calculated a multiwavelet w with taps associated ˜ be another multiwavelet with taps C˜k = U Ck U and with U Ck and Dk . Let w ˜ k corresponding to the multiwavelet w ˜ are an orthogonal matrix. Then the taps D ˜ k = U Dk U . easily computed as D Theorem 2.3. A piecewise continuous four tap multi-scaling function φ satisfying (a), (b), (c), (d), is not identically zero if and only if det(C0 − I) · det(C0 S + C1 − I) = 0.
(2.2)
Proof. If the multi-scaling function is zero for x = 0, 1, 2, 3 then φ(n) = 0, ∀n ∈ Z, as φ has support in [0, 3]. From the dilation equation it follows that φ(x) = 0 for all dyadic rational numbers x ∈ R, and hence φ(x) = 0 at any point where it is continuous. So, if φ(x) ≡ 0, one of the φ(i) must be non-zero for i = 0, 1, 2 or 3. We can write the dilation equation in block matrix form as NΦ = 0 where
⎛
C0 − I ⎜ C2 N =⎜ ⎝ 0 0
0 C0 S + C1 − I 0 0
0 0 SC1 S + SC0 − I 0
⎞ 0 ⎟ 0 ⎟ ⎠ C1 SC0 S − I
and Φ = (φ(0), φ(1), φ(2), φ(3)) . A non-zero solution Φ exists if and only if detN = det(C0 − I)2 · det(C0 S + C1 − I)2 = 0. Remark 2.4. Condition (2.2) is not invariant under the transformations U = A or U = B of Theorem 2.1. In these cases the existence of the Ci ’s is guaranteed by Theorem 2.1 however, the corresponding multi-scaling function may be identically zero.
3. Multiwavelets with three taps The first 3 tap multi-scaling function satisfying (a)–(d) was discovered by [CL96]. It has the form, 1 1 4 0 1 √2 2 2 −2 √ √ √ , C2 = , C1 = C0 = 7 − 7 4 − 7 − 7 4 0 2 4 and the corresponding multiwavelet is 1 2 1 −4 1 2 −2 0√ 2 , D2 = , D0 = . D1 = 4 −1 −1 4 0 −2 7 4 1 −1
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Let’s suppose that the multi-scaling function φ(x) = (φ1 (x), φ2 (x)) has support in the interval [0, 2] with φ1 (1 − x) = φ1 (1 + x) (symmetric component) and φ2 (1 − x) = −φ2 (1 + x) (antisymmetric component), with x ∈ [0, 1]. These two conditions can be written as, φ(1 − x) = Sφ(1 + x), 1 0 . with S = 0 −1 Further, we suppose that the dilation equation has three taps,
(3.1)
φ(x) = C0 φ(2x) + C1 φ(2x − 1) + C2 φ(2x − 2). i c00 ci01 , i = 0, 1, 2. From now on we use the following notation: Ci = ci10 ci11 From (3.1) it is easy to see that: C2 = SC0 S,
and C1 = SC1 S, 1 c00 0 , and C2 , C0 are related from which it follows that C1 is diagonal: C1 = 0 c111 by: c200 = c000 , c211 = c011 , c201 = −c001 , c210 = −c010 . (3.2) The next step is to impose approximation order two ([Str96]). Let u0 = (a, b) be a non-zero vector. Using the first equation of (2.1) for n = 0 with P (0) = 12 (C0 + C1 + C2 ) we have, u0· P (0) − u0 = 12 a(−2 + 2c000 + c100 ), 12 b(−2 + 2c011 + c111 ) = 0.
(3.3)
We now consider all possible solutions for (a) and (b). • I: b = 0 and a = 0. It can be shown that in this case there are no non-zero solutions satisfying (a)–(d). • II: b = 0 and a = 0. By using (3.3) and the second equation of (2.1) we get c000 = 12 and c100 = 1. Next we apply (2.1) for n = 1 and we find that either c111 = 12 or c001 = 0. Now we examine these two cases: • II.1: c111 = 12 . Applying the necessary and sufficient conditions for the orthogonality of the multiscaling function and multiwavelet from ([Str96]) we have that C2 C0 = 0 C0 C0 + C1 C1 + C2 C2 = 2I. Solving (3.4) gives us another two possibilities c010 = ±c011 ,
1 c001 = ± . 2
(3.4) (3.5)
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• II.1.a): Applying the first of these choices c010 = −c011 and c001 = − 12 , to (3.5) we get the following solution 1 1 1 −√21 1 0 2√ 2√ 2√ C1 = C2 = . C0 = 0 12 ± 47 ∓ 47 ∓ 47 ∓ 47 The multiwavelets Dk can be found as in [CL96], 1 1 −1 0√ − 21 2 2 D D = = D0 = 1 2 ± 14 ∓ 41 ∓ 14 0 − 27
1 2 ∓ 41
.
• II.1.b): In the other case, c010 = c011 and c001 = 12 . Using the same method as before gives us the Chui-Lam multiwavelet [CL96] mentioned earlier. This multiwavelet is also II.1.a) transformed by U = S. • II.2: c001 = 0. Here C2 · C0 = 0 and there are no solutions. • III: a = 0 and b = 0. Now we have c011 = (2 − c111 )/2. By using equations (2.1) for n = 0, we find that c111 = 1 (and c011 = 12 ). If we use (2.1) for n = 1, we get two new cases: c100 = 12 or c010 = 0. • III.1.a): c100 = 12 . Applying the same method as in II, we obtain II.1.a) transformed by U = A. • III.1.b): c001 = c000 , c010 = 12 . Here we find II.1.a) transformed by U = B. • III.2: c010 = 0. As in case II.2, we cannot obtain orthogonal multiwavelets. Conclusion: There would seem to exist eight distinct multi-scaling functions (with their corresponding multiwavelets) that verify conditions (a)–(d). However, in reality there are just two families, all the other cases can be derived from I.1.a by the transformations given in theorem 2.1, I.1.b is I.1.a with U = S, II.I.a with U = A, and II.1.b with U = B. Also, in each of these cases we can show that the eigenvalues of P (0) are 1 and λ with |λ| < 1. So we have uniform convergence of the following infinite product: ([Str96, CDP95, HC94]) ˜ φ(ω) =
∞ > j=1
P
ω 2j
˜ φ(0),
and hence the orthogonality conditions are satisfied.
(3.6)
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4. Multiwavelets with four taps Now we suppose that the multi-scaling function φ(x) = (φ1 (x), φ2 (x)) and the multiwavelet interval [0, 3] with 3 3 = (w w(x) 1 (x), w2 (x)) have support in the φ1 2 − x = φ1 2 + x (symmetric component) and φ2 32 − x = −φ2 32 + x (antisymmetric component). These two conditions can be written as 3 3 − x = Sφ +x . (4.1) φ 2 2 Also, we suppose that the dilation equation has four taps φ(x) = C0 φ(2x) + C1 φ(2x − 1) + C2 φ(2x − 2) + C3 φ(2x − 3). From the symmetry in equation (4.1), it follows that C2 = SC1 S and C3 = SC0 S. The orthogonality conditions (3.4), (3.5) in this case are C0 C0
+
C1 C1
+
C2 C0 + C3 C1 = 0
(4.2)
C2 C2
(4.3)
+
C3 C3
= 2I.
Solving (4.2) we find that c100 =
c001 c111 1 c000 c111 1 c011 c111 , c = , c = . 01 10 c010 c010 c010
(4.4)
Let u0 be a non-zero vector. The first of the approximation order equations (2.1) for n = 0 is just u0 P (0) − u0 = a −1 + c000 + c100 , b(−1 + c011 + c111 ) = 0. (4.5) We will break down all possible solutions of this equation into three cases: • I: a = 0 and b = 0. From (4.5), c111 = 1 − c011 and c000 = 1 − c100 . The second equation of (2.1) for n = 0, gives us c010 = c110 and c001 = c101 . Then applying (2.1) for n = 1 gives c100 = 34 and c011 = 14 . Substituting these in the orthogonality condition (4.2), gives c101 = c110 √ and c110 = ± 43 . Finally, substituting all of the above in the second orthogonality condition (4.3) yields the following values for Ci (cf. [GV95, GV94]): √
√
1 3 3 3 ± ± 4 4 4 4 √ √ C C0 = = 1 1 3 ± 43 ± 43 4 4 √
√
3 3 1 3 ∓ ∓ 4√ 4 4√ 4 C3 = . C2 = 3 1 3 3 ∓ 4 ∓ 4 4 4 The corresponding multiwavelets Dk can be found in [GV95, GV94]. • II: a = 0 and b = 0. Now from (4.5) we have c111 = 1 − c011
(4.6)
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and from the second equation of (2.1) for n = 0 we get c010 = c110 .
(4.7)
If we apply the approximation conditions for n = 1 and the orthogonality relation (4.2) we end up with ? c110 = ± c011 − (c011 )2 .
(4.8)
= 0, (4.8) leads to two further cases: & • II.1: c110 = − c011 − (c011 )2 . Substituting the value of c110 in (4.2) we find that When
c011
c100 = (1 + 6 c000 − 4 c011 )/2
or c100 = (−1 + 2 c000 + 4 c011 )/6
(4.9)
Now we solve for each of these in turn. • II.1.a): c100 = (1 + 6 c000 − 4 c011 )/2. Combining the orthogonality condition (4.3), and equations (4.6), (4.7), (4.8), (4.9) and with one of the conditions for a non-zero multi-scaling function, det(C0 −Id) = 0, (2.2), leads to the following solution. II.1.a).(i):
C0 = C2 =
0.373599 −0.499841 0.595545 0.499841
−0.610778 0.512626 0.364281 0.487374
C1 = C3 =
0.595545 −0.499841 0.373599 0.499841
−0.364281 0.487374 0.610778 . 0.512626
Alternatively, with the other condition for a non-zero multi-scaling function, det(C1 + C0 S − Id) = 0, (2.2), we find the multi-scaling function from I. • II.1.b): c100 = (−1 + 2 c000 + 4 c011 )/6. Proceeding in the same way as in II.1.a we find there is only one solution to both of the determinant conditions (2.2), namely the solution from I with the “−” sign. & • II.2: c110 = c011 − (c011 )2 . Using (4.2) we get two possible values for c101 : c100 = (1 + 6 c000 − 4 c011 )/2
or c100 = (−1 + 2 c000 + 4 c011 )/6.
• II.2.a): c100 = (1 + 6 c000 − 4 c011 )/2. II.2.a).(i): Combining (4.3) with the above we find II.1.a(i) similarity transformed by S. II.2.a).(ii): This is the solution already found in I. • II.2.b): c100 = (−1 + 2 c000 + 4 c011 )/6. Applying(4.3) we find that, 2 1 − 4 c011 −1 + c011 − 36 −1 + c011 c011 2 2 −4 c000 1 + 8 c011 + 4 c000 1 − 5 c011 + 4 c011 = 0. and the solution is I with the “+” sign.
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• II.3: When c110 = 0, there are no regular solutions with properties (a)–(d). • III: b = 0 and a = 0. From (2.1) we have that c000 = 1−c100 . The approximation conditions (2.1) for n = 0 now give c101 = c001 . Further, the approximation conditions for n = 1 combined with the above leads to the following: 8c001 c010 + −3 + 4 c100 −1 + 2 c011 + 2 c111 1 c10 = 8c001 From the orthogonality relation (4.2) we can derive the following conditions 2 (4.10) −2 c001 + −1 + c100 c100 = 0 2 − 8 c10 −8 c001 c010 + 8 c001 c011 + c111 01 (4.11) 2 −1 + 2 c011 + 2 c111 = 0 + 3 − 7 c100 + 4 c100 2
−2 c010 + 2 c011 c111 − Equation (4.10) has solutions
c010 (−3+4 c100 ) (−1+2 c011 +2 c111 ) 4 c001
= 0.
(4.12)
? c001 = ± c100 − (c100 )2 .
We now consider each of these in turn (c100 = 0). & • III.1: c001 = − c100 − (c100 )2 . If we use (4.11), we have that c010 = 2 − −3 + 6 c011 + 7 c100 − 6 c011 c100 − 4 c100 + 6 c111 − 6 c100 c111 ? . 2 8 c100 − c100 then combining this with (4.12), we find that c111 = (3 + 2 c011 − 4 c100 )/6, or c111 = (−3 + 6 c011 + 4 c100 )/2. So we can break the solution down into two further cases: • III.1.a): c111 = (3 + 2 c011 − 4 c100 )/6. Applying the above to the orthogonality condition (4.3), we find a one-parameter family of solutions, ⎛ ⎞ ? 1 1 − c1 2 1 − c − c 00 00 00 ⎟ ⎜ C0 = ⎝ c011 (−1+c100 ) ⎠ √1 12 c011 c00 −c00 ⎛ ⎞ ? 1 1 − c1 2 c − c 00 00 00 ⎟ ⎜ C1 = ⎝ c100 (−3−2 c011 +4 c100 ) , 3+2 c011 −4 c100 ⎠ √ 6
c100 −c100 2
6
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G. Knowles
0.75
0.5
0.25
0.2
0.4
0.6
0.8
1
-0.25
-0.5
-0.75
Figure 1. Graph of the equation (4.13). ⎛ c100
⎜ C2 = ⎝ (3+2 c011 −4 c100 ) c100 √1 12 ⎛
6
11
1 − c100
c00 −c00
00
6
c00 −c00
C3 = ⎝ c0 −c0 c1 √11 1 11 1 002 where
⎞ ? 2 c100 − c100 ⎟ 3+2 c0 −4 c1 ⎠
? ⎞ 2 c100 − c100 ⎠ c011
2 4 c011 c100 −3 + 4 c100 + c011 −36 + 32 c100 2 +c100 27 − 12 c100 − 16 c100 = 0.
(4.13)
The graph of the set of points (c100 , c011 ) that satisfies the previous condition can be seen in Figure 1 where the x-axis is the variable c100 and the y-axis is the variable c011 . We can see from the graph that apart from the points c100 = {0, 1} all of these solutions exist and are real. Further, for all of these solutions det(C1 + C0 S − Id) = 0, so the multi-scaling function φ is non-zero. The multiwavelet can be found by solving the orthogonality conditions (3.5), ⎛? ⎞ 1 1 − c1 2 c −c 00 00 ⎜ 00 √ ⎟ D0 = ⎝ 3+2 c0 −4 c1 (3+2 c011 −4 c100 ) c100 ⎠ 11 00 √ 1 − 6 6
1−c00
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⎛ ? ⎞ 2 1 1 1 − c00 − c00 1 − c00 ⎠ ? D1 = ⎝ 1−c100 0 0 −c11 c11 1 c00 ⎛ ? ⎞ 2 − c100 − c100 −1 + c100 ⎠ ? D2 = ⎝ 1−c100 c011 c011 1 c00 ⎛? ⎞ 1 1 − c1 2 c c 00 00 00 ⎜ √ ⎟ D3 = ⎝ 3+2 c0 −4 c1 (3+2 c011 −4 c100 ) c100 ⎠ . 11 00 √ − − 1 6 6
1−c00
• III.1.b): c111 = (−3 + 6 c011 + 4 c100 )/2. The solutions for Ci are ⎛ C0 =
1 − c100 ⎜ ⎝ (−1+c100 ) (−3+6 c011 +4 c100 ) 2
√
c100 −c100 2
⎞ ? 2 1 1 − c00 − c00 ⎟ ⎠ c011
? ⎞ 2 − c100 − c100 ⎠ C1 = ⎝ c0 c1 − √ 111 001 2 − 32 + 3 c011 + 2 c100 c00 −c00 ? ⎛ ⎞ 2 c100 − c100 c100 ⎠, C2 = ⎝ c0 c1 √ 111 001 2 − 23 + 3 c011 + 2 c100 c −c ⎛ 00 00 ⎞ ? 1 1 − c1 2 1 − c c 00 00 00 ⎟ ⎜ C3 = ⎝ −((−1+c100 ) (−3+6 c011 +4 c100 )) ⎠ 0 √1 12 c11 ⎛
c100
2
where
c00 −c00
2 3 2 −9 + 37 c100 − 44 c100 + 16 c100 + c011 −36 + 32 c100 2 +12 c011 3 − 7 c100 + 4 c100 = 0.
(4.14)
The graph of the set of points (c100 , c011 ) that satisfies the previous condition can be seen in Figure 2 where the x-axis is the variable c100 and the y-axis is the variable c011 . As before these solutions are exist and are real apart from when c100 = {0, 1}. We also have that det(C1 + C0 S − Id) = 0 so the multi-scaling functions φ corresponding to these Ci ’s are non-zero. The multiwavelet can be found by solving the orthogonality conditions (3.5), ⎛ ⎞ (√1−c100 ) c100 1 1 − c 00 ⎜ 1 12 ⎟ D0 = ⎝ c00 −c00 (1−c100 )√ (−3+6 c011 +4 c100 ) ⎠ 0 c11 1 1 2 2
c00 −c00
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G. Knowles
0.4
0.2
0.2
0.4
0.6
0.8
1
-0.2
Figure 2. Graph of the equation (4.14). ⎛ ? D1 = − ⎝ 3
c100 − c100
2
−c100
⎞
⎠ c0 c1 − 3 c011 − 2 c100 − √ 111 001 2 c00 −c00 ⎛ ? ⎞ 2 c100 − c100 c100 ⎠ D2 = − ⎝ 3 c0 c1 − 2 + 3 c011 + 2 c100 − √ 111 001 2 c00 −c00 ⎛ ⎞ 1 1 (√1−c00 ) c00 1 −1 + c 00 ⎜ c1 −c1 2 ⎟ D3 = ⎝ 00 00 ⎠. 1 0 1 1−c −3+6 c ( ) ( 00 11 +4 c00 ) 0 √ −c11 2 1 1 2
2
c00 −c00
& • III.2.3: c001 = c100 − (c100 )2 and c001 = 0 Here we find two parametrised solutions, corresponding to those of Section III.1 transformed by U = S, A, B. The derivation follows that of III.1. Conclusion: We find one distinct case and two one parameter families of multiwavelets of four taps that verifies conditions (a)–(d), For all these cases, it can be seen that the eigenvalue of P (0) λ = 1 satisfies |λ| < 1. So we have uniform convergence of the infinite product (3.6). Acknowledgement I would like to thank the referee for his valuable comments, and A. Mir for help on an early version of this paper.
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References [CDP95] A. Cohen, I. Daubechies, and G. Plonka. Regularity of refinable function vectors. J. Fourier Anal. Appl, 3(1997), 295–324. [CMP98] M. Cotronei, L. Montefusco and L. Puccio. Image compression through embedded multiwavelet transform coding. IEEE Transactions on Image Processing, 9(2000), 184–189. [CMP97] M. Cotronei, L. Montefusco and L. Puccio. Multiwavelet analysis and signal processing. IEEE Trans. on Circuits and Systems II, Special Issue on Multirate Systems, Filter Banks, Wavelets, and Applications, 1997. [CL96] C. Chui and J. Lian. A study on orthonormal multiwavelets. Appl. Numer. Math., 20(1996), 273–298. [D92] I. Daubechies. Ten lectures on wavelets. CBMS-NSF Reg. Conf. Series on App. Math., SIAM, 61, 1992. [GHM94] J. Geronimo, D. Hardin and P. Massopust. Fractal functions and wavelet expansions based on several scaling functions. J. Approx. Th., 78(1994), 373–401. [Haa10] A. Haar. Zur Theorie der orthogonalen Funktionensysteme. Math. Annal., 69(1910), 331–371. [HC94] C. Heil and D. Colella. Matrix refinement equations: Existence and uniqueness. J. Fourier Anal. Appl, 2(1996), 363–377. [LC04] J. Lian and C. Chui. Balanced multi-wavelets with short filters. IEEE Signal Processing Letters, 11(2004), 75–78. [Mey85] Y. Meyer. Principe d’incertitude, bases hilbertiannes et alg`ebres d’op´erateurs. S´eminaire Bourbaki, 662, 1985. [GV95] G. Strang and V. Strela. Short Wavelets and Matrix Dilation Equations. IEEE Trans. on Signal Proc., 43(1995), 108–115. [GV94] G. Strang and V. Strela. Orthogonal Multiwavelets with Vanishing Moments. Optical Engineering, 33(1994), 2104–2107. [StTr97] G. Strang and Truong Nguyen. Wavelets and filter banks. Wellesley-Cambridge Press, 1997. [Str96] V. Strela. Multiwavelets: Theory and Applications. PhD thesis, M.I.T, June 1996. [STT97] Li-Xin Shen, Jo Yew Tham, and Hwee Huat Tan. Symmetric-antisymmetric orthonormal mulitwavelets and related scalar wavelets. Applied and Computational Harmonic Analysis, 8(2000), 258–279. [TSLH97] Jo Yew Tham, Li-Xin Shen, Seng Luan Lee, and Hwee Huat Tan. A general approach for analysis and application of discrete multiwavelet transforms. IEEE Trans. Signal Processing, 48(2000), 457–464. [VK95] M. Vetterli and J. Kovacevic. Wavelets and subband coding. Prentice Hall, 1995. Greg Knowles School of Computer Science, Engineering and Mathematics Flinders University, Australia e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 285–292 c 2009 Birkh¨ auser Verlag Basel/Switzerland
On the Range of a Vector Measure Mienie de Kock and Daniele Puglisi Abstract. Let C be a countable subset of c0 that lies in the range of a (c0 )∗∗ valued measure, then C lies in the range of a c0 -valued measure. We extend this result to C(K), where K is a compact Hausdorff space, i.e., we let C be a countable subset of C(K) that lies in the range of a C(K)∗∗ -valued measure, then C lies in the range of a C(K)-valued measure. We will also see that in any separable Banach space the result still holds. Mathematics Subject Classification (2000). Primary 46B03. Keywords. Range of a vector measure.
1. Introduction In [10], Professor A. Sofi asks (Problem 6) to which Banach spaces X is it so that if C is a countable subset of X that lies in the range of a countably additive X ∗∗ -valued measure with a σ-field domain, then there is a countably additive Xvalued measure with a σ-field domain, whose range also contains C. Of course, if X is complemented in X ∗∗ , then the answer is plain and easy. In this note we show that if X is c0 , X is a separable Banach space or X is a C(K)-space for a compact Hausdorff space K, then any countable subset C of X that lies in the range of an X ∗∗ -valued countably additive measure on a σ-field lies in the range of an X-valued countably additive measure on the same σ-field. The “techniques” are all Banach space techniques. We never called on a change of domain. This problem is related to the following considered by F.J. Freniche (see [3]): Given a vector measure μ with values in the bidual X ∗∗ of the Banach space X, under what conditions can we say μ actually takes its values inside X? The author shows that if X is a Banach space such that its dual closed unit ball is The first author was partially supported by the Tarleton State University – Central Texas MiniSeed Grant 2009. The second author was supported by National Sciences Foundation Focused Research Group Grant, FRG: Fourier Analytic and Probabilistic Method in Geometric Functional Analysis and Convexity. Grant number DMS-0652684.
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weak∗ sequentially compact and if the X ∗∗ -valued measure μ satisfies the Geitz’s condition, then its range is contained in X. Therefore, if X is a Banach space whose dual unit ball is weak∗ sequentially compact and if C is a countable subset of X that lies in the range of an X ∗∗ -valued measure that verifies the Geitz’s condition, then C lies in the range of an X-valued measure on the same σ-field. The closely related bounded variation version of the problem as treated in this paper is already known to have been resolved in the negative. A counterexample to this effect was constructed by B. Marchena and C. Pineiro (see [6]), and later improved by M.A. Sofi (see [10]).
2. Preliminaries Let X and Y denote Banach spaces, and let F : Σ → X denote the vector measure defined on the σ-field Σ. Let K denote a scalar field. Then we let c0 (Γ) consist of all the functions f : Γ → K, such that for every > 0, {γ ∈ Γ : |f (γ)| ≥ } is finite. Note that each f ∈ c0 (Γ) has countable support and is bounded and that f c0 (Γ) = f ∞ . If Γ = N, we use the usual notation and write c0 , which consists of all sequences converging to zero. Likewise, l∞ consists of all bounded sequences of scalars. We say that a set A lies in the range of a Y -valued measure if there is a σ-field Σ and a countably additive measure F : Σ → Y so that A ⊆ F (Σ). A set C is absolutely convex if and only if for any points x1 and x2 in C, and any numbers λ1 and λ2 satisfying |λ1 | + |λ2 | ≤ 1, the sum λ1 x1 + λ2 x2 belongs to C. Since the intersection of any collection of absolutely convex sets is a convex set, then for any subset A we denote the absolutely convex hull to be the intersection of all absolutely convex sets containing A. We denote by span A the linear span of A, that is the intersection of all vector subspaces containing A. A space is injective if every isomorphic embedding of it in an arbitrary Banach space Y is the range of a bounded linear projection defined on Y . A topological space is zero-dimensional, if its topological dimension is zero, or equivalently, if it has a base consisting of clopen sets. A Banach space is weakly compactly generated whenever it is the closed linear span of one of its weakly compact subsets. We use the same notation as in [2]. We will need the following theorems in order to prove the main results. The proof of the first theorem can be found in [2], page 14. Theorem 2.1 (Bartle-Dunford-Schwartz). Let F : Σ → X be a countably additive vector measure on a σ-field Σ. Then the range of F is relatively weakly compact. Theorem 2.2 (Rosenthal). Any weakly compact subset of ∞ is norm separable.
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Sobczyk (see [9]) proved the following theorem that states that c0 is complemented in every separable space in which it resides, although the norm of the projection does not need to be equal to one. Theorem 2.3 (Sobczyk). If X is a separable Banach space and Y ⊆ X is a closed subspace isometric to c0 , then there is a continuous linear projection P from X onto Y with P ≤ 2. The proof of the following theorem can be found in [1]. Theorem 2.4 (Amir and Lindenstrauss). Let X be a weakly compactly generated Banach space. If X0 is a separable subspace of X and Y0 is a separable subspace of X ∗ , then there is a projection P : X → X whose range is separable such that X0 ⊆ P (X) and Y0 ⊆ P ∗ (X ∗ ). A proof of the following two theorems can be found in [8] and [7]. Theorem 2.5 (Miljutin). If K is an uncountable compact metric space, then the space C(K) is isomorphic to C(Δ), where Δ denotes the Cantor discontinuum. Theorem 2.6 (Pelczynski). 1. Let K be a zero-dimensional compact metric space. If a separable Banach space X contains a subspace Y that is isometrically isomorphic to C(K), then there are a subspace Z of Y and a projection P : X → Z (onto Z) such that Z is isometrically isomorphic to C(K) and P = 1. 2. Let K be a compact metric space. If a separable Banach space X contains a subspace Y that is isomorphic to C(K), then there is a subspace Z of Y such that Z is isomorphic to C(K) and Z is complemented in X.
3. Main results Section I Theorem 3.1. If C is a countable subset of c0 that lies in the range of a c∗∗ 0 -valued measure, then C lies in the range of a c0 -valued measure. Proof. Let F : Σ → ∞ be a countably additive measure defined on the σ-field Σ such that C ⊆ F (Σ). By 2.1 we have that F (Σ) is a relatively weakly compact subset of ∞ . By 2.2 we have that F has a norm separable range. So F (Σ) generates a separable closed linear subspace of ∞ ; we can enlarge our set by letting X be the closed linear span (in ∞ ) of F (Σ) ∪ c0 . In so doing, we obtain a separable Banach space X that contains C, itself a subset of c0 . Now we are in a position to call on 2.3: The result is a bounded linear projection P from X onto c0 . We also have that C ⊆ P ◦ F (Σ) and P ◦ F : Σ → c0 is a c0 -valued measure whose range contains C. Theorem 3.2. If C is a countable subset of c0 and C ⊆ F (Σ), where F : Σ → X, for F a countably additive vector measure mapping from a σ-field Σ to a Banach space
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X containing c0 . Then there exists a countably additive vector measure G : Σ → c0 such that C ⊆ G(Σ). Proof. Let W = span[F (Σ)∪c0 ]. Then W ⊂ X and is weakly compactly generated containing c0 . By 2.4 there exists a continuous projection P : W → W such that c0 ⊂ P (W ) = S, with S separable. Now 2.3 yields a projection Q : S → S with Q ≤ 2 so that Q ◦ S = c0 . Finally, the set-function G = Q ◦ P ◦ F : Σ → c0 defines a vector measure whose range contains C.
Section II Theorem 3.3. Let S be a separable Banach space, and suppose C is a countable subset of S that lies in the range of an S ∗∗ -valued measure. Then C lies in the range of an S-valued measure. Proof. Let F : Σ −→ S ∗∗ be a countable additive measure on the σ-field Σ such that C ⊆ F (Σ). By 2.1 F (Σ) is relatively weakly compact. Let K denote the absolutely convex hull of F (Σ) in S ∗∗ ; the set K is the unit ball of its linear span YK if we consider on YK the norm · K given by the Minkowski functional of K. By 2.4 (see [1]) we can assume that YK is separable in the S ∗∗ norm topology. Moreover, the inclusion map YK → S ∗∗ is weakly compact and has separable range. Observe that we may assume that YK is a Banach space since if it is not complete, then we can complete it and the previous inclusion would remain weakly compact with separable range and with the same norm. A close look at the still marvellous factorization scheme of Davis, Figiel, Johnson and Pe lczynski tells us that there is a separable reflexive Banach space R and bounded linear operators (with a ≤ 1) a : YK −→ R and b : R −→ S ∗∗ so that b ◦ a is the inclusion YK → S ∗∗ of YK into S ∗∗ . Since S is separable, let D = {x∗n : n ∈ N} be a weak∗ -dense sequence consisting of linearly independent elements of S ∗ . Let F be a countable · dense subset of YK ( · is the norm in S ∗∗ ) and look at F ∪ C = {zn∗∗ : n ∈ N}, which lies inside YK . Let rn = a(zn∗∗ ) for each n ∈ N. Since a : YK −→ R is a bounded linear operator, a is also · -continuous and so {rn : n ∈ N} is norm dense in R. Let Fn = span{z1∗∗ , . . . , zn∗∗ } ⊆ S ∗∗ and @n = span{x∗ , . . . , x∗ } ⊆ S ∗ F 1 n By the Principle of Local reflexivity we can find for each n ∈ N an injective linear map Tn : Fn −→ S such that
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(A) Tn |Fn ∩S = id|Fn ∩S , (B) Tn ≤ 1 , Tn−1 ≤ 1 + n1 , @n and for r ∈ span{a(z1∗∗ ), . . . , a(zn∗∗ )} (C) x∗ ( Tn (b(r) ) ) = x∗ ( b(r) ), x∗ ∈ F or, equivalently @n , z ∗∗ ∈ Fn . (C) x∗ Tn z ∗∗ = z ∗∗ (x∗ ), x∗ ∈ F n Let Rn = a(Fn ). Define Un : Rn −→ S by T
b
n Rn −→ Fn −→ S. A BC D
Un
Let
(n : S ∗ −→ R∗ U n
be given by (n x∗ (r) = U
x∗ Un (r), 0,
if r ∈ Rn ; otherwise.
(n is homogeneous and it can be viewed as a point of (BR∗ , weak∗ )D , a U @n |D ∈ (BR∗ , weak∗ )D ). compact metrizable space. (To be more precise, U (n )n in this way we see that (U @n )n has a limit point U ( ∈ (BR∗ , weak∗ )D Viewing (U ∗ that can be extended to all of S with values still in BR∗ using the fact that D is linearly independent. The compact metrizable nature on (BR∗ , weak∗ )D tells us (n )n ; ( is actually a pointwise limit (over D) of a subsequence (U (n )k of (U that U k ∗ ∗ ∗ ( to a map from S to R . the density of D in S allows us to extend U ( Step 1. Now we note that U is actually a bounded linear operator. • Linearity: Let x∗ , y ∗ ∈ S ∗ , λ, μ ∈ K and r ∈ R. Consider n ∈ N such that r ∈ Rn for each n ≥ n. Then ( (λx∗ + μy ∗ ), r = lim U (n (λx∗ + μy ∗ ), r
U k→∞
k
∗
= lim λx + μy ∗ , Unk (r) k→∞
= λ lim x∗ , Unk (r) + μ lim y ∗ , Unk (r) k→∞
k→∞
(n (x∗ ), r + μ lim U (n y ∗ , r = λ lim U k k k→∞
k→∞
( (y ∗ ), r . ( (x∗ ), r + μ U = λ U • Continuity: Let x∗ ∈ S ∗ , and r ∈ BR . Consider n ∈ N such that r ∈ Bn for each n ≥ n. Then ( (x∗ ), r = lim U (n (x∗ ), r
U k
k→∞
∗
= lim x , Unk (r) k→∞
≤ lim x∗ Unk (r) k→∞
(since Un ≤ 1 ∀n ∈ N)
≤ x∗
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' Step 2. For all r ∈ n∈N Rn , {Un (r)}n∈N is relatively weakly compact and from this we see that {Un (r)}n∈N is relatively weakly compact for each r ∈ R. This follows directly from (C) above and x∗ Un (r) = Tn (b(r))x∗ = b(r)x∗ @n . for any r ∈ span{a(z1∗∗ ), . . . , a(zn∗∗ )} and x∗ ∈ F ( is weak∗ -weak ∗ continuous. Step 3. U weak∗
Let (s∗n )n be a sequence in S ∗ such that s∗n −→ s∗ and let r ∈ R. Then ( (s∗n ), r = lim lim U (n (s∗n ), r lim U k
n→∞
n→∞ k→∞
(n (s∗ ), r = lim lim U n k n→∞ k→∞
(since we are considering nk big enough)
= lim lim s∗n , Unk (r)
(∗)
= lim lim s∗n , Unk (r)
n→∞ k→∞
k→∞ n→∞
(n (s∗n ), r = lim lim U k k→∞ n→∞
(n (s∗ ), r = lim U k k→∞
( (s∗ ), r = U where in (∗) we used the double limit Grothendieck theorem (see [5] Corollarie 1 to Th´eor`em 7) and Step 2. So S( is a weak∗ -weak∗ bounded linear operator. That means there is a bounded linear operator V : R −→ S such that ( V∗ =U Now consider the new σ-additive measure F F( : Σ −→ F ∪ C
a, V
R −→ S Next we show that C ⊆ F( (Σ).
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Indeed, for all x ∈ C ( (x∗ )
V (a(x)), x∗ = a(x), U (n (x∗ ) = lim a(x), U k k→∞
= lim Unk (a(x)), x∗ k→∞
= lim Tnk (b ◦ a(x)), x∗ k→∞
= lim Tnk (x), x∗ k→∞ ∗
( by (A) above) = x, x
Comment 3.4. The second author believes that, assuming that Martin’s axiom (see [4]) is true and a transfinite induction argument instead of a natural induction argument, the separability hypothesis of the previous theorem can be removed. Section III The reader might realize that when a Banach space X is complemented in X ∗∗ , then the problem has a positive solution. In fact, consider the projection P : X ∗∗ → X and compose it with the vector measure F : Σ → X ∗∗ to obtain the countably additive X-valued measure P ◦ F : Σ → X. Theorem 3.5. Let C be a countable subset of C(K), where K is a compact Hausdorff space. If C lies in the range of a C(K)∗∗ -valued measure, then C lies in the range of a C(K)-valued measure. Proof. First, we recall a construction due to Eilenberg: If k1 , k2 ∈ K, then we say k1 ∼ k2 if f (k1 ) = f (k2 ) for each f ∈ C; of course ∼ is an equivalence relation on K and between equivalence classes [k1 ] and [k2 ], we can define a metric d([k1 ], [k2 ]) by |fn (k1 ) − fn (k2 )| d([k1 ], [k2 ]) = (fn + 1)2n fn ∈C
remembering that C = {fn : n ∈ N} is countable. Each f ∈ C “lifts” to an f˜ ∈ C(K0 ), K0 the metric space of equivalence classes; the map q : K → K0 that takes k to [k] is a continuous surjection. So K0 is a compact metric space and q : K K0 is a surjective continuous map; q induces an isometric linear embedding q ◦ : C(K0 ) → C(K), where q ◦ (f˜)(k) = f˜([k]), for any f˜ ∈ C(K0 ). It is important to realize that if C˜ is the result of lifting members of C in C(K) to ˜ = C. Here is the setup. members of C(K0 ), then q ◦ (C) C˜ ⊆ C(K0 ) ⊆ C(K) and C ⊆ F (Σ) ⊆ C(K)∗∗ for some countably additive F : Σ → C(K)∗∗ with a σ-field domain Σ. Because C(K0 )∗∗ is isometrically isomorphic to a subspace of C(K)∗∗ ; but C(K0 )∗∗ is an injective Banach space so there is a bounded linear projection P : C(K)∗∗ →
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C(K)∗∗ on C(K)∗∗ with range P (C(K)∗∗ ) = C(K0 )∗∗ . So we have a C(K0 )∗∗ such that the countable set C˜ lies in P ◦ F (Σ). The previous theorem tells us there is a C(K0 )-valued countably additive measure G on Σ so that C˜ ⊆ G(Σ). Look at: ˜ ⊆ q ◦ (G(Σ)) ⊆ q ◦ (C(K0 )) ⊆ C(K). C = q ◦ (C) Acknowledgement The paper was presented at the conference “Third Meeting on Vector Measures, Integration and Applications, Eichst¨ att, Germany, 2008” by Prof. J. Diestel. We would like to thank him deeply for his generous help and useful suggestions.
References [1] D. Amir and J. Lindenstrauss: The structure of weakly compact sets in Banach spaces. Ann. of Math. 88 2 (1968), 35–46. [2] J. Diestel and J.J. Uhl, Jr.: Vector Measures. Mathematical Surveys and Monographs 15, Providence, Rhode Island, 1977. [3] F.J. Freniche: Some remarks on the average range of a vector measure. Proc. Amer. Math. Soc. 107 1 (1989), 119–124. [4] D.H. Fremlin: Consequences of Martin’s axiom. Cambridge Tracts in Mathematics 84, Cambridge University Press, 1984. [5] A. Grothendieck: Crit` eres de compacit´ e dans les espaces fonctionnels g´en´eraux. (French) Amer. J. Math. 74 (1952), 168–186. [6] B. Marchena and C. Pineiro: A note on sequences lying in the range of a vector measure valued in the bidual. Proc. Amer. Math. Soc. 126 10 (1998), 3013–3017. [7] A.A. Miljutin: Isomorphisms of spaces of continuous functions on compacts of the power continuum. Teor. Funk. Funkcional. Analiz. i Prilozen. 2 (1996), 150–156 (Russian). [8] A. Pelczynski: On C(S)-subspaces of separable Banach spaces. Studia Math. T. XXXI (1968), 513–522. [9] A. Sobczyk: Projection of the space (m) on its subspace (c0 ). Bull. Amer. Math. Soc. 47 (1941), 938–947. [10] M.A. Sofi: Structural Properties of Banach and Fr´ echet spaces determined by the range of vector measures. Extracta Math. 22 3 (2007), 257–296. Mienie de Kock Department of Mathematics Texas A&M University-Central Texas Killeen, TX 76549, USA e-mail: [email protected] Daniele Puglisi Department of Mathematics University of Missouri Columbia, MO, 65211-4100, USA e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 293–302 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Measure and Integration: Characterization of the New Maximal Contents and Measures Heinz K¨onig Dedicated to the Memory of Igor Kluv´ anek
Abstract. The work of the author in measure and integration is based on parallel extension theories from inner and outer premeasures to their maximal extensions, both times in three different columns (finite, sequential, nonsequential). The present paper characterizes those contents and measures which occur as these maximal extensions. Mathematics Subject Classification (2000). 28A10, 28A12, 28C15. Keywords. Inner and outer premeasures and their maximal extensions, complete, saturated, and SC contents and measures, tame inner and outer premeasures, quasi-Radon measures.
The present article is devoted to the foundational part of the theory of measure and integration developed in the author’s book [4] and in a series of subsequent papers, summarized in [6] and [7]. It consists of parallel inner and outer extension theories which proceed from the inner • premeasures ϕ : S → [0, ∞[ to their maximal inner • extensions Φ = ϕ• |C(ϕ• ), and from the outer • premeasures ϕ : S → [0, ∞] to their maximal outer • extensions Φ = ϕ• |C(ϕ• ), both times in the three parallel procedures • = : the finite one, based on finite formations, • = σ : the sequential one, based on countable formations, • = τ : the nonsequential one, based on arbitrary formations. The set functions Φ thus produced are contents on algebras in case • = and measures on σ algebras for • = στ . As a rule the main theorems in the new theory start from assumptions on certain initial inner or outer • premeasures ϕ, and the assertions are for their maximal • extensions Φ, or for related entities. The essential point in this set-up
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is the both decisive and flexible position of the basic data ϕ. It so happened that another issue did not come to the surface so far: the problem to characterize those contents and measures which occur as the maximal • extensions Φ of the different kinds. The present article wants to obtain such a characterization. It is of course of interest, in particular for the comparison with the more traditional extension procedures in measure and integration. In both the inner and the outer situation the characterization has a common form in the three cases • = στ . The main characteristic properties are the familiar notions complete and saturated, but the top one is a new notion named SC, which is close to, but not equivalent to the combination of complete and saturated. These properties will be discussed in Section 1. Section 2 will then be devoted to the characterizations in question. In Section 3 we shall add another characterization theorem, which is under the r´egime of local finiteness: On the side of the new theory there are the inner • premeasures ϕ which are of local finiteness type with respect to certain • complemental pairs of lattices in the sense of [6] Section 4, while on the traditional side there are the quasi-Radon measures of Fremlin [2], [3] in case • = τ and their relatives for • = σ. Section 4 will then be devoted to the comparison quoted above.
1. The relevant properties of contents and measures We start to recall the basic concepts and facts. Our main reference will be the survey article [6]. Let X be a nonvoid set, which carries the set systems under consideration. For an isotone set function ϕ : S → [0, ∞] with ϕ(∅) = 0 on a lattice S with ∅ ∈ S the inner and outer • envelopes ϕ• , ϕ• : P(X) → [0, ∞] for • = στ are in the usual terms ϕ• (A) = sup{ inf ϕ(M ) : M ⊂ S nonvoid • with M ↓⊂ A}, M∈M
ϕ• (A) = inf{ sup ϕ(M ) : M ⊂ S nonvoid • with M ↑⊃ A}, M∈M
with inf ∅ := ∞. It follows that ϕ ϕ . If moreover ϕ is submodular, then [ϕ < ∞] := {S ∈ S : ϕ(S) < ∞} ⊂ S is a lattice as well, and we can define ϕ◦ : P(X) → [0, ∞] to be ϕ◦ = (ϕ|[ϕ < ∞]) . Thus ϕ◦ ϕ , and ϕ◦ (A) = ϕ (A) when ϕ (A) < ∞. We start with a basic fact [8] Section 2. 1.1 Lemma. Let ϕ : S → [0, ∞] be a content on a ring S. Then ϕ(S) = ϕ (S ∩ E) + ϕ (S ∩ E ) for S ∈ S and E ⊂ X, and hence ϕ(S) = ϕ◦ (S ∩ E) + ϕ (S ∩ E ) for S ∈ [ϕ < ∞] and E ⊂ X. For the sake of completeness we include a proof. ) For A ∈ S with S∩E ⊂ A we have S ∩ A ⊂ S ∩ E and hence ϕ(S) = ϕ(S ∩ A ) + ϕ(S ∩ A) ϕ (S ∩ E) + ϕ(A).
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It follows that ϕ(S) ϕ (S ∩ E) + ϕ (S ∩ E ). ) For A ∈ S with A ⊂ S ∩ E we have S ∩ E ⊂ S ∩ A and hence ϕ(S) = ϕ(A) + ϕ(S ∩ A ) ϕ(A) + ϕ (S ∩ E ). It follows that ϕ(S) ϕ (S ∩ E) + ϕ (S ∩ E ).
Next we recall for a set function Θ : P(X) → [0, ∞] with Θ(∅) = 0 the Carath´eodory class C(Θ) := {A ⊂ X : Θ(M ) = Θ(M ∩ A) + Θ(M ∩ A ) ∀M ⊂ X}, the members of which are called measurable Θ. One verifies that Θ|C(Θ) is a content on the algebra C(Θ). 1.2 Remark. Let α : A → [0, ∞] be a content on an algebra A. For A ⊂ X then ) A ∈ C(α ) ⇐⇒ α(S) α (S ∩ A) + α (S ∩ A ) for all S ∈ [α < ∞], ◦) A ∈ C(α◦ ) ⇐⇒ α(S) α◦ (S ∩ A) + α◦ (S ∩ A ) for all S ∈ [α < ∞]. Proof. To be shown is ⇐=. ) The right side holds true for all S ∈ A. We fix M ⊂ X and use this fact for the S ∈ A with S ⊃ M , which furnishes α (M )
α (M ∩ A) + α (M ∩ A ). It follows that α (M ) = α (M ∩ A) + α (M ∩ A ) since α is submodular. ◦) is obtained as before. 1.3 Proposition. Let α : A → [0, ∞] be a content on an algebra A. Then 1) C(α ) = C(α◦ ) ⊃ A. 2) If E ∈ C(α ) = C(α◦ ) is upward enclosable [α < ∞] then α (E) = α◦ (E). Proof. i) The inclusions A ⊂ C(α ) and A ⊂ C(α◦ ) are obvious from 1.2. ii) Assume that E ∈ C(α ). For S ∈ [α < ∞] then 1.1 and 1.2.) furnish α(S) = α◦ (S ∩ E) + α (S ∩ E ) α (S ∩ E) + α (S ∩ E ) α(S) < ∞, and hence α◦ (S ∩ E) = α (S ∩ E). Likewise of course α◦ (S ∩ E ) = α (S ∩ E ). Thus 1.1 and 1.2.◦) show that E ∈ C(α◦ ). Moreover α◦ (E) = α (E) when there exists an S ∈ [α < ∞] with S ⊃ E. iii) Assume that E ∈ C(α◦ ). For S ∈ [α < ∞] as above 1.1 and 1.2.◦) furnish α (S ∩E ) = α◦ (S ∩E ) and α (S ∩E) = α◦ (S ∩E), and thus 1.1 and 1.2.) show that E ∈ C(α ). After this we define a content α : A → [0, ∞] on an algebra A to be SC iff C(α ) = C(α◦ ) = A. We recall that α is called complete iff Q ∈ A with α(Q) = 0 implies that all P ⊂ Q are in A, and is called saturated iff [α < ∞]-A ⊂ A, with - the transporter as in [4, 6, 7]. We mention that Fremlin [2] 64G and [3] 211H defines a measure α on a σ algebra A to be locally determined iff it is both saturated and semifinite (which means that α is inner regular [α < ∞]). 1.4 Theorem. Let α : A → [0, ∞] be a content on an algebra A. Then 1) α is SC =⇒ α complete and saturated. 2) The converse need not be true. 3) Assume that A is a σ algebra. Then α is SC ⇐= α complete and saturated.
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Proof of 1). Assume that α is SC. i) To see that α is complete let P ⊂ Q ∈ A with α(Q) = 0. Then α (P ) α (Q) = 0 and hence α (P ) = 0. Therefore for S ∈ [α < ∞] we have α (S ∩ P ) + α (S ∩ P ) = α (S ∩ P ) α(S), so that 1.2.) implies that P ∈ C(α ) = A. ii) To see that α is saturated let A ∈ [α < ∞]-A. For S ∈ [α < ∞] thus S ∩ A ∈ A and hence S ∩ A = S \ (S ∩ A) ∈ A. It follows that α(S) = α(S ∩ A) + α(S ∩ A ). Thus 1.2 furnishes A ∈ C(α ) = C(α◦ ) = A. Assertion 2) will be proved with the counterexample below. Proof of 3). Assume that α is complete and saturated, and that A is a σ algebra. We shall prove that C(α◦ ) ⊂ A and fix E ∈ C(α◦ ). It suffices to prove that E ∈ [α < ∞]-A. Thus we fix S ∈ [α < ∞] and claim that S ∩ E ∈ A. i) We know that α(S) = α◦ (S ∩ E) + α◦ (S ∩ E ). From the definition of α◦ we obtain sequences of Pn , Qn ∈ [α < ∞] with Pn ⊂ S ∩ E and α(Pn ) → α◦ (S ∩ E), Qn ⊂ S ∩ E and α(Qn ) → α◦ (S ∩ E ). We can achieve that Pn ↑ some P ⊂ S ∩ E and Qn ↑ some Q ⊂ S ∩ E . Then P, Q ∈ A since A is a σ algebra, and α(Pn ) α(P ) α◦ (S ∩ E) α(Qn ) α(Q) α◦ (S ∩ E )
implies that implies that
α(P ) = α◦ (S ∩ E), α(Q) = α◦ (S ∩ E ),
in particular P, Q ∈ [α < ∞]. ii) From P ∩ Q = ∅ we obtain α(P ∪ Q) = α(P ) + α(Q) = α◦ (S ∩ E) + α◦ (S ∩ E ) = α(S), so that P ∪ Q ∈ [α < ∞] fulfils P ∪ Q ⊂ S and α S \ (P ∪ Q) = 0. But S \ (P ∪ Q) = (S ∩ E) \ P ∪ (S ∩ E ) \ Q , so that (S ∩ E) \ P ⊂ S \ (P ∪ Q), and hence (S ∩ E) \ P ∈ A since α is complete. It follows that S ∩ E ∈ A. 1.5 Example. Let X = [0, 1] and λ : L → [0, ∞[ be the Lebesgue measure on X (which is known to be complete). We define A ⊂ L to consist of those A ∈ L which fulfil either [0, δ] ⊂ A or [0, δ] ∩ A = ∅ for some 0 < δ < 1. Then A is an algebra, but not a σ algebra. And α := λ|A is a finite content on A and hence saturated, and moreover upward σ continuous. The completeness of λ combined with the definition of A implies at once that α is complete. But α is not SC: To see this consider A = {0}. We have A ∈ L but A ∈ A. However, for S ∈ A we have α (S ∩ A) α (A) = 0 and hence α (S ∩ A) + α (S ∩ A ) = α (S ∩ A ) α(S), so that 1.2.) implies that A ∈ C(α ). Thus we have indeed C(α ) = A.
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2. The characterization theorems We continue to assume a nonvoid set X and • = στ . 2.1 Inner Remark. Let S be a lattice with ∅ ∈ S and ϕ : S → [0, ∞[ be isotone with ϕ(∅) = 0. By definition and the inner • extension theorem [6] 3.5 then ϕ is an inner • premeasure iff there exist contents α : A → [0, ∞] on algebras A ⊃ S which are inner • extensions of ϕ. Of these α a unique one is SC: this is the maximal α = ϕ• |C(ϕ• ). We recall that an inner • extension of ϕ in the sense of [6] Section 3 is defined to be a content α : A → [0, ∞] on a ring A which is an extension of ϕ and satisfies S ⊂ S• ⊂ A with α is inner regular S• , α|S• is downward • continuous (note that α|S• < ∞). Proof. The α : A → [0, ∞] in question are restrictions of ϕ• |C(ϕ• ). Thus S ⊂ S• ⊂ [α < ∞] ⊂ A ⊂ C(ϕ• ) and α = ϕ• |A. On [α < ∞] we have ϕ• = α = α|[α < ∞] = α◦ , and hence ϕ• = α◦ partout since both sides are inner regular [α < ∞]. It follows that α is SC ⇔ A = C(α◦ ) = C(ϕ• ) ⇔ α = ϕ• |C(ϕ• ). 2.2 Inner Characterization Theorem. Let α : A → A be a content on an algebra A. Then α = ϕ• |C(ϕ• ) for some inner • premeasure ϕ : S → [0, ∞[ ⇐⇒ there exists a lattice S with ∅ ⊂ S ⊂ A and α|S < ∞ such that α is an inner • extension of α|S, and α is SC . This is an immediate consequence of 2.1. We continue with an additional equivalence in the cases • = σ. 2.2 Continuation. Moreover in case • = : ⇐⇒ α is semifinite and SC; in case • = σ: ⇐⇒ α is a measure on the σ algebra A, and is semifinite and SC (that is complete and saturated). Proof. Both times =⇒ is clear, and ⇐= results for S := [α < ∞].
We turn to the outer counterpart. 2.3 Outer Remark. Let S be a lattice with ∅ ∈ S and ϕ : S → [0, ∞] be isotone with ϕ(∅) = 0. By definition and the outer • extension theorem [6] 3.1 then ϕ is an outer • premeasure iff there exist contents α : A → [0, ∞] on algebras A ⊃ S which are outer • extensions of ϕ. Of these α a unique one is SC: this is the maximal α = ϕ• |C(ϕ• ).
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We recall that an outer • extension of ϕ in the sense of [6] Section 3 is defined to be a content α : A → [0, ∞] on a ring A which is an extension of ϕ and satisfies S ⊂ S• ⊂ A with α is outer regular S• , α|S• is upward • continuous. Proof. The α : A → [0, ∞] in question are restrictions of ϕ• |C(ϕ• ). Thus S ⊂ S• ⊂ A ⊂ C(ϕ• ) and α = ϕ• |A. On A we have ϕ• = α = α , and hence ϕ• = α partout since both sides are outer regular A. It follows that α is SC ⇔ A = C(α ) = C(ϕ• ) ⇔ α = ϕ• |C(ϕ• ). 2.4 Outer Characterization Theorem. Let α : A → A be a content on an algebra A. Then α = ϕ• |C(ϕ• ) for some outer • premeasure ϕ : S → [0, ∞] ⇐⇒ there exists a lattice S with ∅ ⊂ S ⊂ A such that α is an outer • extension of α|S, and α is SC. This is an immediate consequence of 2.3 as before. We continue with an additional equivalence in the cases • = σ. 2.4 Continuation. Moreover in case • = : ⇐⇒ α is SC; in case • = σ: ⇐⇒ α is a measure on the σ algebra A, and is SC (that is complete and saturated). Proof. Both times =⇒ is clear, and ⇐= results for S := A.
2.5 Remark. Assume that α = ϕ• |C(ϕ• ) for an inner • premeasure ϕ : S → [0, ∞[, or α = ϕ• |C(ϕ• ) for an outer • premeasure ϕ : S → [0, ∞]. Then α need not be the completion of the restriction α|Aσ(S) of α to the generated σ algebra Aσ(S) when • = στ , but can be much more comprehensive, and the like for • = . As an example let S consist of the finite subsets of an uncountable set X and ϕ : S → [0, ∞[ be the cardinality restricted to S. In all cases then ϕ• = ϕ• = card, so that C(ϕ• ) = C(ϕ• ) = P(X) and α = card. Now Aσ(S) consists of the countable and the cocountable subsets of X. Thus if E ⊂ X is neither countable nor cocountable, then for each A ∈ Aσ(S) the difference set AΔE = (A ∩ E) ∪ (A ∩ E ) is uncountable and hence has α(AΔE) = ∞.
3. Another inner characterization theorem The topic of the present section came up in the frame of Radon measures. Let X be a Hausdorff topological space with the obvious set systems Op(X) and Comp(X) and the Borel σ algebra Bor(X). A measure α : A → [0, ∞] on a σ algebra A ⊃ Bor(X) is called Radon iff α|Comp(X) < ∞ and α is inner regular Comp(X). This is the actual definition initiated – as far as the author is
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aware – in Berg-Christensen-Ressel [1] Chapter 2, while the traditional definition fortified α|Comp(X) < ∞ to local finiteness: Each point of X and hence each A ∈ Comp(X) is contained in some U ∈ Op(X) with α(U ) < ∞. The traditional definition is still in frequent use, for example in Fremlin [3]. In the present new development of measure and integration, as before on a nonvoid set X and for • = στ , the counterpart of the Radon measures in the actual sense can be viewed to be the inner • premeasures ϕ : S → [0, ∞[ (for • = τ or for • = στ ). A counterpart of the Radon measures in the traditional sense can then be obtained in form of certain particular inner • premeasures ϕ : S → [0, ∞[ in the context of the • complemental pairs in the sense of [6] Section 4. These particular inner • premeasures ϕ will be the heroes of the present section. We start to recall the relevant concepts and facts. We define a pair of lattices S and T with ∅ to be • complemental iff T ⊂ (S-S• )⊥ and S ⊂ (T-T• )⊥, with M⊥ := {M : M ∈ M} for M a nonvoid set system. In this situation an inner • premeasure ϕ : S → [0, ∞[ is called • tame for S and T iff ϕ• is outer regular T• at S; note that T• ⊂ (S-S• )⊥ ⊂ C(ϕ• ). Equivalent is the much simpler condition that each S ∈ S is contained in some T ∈ T• with ϕ• (T ) < ∞, which is a certain local finiteness condition. Likewise an outer • premeasure ψ : T → [0, ∞] is called • tame for S and T iff ψ • |S < ∞ and ψ • is inner regular S• at T; as above note that S• ⊂ (T-T• )⊥ ⊂ C(ψ • ). After these definitions we recall [6] 4.6, which asserts that the two kinds of set functions ϕ : S → [0, ∞[ and ψ : T → [0, ∞] are in one-to-one correspondence via ψ = ϕ• |T and ϕ = ψ • |S, and henceforth are called • complemental pairs for S and T. For these pairs one has ϕ• ψ • , with ϕ• = ψ • on S• and T• and [ψ • |C(ψ • ) < ∞], and C(ϕ• ) = C(ψ • ). In the concrete situation of Radon measures this correspondence is due to Laurent Schwartz [9]. In the context of the present paper we shall specialize the lattices T with ∅ ∈ T to those with ∅, X ∈ T and T = T• , for short called the • topologies, because in case • = τ these are the familiar topologies. Then the relation that S and T be • complemental reads S ⊂ S• ⊂ T⊥ ⊂ S-S• . After this we define a content α : A → [0, ∞] on an algebra A to be • quasiRadon for a • topology T iff it is SC and satisfies T ⊂ A (and hence T⊥ ⊂ A) with α is inner regular H := {H ∈ T⊥ : H is enclosable [α|T < ∞]}, α|T is upward • continuous; note that H is a lattice with ∅ ∈ H = H• ⊂ A. We shall see next that for • = στ these α are measures on σ algebras, and then conclude from the respective definition in [6] Section 4 that in case • = τ we obtain the quasi-Radon measures in the sense of Fremlin [2, 3]. 3.1 Theorem. Let α : A → [0, ∞] be a content on an algebra A which is • quasiRadon for the • topology T. Then the above H and T form a • complemental pair ⊂ A. Moreover ξ := α|H < ∞ is an inner • premeasure, η := α|T is an outer • premeasure,
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and the two are • tame and form a • complemental pair for H and T. We have α = ξ• |C(ξ• ) (but α need not be = η • |C(η • )). We note that α = ξ• |C(ξ• ) combined with ξ = η • |H shows that the restriction η = α|T determines α. Also note that ξ• = ξ and η • = η from [6] 2.2.4) since H = H• and T = T• . An example for the final assertion can be obtained from [5] Example 4.8 due to Dowker. Proof. 1) We claim that H ⊂ T⊥ ⊂ H-H, so that H and T are • complemental. To see the second inclusion let M ∈ T⊥. For H ∈ H ⊂ T⊥ then M ∩ H ∈ T⊥, and hence M ∩ H ∈ H since M ∩ H ⊂ H is enclosable [α|T < ∞]. Thus M ∈ H-H. 2) We claim that ξ = α|H is downward • continuous. To see this (for • = στ ) let M ⊂ H be nonvoid • with M ↓ D ∈ H. To be shown is inf{α(M ) : M ∈ M } = α(D). In view of directedness we can assume that all M ∈ M are contained in some fixed T ∈ T with α(T ) < ∞. Then {T \ M : M ∈ M} ⊂ T is nonvoid • with ↑ T \ D ∈ T, and the claim reads sup{α(T \ M ) : M ∈ M} = α(T \ D) and hence is true. 3) The definition of • quasi-Radon and 2) assert that α is an inner • extension of ξ = α|H. Thus ξ is an inner • premeasure. Since α is SC we obtain from 2.1 that α = ξ• |C(ξ• ). Moreover ξ is • tame for H and T, since each H ∈ H is contained in some T ∈ T with α(T ) = ξ• (T ) < ∞. 4) Now η := ξ• |T = α|T is the unique outer • premeasure η : T → [0, ∞] which is • tame for H and T and such that ξ and η form a • complemental pair for H and T. This completes the proof. 3.2 Theorem. Let ϕ : S → [0, ∞[ be an inner • premeasure and T be a • topology, and assume that S and T are • complemental and ϕ is • tame for S and T. Then α := ϕ• |C(ϕ• ) is • quasi-Radon for T. Proof. We have α = ϕ• |A on A := C(ϕ• ). 1) α is SC in view of 2.1. 2) We have T ⊂ (S-S• )⊥ ⊂ C(ϕ• ) = A from [6] 3.5 and hence T⊥ ⊂ A, so that the above H is well defined. Moreover S• ⊂ H, since by assumption on the one hand S ⊂ S• ⊂ T⊥, and on the other hand each S ∈ S• is contained in some T ∈ T with α(T ) = ϕ• (T ) < ∞. It follows that α = ϕ• |A is inner regular H. 3) α = ϕ• |C(ϕ• ) has α|(S-S• )⊥ upward • continuous. This is clear for • = σ and in [6] 3.6.ii) for • = τ . Thus T ⊂ (S-S• )⊥ implies that α|T is upward • continuous. It follows that α is • quasi-Radon for T. 3.3 Inner Characterization Theorem. Let α : A → [0, ∞] be a content on an algebra A. For a • topology T then α = ϕ• |C(ϕ• ) for some inner • premeasure ϕ : S → [0, ∞[ such that S and T are • complemental and ϕ is • tame for S and T ⇐⇒ α is • quasi-Radon for T. This is an immediate consequence of 3.1 and 3.2. We remark that the result in the case • = τ restricted to measures on σ algebras has been formulated without proof earlier in [6] 4.9.
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4. Application to the inner measure constructions The present inner characterization Theorems 2.2 and 3.3 illuminate the connection between the inner • extension Theorem [4] 6.31 = [6] 3.5 = [7] Section 4 of the present author and the so-called inner measure constructions of Fremlin [3], the basic results of which are lemma 413H for the case • = , theorem 413J for the case • = σ, theorem 415K for the case • = τ , where the last one is restricted to the topological situation and to local finiteness. Both times these are the basic results for the fundamental task of extension of basic set functions: One assumed an isotone set function ϕ : S → [0, ∞[ on a lattice S with ∅ ∈ S and ϕ(∅) = 0 and formulated certain conditions on ϕ in order that it possesses a (unique) extension which is (at least) a content α : A → [0, ∞] on an algebra A and has certain desired properties (in the interest of a common set-up we pass over certain technical deviations). In this context then the present inner characterization Theorems 2.2 and 3.3 can be read so as to assert that (at least in the common cases • = στ ) the desired properties in the two theories are equivalent – which is not at all visible at first sight. After this we turn to the two collections of conditions imposed upon ϕ. In Fremlin [3] the conditions are (α) the crude tightness ()
ϕ(B) = ϕ(A) + ϕ (B \ A) for all A ⊂ B in S,
and (β) ϕ to be (downward) • continuous at ∅, and in addition in case • = τ , which is restricted to the topological situation and to local finiteness, an appropriate local finiteness condition (γ) with the topology T in question. In the new context the conditions can be formulated as (αβ) the • tightness (•)
ϕ(B) = ϕ(A) + ϕ• (B \ A) for all A ⊂ B in S,
and under local finiteness with the • topology T as above (γ) ϕ to be • tame for S and T. Then the results in the two theories show a remarkable difference: In Fremlin [3] the conditions (α)(β) and (α)(β)(γ) are sufficient conditions for the aims in question, but in cases • = στ they are not equivalent ones, at least without the additional requirement S = S• , whereas in the new situation of the author the conditions (αβ) and (αβ)(γ) are equivalent conditions in all cases • = στ and for all S. The reason is that the crude envelope ϕ is the appropriate one for the case • = but not for • = στ , where the respective • envelope ϕ• attains its place: see the subsequent simple example extracted from [4] 6.32, and for the entire context [7] Sections 3–5. One notes that condition (α) of Fremlin [3] is identical with that in Topsøe [10], [11] from 1970. In the meantime the new development due to the present author had made clear that for • = στ the new inner and outer • envelopes ϕ• and ϕ• are the adequate ones.
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4.1 Example. We take X = R and S = Op(R), and ϕ = δa |S for some fixed a ∈ R. Thus ϕ is isotone with ϕ(∅) = 0, and modular and downward • continuous for • = στ . In the cases • = στ one verifies that ϕ• = δa partout, so that the • tightness condition (•) is fulfilled. Hence the inner • extension theorem [6] 3.5 asserts that ϕ is an inner • premeasure, so that the equivalent desired properties in the two theories are fulfilled. But ϕ ({a}) = 0, which implies that the crude tightness condition () is violated for B ∈ S with a ∈ B and A = B \ {a}.
References [1] C. Berg, J.P.R. Christensen and P. Ressel, Harmonic Analysis on Semigroups. Springer 1984. [2] D.H. Fremlin, Topological Riesz Spaces and Measure Theory. Cambridge Univ. Press 1974. [3] D.H. Fremlin, Measure Theory Vol. 1–4. Torres Fremlin 2000–2003 (in a numbered reference the first digit indicates its volume). http://www.essex.ac.uk/maths/staff/ fremlin/mt.htm. [4] H. K¨ onig, Measure and Integration: An Advanced Course in Basic Procedures and Applications. Springer 1997. [5] H. K¨ onig, Measure and Integration: Mutual generation of outer and inner premeasures. Annales Univ. Saraviensis Ser. Math. 9(1998) No. 2, 99–122. [6] H. K¨ onig, Measure and Integration: An attempt at unified systematization. Rend. Istit. Mat. Univ. Trieste 34(2002), 155–214. Preprint No. 42 under http://www.math. uni-sb.de. [7] H. K¨ onig, Measure and Integral: New foundations after one hundred years. In: Functional Analysis and Evolution Equations (The G¨ unter Lumer Volume). Birkh¨ auser 2007, pp. 405–422. Preprint No. 175 under http://www.math.uni-sb.de. [8] J. L o´s and E. Marczewski, Extensions of measure. Fund.Math. 36(1949), 267–276. [9] L. Schwartz, Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford Univ. Press 1973. [10] F. Topsøe, Compactness in spaces of measures. Studia Math. 36(1970), 195–212. [11] F. Topsøe, Topology and Measure. Lect. Notes Math. 133, Springer 1970. Heinz K¨ onig Universit¨ at des Saarlandes Fakult¨ at f¨ ur Mathematik und Informatik D-66123 Saarbr¨ ucken, Germany e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 303–311 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Vector Measures of Bounded γ-variation and Stochastic Integrals Jan van Neerven and Lutz Weis Abstract. We introduce the class of vector measures of bounded γ-variation and study its relationship with vector-valued stochastic integrals with respect to Brownian motions. Mathematics Subject Classification (2000). 46G10, 60H05. Keywords. Vector measures, bounded randomised variation, stochastic integration.
1. Introduction It is well known that stochastic integrals can be interpreted as vector measures, the identification being given by the identity F (A) = φ dB. A
Here, the driving process B is a (semi)martingale (for instance, a Brownian motion), and φ is a stochastic process satisfying suitable measurability and integrability conditions. This observation has been used by various authors as the starting point of a theory of stochastic integration for vector-valued processes. Let X be a Banach space. In [5] we characterized the class of functions φ : (0, 1) → X which are stochastically integrable with respect to a Brownian motion (Wt )t∈[0,1] as being the class of functions for which the operator Tφ : L2 (0, 1) → X, 1 f (t)φ(t) dt, Tφ f := 0
The first named author is supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).
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belongs to the operator ideal γ(L2 (0, 1), X) of all γ-radonifying operators. Indeed, we established the Itˆo isomorphism 2 1 f dW = Tf 2γ(L2 (0,1),X) . E 0
The linear subspace of all operators in γ(L2 (0, 1), X) of the form T = Tf for some function f : (0, 1) → X is dense, but unless X has cotype 2 it is strictly smaller than γ(L2 (0, 1), X). This means that in general there are operators T ∈ γ(L2 (0, 1), X) which are not representable by an X-valued function. Since the space of test functions D(0, 1) embeds in L2 (0, 1), by restriction one could still think of such operators as X-valued distributions. It may be more intuitive, however, to think of T as an X-valued vector measure. We shall prove (see Theorem 2.3 and the subsequent remark) that if X does not contain a closed subspace isomorphic to c0 , then the space γ(L2 (0, 1), X) is isometrically isomorphic in a natural way to the space of X-valued vector measures on (0, 1) which are of bounded γ-variation. This gives a ‘measure theoretic’ description of the class of admissible integrands for stochastic integrals with respect to Brownian motions. The condition c0 ⊆ X can be removed if we replace the space of γ-radonifying operators by the larger space of all γ-summing operators (which contains the space of all γ-radonifying operators isometrically as a closed subspace). Vector measures of bounded γ-variation behave quite differently from vector measures of bounded variation. For instance, the question whether an X-valued vector measure of bounded γ-variation can be represented by an X-valued function is not linked to the Radon-Nikod´ ym property, but rather to the type 2 and cotype 2 properties of X (see Corollaries 2.5 and 2.6). In Section 3 we consider yet another class of vector measures whose variation is given by certain random sums, and we show that a function φ : (0, 1) → X is stochastically integrable with respect to a Brownian motion (Wt )t∈[0,1] on a probability space (Ω, P) if and only if the formula F (A) := A φ dW defines an L2 (Ω; X)-valued vector measure F in this class.
2. Vector measures of bounded γ-variation Let (S, Σ) be a measurable space, X a Banach space, and (γn )n1 a sequence of independent standard Gaussian random variables defined on a probability space (Ω, F , P). Definition 2.1. We say that a countably additive vector measure F has bounded γ-variation with respect to a probability measure μ on (S, Σ) if F Vγ (μ;X) < ∞, where N 1 F (An ) 2 2 γn & , F Vγ (μ;X) := sup E μ(An ) n=1 the supremum being taken over all finite collections of disjoint sets A1 , . . . , AN ∈ Σ such that μ(An ) > 0 for all n = 1, . . . , N .
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It is routine to check (e.g., by an argument similar to [4, Proposition 5.2]) that the space Vγ (μ; X) of all countably additive vector measures F : Σ → X which have bounded γ-variation with respect to μ is a Banach space with respect to the norm · Vγ (μ;X) . Furthermore, every vector measure which is of bounded γ-variation is of bounded 2-semivariation. In order to give a necessary and sufficient condition for a vector measure to have bounded γ-variation we need to introduce the following terminology. A bounded operator T : H → X, where H is a Hilbert space, is said to be γsumming if there exists a constant C such that for all finite orthonormal systems {h1 , . . . , hN } in H one has N 2 E γn T hn C 2 . n=1
The least constant C for which this holds is called the γ-summing norm of T , notation T γ∞ (H,X) . With respect to this norm, the space γ∞ (H, X) of all γsumming operators from H to X is a Banach space which contains all finite rank operators from H to X. In what follows we shall make free use of the elementary properties of γ-summing operators. For a systematic exposition of these we refer to [2, Chapter 12] and the lecture notes [4]. Theorem 2.2. Let A be an algebra of subsets of S which generates the σ-algebra Σ, and let F : A → X be a finitely additive mapping. If, for some 1 p < ∞, T : Lp (μ) → X is a bounded operator such that F (A) = T 1A,
A∈A,
then F has a unique extension to a countably additive vector measure on Σ which is absolutely continuous with respect to μ. If T : L2 (μ) → X is γ-summing, then the extension of F has bounded γ-variation with respect to μ and we have F Vγ (μ;X) T γ∞ (L2 (μ),X) . Proof. We define the extension F : Σ → X by F (A) :='T 1A, A ∈ Σ. To see that F is countably additive, consider a disjoint union A = n1 An with An , A ∈ Σ. = 1A in Lp (μ) and therefore Then limN →∞ 1' N n=1 An lim
N →∞
N n=1
F (An ) = lim T N →∞
N
1An = T 1A = F (A).
n=1
The absolute continuity of F is clear. To prove uniqueness, suppose F˜ : Σ → X is another countably additive vector measure extending F . For each x∗ ∈ X ∗ ,
F˜ , x∗ and F, x∗ are finite measures on Σ which agree on A , and therefore by Dynkin’s lemma they agree on all of Σ. This being true for all x∗ ∈ X ∗ , it follows that F˜ = F by the Hahn-Banach theorem. Suppose next that T : L2 (μ) → X is γ-summing, and consider a finite collection of disjoint sets A1 , . . . , AN in Σ such that μ(An ) > 0 for all n = 1, . . . , N .
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The functions fn = 1An /
& μ(An ) are orthonormal in L2 (μ) and therefore
N N 2 F (An ) 2 & E γn γn T fn T 2γ∞(L2 (μ),X) . = E μ(An ) n=1 n=1
It follows that F has bounded γ-variation with respect to μ and that F Vγ (μ;X) T γ∞ (L2 (μ),X) . Theorem 2.3. For a countably additive vector measure F : Σ → X the following assertions are equivalent: (1) F has bounded γ-variation with respect to μ; (2) There exists a γ-summing operator T : L2 (μ) → X such that F (A) = T 1A,
A ∈ Σ.
In this situation we have F Vγ (μ;X) = T γ∞ (L2 (μ),X) . Proof. (1)⇒(2): Suppose that F has bounded γ-variation with respect to μ. For N a simple function f = n=1 cn 1An , where the sets An ∈ Σ are disjoint and of positive μ-measure, define N T f := cn F (An ). n=1
By the Cauchy-Schwarz inequality, for all x∗ ∈ X ∗ we have N N # &
F (An ), x∗ ## # γm cm μ(Am ) · γn & | T f, x∗ | = #E # μ(An ) m=1 n=1 N N #2 12 # # &
F (An ), x∗ ##2 12 # # # E# E# γn cn μ(An )# γn & # μ(An ) n=1 n=1
N
12 |cn |2 μ(An ) F Vγ (μ;X) x∗
n=1
= f L2 (μ) F Vγ (μ;X) x∗ . It follows that T is bounded and T L (L2 (μ),X) F Vγ (μ;X) . To prove that T is γ-summing we shall first make the simplifying assumption that the σ-algebra Σ is countably generated. Under this assumption .there exists an increasing sequence of finite σ-algebras (Σn )n1 such that Σ = n1 Σn . Let Pn be the orthogonal projection in L2 (μ) onto L2 (Σn , μ) and put Tn := T ◦ Pn . These operators are of finite rank and we have limn→∞ Tn → T in the strong operator topology of L (L2 (μ), X). Fix an index n 1 for the moment. Since Σn is finitely generated there 'N exists a partition S = j=1 Aj , where the disjoint sets A1 , . . . , AN generate Σn . Assuming that μ(Aj ) > 0 for all j = 1, . . . , M and μ(Aj ) = 0 for j = M +1, . . . , N ,
Vector Measures of Bounded γ-variation the functions gj = 1Aj / L2 (Σn , μ) and
307
& μ(Aj ), j = 1, . . . , M , form an orthonormal basis for
Tn 2γ∞ (L2 (μ),X) = Tn 2γ∞ (L2 (Σn ,μ),X) M M 2 F (Aj ) 2 = E γj T gj = E γj & F 2Vγ (μ;X) , μ(A ) n j=1 j=1
the first identity being a consequence of [4, Corollary 5.5] and the second of [4, Lemma 5.7]. It follows that the sequence (Tn )n1 is bounded in γ∞ (L2 (μ), X). By the Fatou lemma, if {f1 , . . . , fk } is any orthonormal family in L2 (μ), then k k 2 2 γj T fj lim inf E γj Tn fj Tn 2γ∞ (L2 (μ),X) F 2Vγ (μ;X) . E n→∞
j=1
j=1
This proves that T is γ-summing and T γ∞(L2 (μ),X) F Vγ (μ;X) . It remains to remove the assumption that Σ is countably generated. The preceding argument shows that if we define T in the above way, then its restriction to L2 (Σ , μ) is γ-summing for every countably generated σ-algebra Σ ⊆ Σ, with a uniform bound T γ∞ (L2 (Σ ,μ),X) F Vγ (μ;X) . Since every finite orthonormal family {f1 , . . . , fk } in L2 (μ) is contained in L2 (Σ , μ) for some countably generated σ-algebra Σ ⊆ Σ, we see that k 2 γj T fj T 2γ∞(L2 (Σ ,μ),X) F 2Vγ (μ;X) . E j=1
It follows that T is γ-summing and T γ∞ (L2 (μ),X) F Vγ (μ;X) . (2)⇒(1): This implication is contained in Theorem 2.2.
By a theorem of Hoffmann-Jørgensen and Kwapie´ n [3, Theorem 9.29], if X is a Banach space not containing an isomorphic copy of c0 , then for any Hilbert space H one has γ∞ (H, X) = γ(H, X), where by definition γ(H, X) denotes the closure in γ∞ (H, X) of the finite rank operators from H to X. Since any operator in this closure is compact we obtain: Corollary 2.4. If X does not contain an isomorphic copy of c0 and F : Σ → X has bounded γ-variation with respect to μ, then F has relatively compact range. Using the terminology of [5], a theorem of Rosi´ nski and Suchanecki [6] asserts that if X has type 2 we have a continuous inclusion L2 (μ; X) → γ(L2 (μ), X) and that if X has cotype 2 we have a continuous inclusion γ∞ (L2 (μ), X) → L2 (μ; X). In both cases the embedding is contractive, and the relation between the operator T and the representing function φ is given by f φ dμ, f ∈ L2 (μ). Tf = S
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If dim L2 (μ) = ∞, then in the converse direction the existence of a continuous embedding L2 (μ; X) → γ∞ (L2 (μ), X) (respectively γ(L2 (μ), X) → L2 (μ; X)) actually implies the type 2 property (respectively the cotype 2 property) of X. Corollary 2.5. Let X have type 2. For all φ ∈ L2 (μ; X) the formula F (A) := φ dμ, A ∈ Σ, A
defines a countably additive vector measure F : Σ → X which has bounded γvariation with respect to μ. Moreover, F Vγ (μ;X) φL2 (μ;X) . If dim L2 (μ) = ∞, this property characterises the type 2 property of X. Proof. By the theorem of Rosi´ nski and Suchanecki, φ represents an operator T ∈ γ(L2 (μ), X) such that T 1A = A φ dμ = F (A) for all A ∈ Σ. The result now follows from Theorem 2.2. The converse direction follows from Theorem 2.3 and the preceding remarks. Corollary 2.6. Let X have cotype 2. If F : Σ → X has bounded γ-variation with respect to μ, there exists a function φ ∈ L2 (μ; X) such that F (A) = φ dμ, A ∈ Σ. A
Moreover, φL2 (μ;X) F Vγ (μ;X) . If dim L (μ) = ∞, this property characterises the cotype 2 property of X. 2
Proof. By Theorem 2.3 there exists an operator T ∈ γ∞ (L2 (μ), X) such that F (A) = T 1A for all A ∈ Σ. Since X has cotype 2, X does not contain an isomorphic copy of c0 and therefore the theorem of Hoffmann-Jørgensen and Kwapie´ n implies that T ∈ γ(L2 (μ), X). Now the theorem of Rosi´ nski and Suchanecki shows that T is represented by a function φ ∈ L2 (μ; X). The converse direction follows from Theorem 2.2 and the remarks preceding Corollary 2.5.
3. Vector measures of bounded randomised variation Let (S, Σ) be a measurable space and (rn )n1 a Rademacher sequence, i.e., a sequence of independent random variables with P(rn = ±1) = 12 . Definition 3.1. A countably additive vector measure F : Σ → X is of bounded randomised variation if F V r (μ;X) < ∞, where N 2 12 rn F (An ) , F V r (μ;X) = sup E n=1
the supremum being taken over all finite collections of disjoint sets A1 , . . . , AN ∈ Σ.
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Clearly, if F is of bounded variation, then F is of bounded randomised variation. The converse fails; see Example 1. If X has finite cotype, standard comparison results for Banach space-valued random sums [2, 3] imply that an equivalent norm is obtained when the Rademacher variables are replaced by Gaussian variables. It is routine to check that the space V r (μ; X) of all countably additive vector measures F : Σ → X of bounded randomised variation is a Banach space with respect to the norm · V r (μ;X) . In Theorem 3.2 below we establish a connection between measures of bounded randomised variation and the theory of stochastic integration. For this purpose we need the following terminology. A Brownian motion on (Ω, F , P) indexed by another probability space (S, Σ, μ) is a mapping W : Σ → L2 (Ω) such that: (i) For all A ∈ Σ the random variable W (A) is centred Gaussian with variance E(W (A))2 = μ(A); (ii) For all disjoint A, B ∈ Σ the random variables W (A) and W (B) are independent. A strongly μ-measurable function φ : S → X is stochastically integrable with respect to W if for all x∗ ∈ X ∗ we have φ, x∗ ∈ L2 (μ) (i.e., f belongs to L2 (μ) scalarly) and for all A ∈ Σ there exists a strongly measurable random variable YA : Ω → X such that for all x∗ ∈ X ∗ we have
YA , x∗ = φ, x∗ dW A
almost surely. Note that each YA is centred Gaussian and therefore belongs to L2 (Ω; X) by Fernique’s theorem; the above equality then holds in the sense of L2 (Ω). We define the stochastic integral of φ over A by A φ dW := YA . For more details and various equivalent definitions we refer to [5]. Theorem 3.2. Let W : Σ → L2 (Ω) be a Brownian motion. For a strongly μmeasurable function φ : S → X the following assertions are equivalent: (1) φ is stochastically integrable with respect to W ; (2) φ belongs to L2 (μ) scalarly and there exists a countably additive vector measure F : Σ → X, of bounded γ-variation with respect to μ, such that for all x∗ ∈ X ∗ we have
F (A), x∗ = φ, x∗ dμ, A ∈ Σ; A
(3) φ belongs to L2 (μ) scalarly and there exists a countably additive vector measure G : Σ → L2 (Ω; X) of bounded randomised variation such that for all x∗ ∈ X ∗ we have
G(A), x∗ = φ, x∗ dW, A ∈ Σ. A
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In this situation we have F Vγ (μ;X) = GV r (μ;L2 (Ω;X))
2 12 = E φ dW . S
Proof. (1)⇔(2): This equivalence is immediate from Theorem 2.3 and the fact, proven in [5], that φ is stochastically integrable with respect to W if and only there exists an operator T ∈ γ(L2 (μ), X) such that Tf = f φ dμ, f ∈ L2 (μ). S
In this case we also have
2 12 T γ(L2(μ),X) = E φ dW . S
In view of Theorem 2.3, this proves the identity 2 12 . F Vγ (μ;X) = E φ dW S
(1)⇒(3): Define G : Σ → L (Ω; X) by G(A) := φ dW, 2
A ∈ Σ.
A
By the γ-dominated convergence theorem [5], G is countably additive. To prove that G is of bounded randomised variation we consider disjoint sets A1 , . . . , AN ∈ ˜ F˜ , P), ˜ then by Σ. If (˜ rn )n1 is a Rademacher sequence on a probability space (Ω, randomisation we have N N 2 2 ˜ ˜ E r˜n G(An ) 2 = EE r˜n φ dW n=1
L (Ω;X)
An
n=1
N
= E
n=1
An
2 2 φ dW E φ dW , S
'N with equality if n=1 An = S. In the second identity we used that the X-valued random variables An φ dW are independent and symmetric. The final inequality follows by, e.g., covariance domination [5] or an application of the contraction principle. It follows that G is a countably additive vector measure of bounded randomised variation and 2 12 GV r (μ;X) = E φ dW . S
(3)⇒(1): This is immediate from the definition of stochastic integrability. Example 1. If W is a standard Brownian motion on (Ω, F , P) indexed by the Borel interval ([0, 1], B, m), then W is a countably additive vector measure with values in L2 (Ω) which is of bounded randomised variation, but of unbounded
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variation. The first claim follows from Theorem 3.2 since W (A) = A 1 dW for all Borel sets A. To see that W is of unbounded variation, note that for any partition 0 = t0 < t1 < · · · < tN −1 < tN = 1 we have N
W ((tn−1 , tn ))L2 (Ω) =
n=1
N & tn − tn−1 . n=1
The supremum over all possible partitions of [0, 1] is unbounded.
References [1] J. Diestel and J.J. Uhl, Vector measures. Mathematical Surveys, Vol. 15, Amer. Math. Soc., Providence (1977). [2] J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators. Cambridge Studies in Adv. Math., Vol. 34, Cambridge, 1995. [3] M. Ledoux and M. Talagrand, Probability in Banach spaces. Ergebnisse d. Math. u. ihre Grenzgebiete, Vol. 23, Springer-Verlag, 1991. [4] J.M.A.M. van Neerven, Stochastic evolution equations. Lecture notes of the 11th International Internet Seminar, TU Delft, downloadable at http://fa.its.tudelft.nl/∼isemwiki. [5] J.M.A.M. van Neerven and L. Weis, Stochastic integration of functions with values in a Banach space. Studia Math. 166 (2005), 131–170. [6] J. Rosi´ nski and Z. Suchanecki, On the space of vector-valued functions integrable with respect to the white noise. Colloq. Math. 43 (1980), 183–201. Jan van Neerven Delft University of Technology Delft Institute of Applied Mathematics P.O. Box 5031 NL-2600 GA Delft, The Netherlands e-mail: [email protected] Lutz Weis University of Karlsruhe Mathematisches Institut I D-76128 Karlsruhe, Germany e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 313–322 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Does a Compact Operator Admit a Maximal Domain for its Compact Linear Extension? Susumu Okada Abstract. Given a Banach-space-valued, compact linear operator defined on a σ-order continuous Banach function space, we determine precisely when this operator admits a maximal compact linear extension. Mathematics Subject Classification (2000). Primary 28B05, 47B07, 47B38; Secondary 46E30. Keywords. Banach function space, compact operator, integration operator, maximal compact extension, vector measure.
1. Introduction Let X(μ) be a (complex) Banach function space (briefly B.f.s.) based on a positive finite measure space (Ω, Σ, μ) and assume that the given lattice norm on X(μ) is σ-order continuous. Given a continuous linear operator T from X(μ) into a (complex) Banach space E, the σ-additive set function mT : A → T (χA ), for A ∈ Σ, is called the vector measure associated with T . The space L1 (mT ) of all C-valued mT -integrable functions is equipped with the topology of uniform convergence of indefinite integrals. Suppose that T is μ-determined, that is, the mT -null and μ-null sets are the same. We then have the continuous inclusion X(μ) ⊆ L1 (mT ). The Curbera-Ricker Optimal Domain Theorem, [1, Corollary 3.3], asserts that L1 (mT ) is the largest σ-order continuous B.f.s. (over (Ω, Σ, μ) ) into which X(μ) is continuously embedded and to which T admits an E-valued, continuous linear extension. Such a maximal continuous linear extensionis unique (up to isomorphism) and realized as the integration operator ImT : f → Ω f dmT , for f ∈ L1 (mT ), (see also [13, Theorem 4.14]). Now consider the case when T : X(μ) → E is compact. It can happen that its maximal continuous linear extension is non-compact. For example, this is the case with the Volterra integral operator on L1 ([0, 1]); see the example at the end of Section 2. A natural question raised by E.A. S´ anchez P´erez is whether or not
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S. Okada
T admits a maximal compact linear extension in a similar manner to its maximal continuous linear extension. To be precise, we say that T admits a maximal compact linear extension if there exists a largest σ-order continuous B.f.s. Y (μ) into which X(μ) is continuously embedded and to which T admits an E-valued, compact linear extension. Our main result is the following one which answers S´ anchez P´erez’s question; its proof will be given in Section 2. Theorem 1.1. Suppose that X(μ) is a σ-order continuous B.f.s. based on a positive, finite measure space (Ω, Σ, μ) and that T : X(μ) → E is a μ-determined, compact linear operator. Then T admits a maximal compact linear extension if and only if the integration operator ImT : L1 (mT ) → E is compact. Section 3 provides an analogue of Theorem 1.1 for weakly compact operators (see Proposition 3.1).
2. Proof of Theorem 1.1 We first formulate some definitions and facts needed to establish Theorem 1.1. Let (Ω, Σ, μ) be a positive, finite measure space. The characteristic function of each set A ∈ Σ is denoted by χA . Define sim Σ :=
n
aj χA(j) : aj ∈ C, A(j) ∈ Σ, j = 1, . . . , n, n ∈ N .
j=1
Let L0 (μ) denote the (complex) vector space of all (equivalence classes of) C-valued Σ-measurable functions on Ω modulo μ-null functions. We equip L0 (μ) with the μa.e. pointwise order for it positive cone so that L0 (μ) is a complex vector lattice. A complex vector sublattice X(μ) of L0 (μ) is called a Banach function space (briefly B.f.s.) based on (Ω, Σ, μ) if the following conditions hold: (i) X(μ) is an order ideal of L0 (μ), that is, f ∈ X(μ) whenever f ∈ L0 (μ) and |f | ≤ |g| for some g ∈ X(μ); (ii) X(μ) is equipped with a lattice norm · X(μ) for which it is complete; and (iii) sim Σ ⊆ X(μ). This definition of B.f.s.’ over C and further studies of such spaces occur in [13, Chapter 2]. The lattice norm · X(μ) on a B.f.s. X(μ) is said to be σ-order continuous if X(μ) % fn ↓ 0 (μ-a.e. pointwise) implies that fn X(μ) → 0 as n → ∞; we also say simply that the space X(μ) is a σ-order continuous B.f.s. It is clear that sim Σ is dense in any σ-order continuous B.f.s. over (Ω, Σ, μ). Examples of σ-order continuous B.f.s.’ include Lp (μ) for 1 ≤ p < ∞, the Lorentz spaces Lp,q (μ) for 1 ≤ q ≤ p < ∞ and certain Orlicz spaces. Given a Banach space E with norm · E , its closed unit ball is denoted by B[E] := {x ∈ E : xE ≤ 1}.
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The dual space E ∗ of E is equipped with the usual dual norm · E ∗ and the duality between E and E ∗ is denoted by x, x∗ for x ∈ E and x∗ ∈ E ∗ . Fix a σ-order continuous B.f.s. X(μ) based on (Ω, Σ, μ). Given a continuous linear operator T from X(μ) into a Banach space E, the E-valued, finitely additive set function A ∈ Σ, mT : A → T (χA ), is actually σ-additive, as a consequence of the continuity of T and the σ-order continuity of X(μ). By definition a vector measure is a Banach-space-valued, σadditive set function defined on a σ-algebra of subsets of a non-empty set. So, we can call mT the vector measure associated with T . Given x∗ ∈ E ∗ , define a scalar measure mT , x∗ : Σ → C by mT , x∗ (A) :=
mT (A), x∗ , for A ∈ Σ. Its variation measure | mT , x∗ | on Σ is necessarily positive and finite. According to [8, Definition 2.1], a Σ-measurable function f : Ω → C is called mT -integrable if (I-1) f is mT , x∗ -integrable for every x∗ ∈ E ∗ , and (I-2) for every A ∈ Σ, there exists a unique element A f dmT ∈ E satisfying , - f dmT , x∗ = f d mT , x∗ , x∗ ∈ E ∗ . A
A
Clearly elements of sim Σ are mT -integrable. According to [8, Theorem 2.4], a Cvalued, Σ-measurable function f on Ω is mT -integrable if and only if there exist ∞ sn ∈ sim Σ, for n ∈ N, converging to f pointwise and for which s dmT n=1 A n is a Cauchy sequence in E for every A ∈ Σ. In this case we have A f dmT = limn→∞ A sn dmT . As an application of this fact, Theorem 3.1 in [1] guarantees that every function f ∈ X(μ) is mT -integrable and T (f ) = Ω f dmT (see also [13, Proposition 4.4] ). The complex vector space L1 (mT ), consisting of all mT -integrable functions on Ω, is equipped with the seminorm · L1 (mT ) defined by |f | d| mT , x∗ |, f ∈ L1 (mT ). f L1 (mT ) := sup x∗ ∈B[E ∗ ]
Ω
Since supA∈Σ A f dmT E ≤ f L1 (mT ) ≤ 4 supA∈Σ A f dmT E for every f ∈ L1 (mT ), [13, (3.31)], the topology given by · L1 (mT ) coincides with that of uniform convergence of indefinite integrals over Σ. Every function f ∈ L1 (mT ) satisfying f L1(mT ) = 0 is called mT -null. We denote by N (mT ) the subspace of L1 (mT ) consisting of all mT -null functions. A set A ∈ Σ is called mT -null if χA ∈ N (mT ). It is easy to see from the definition that a function f ∈ L1 (mT ) is mT -null if and only if f (ω) = 0 for mT -a.e. ω ∈ Ω. Every μ-null set A ∈ Σ is necessarily mT -null because χA (ω) = 0 for μ-a.e. ω ∈ Ω. The converse is not always valid. By definition the operator T is μ-determined when the converse also holds, i.e., the mT -null and μ-null sets coincide. The case when T is not μ-determined can be treated by considering the measure μ restricted to a smaller measurable set; see [13, Proposition 4.28] for the details.
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For the rest of this paper we assume that T is μ-determined. Unless stated otherwise, we identify the seminormed space L1 (mT ) with its quotient space L1 (mT )/N (mT ) so that the seminorm · L1 (mT ) will be identified with its associated quotient norm. The space L1 (mT ) is then a complex vector lattice with respect to the mT -a.e. pointwise order and · L1 (mT ) is a lattice norm on L1 (mT ). Moreover, L1 (mT ) is complete. For this, see [7, Chapter IV] over R while the complex case can be found in [4], [15]. Moreover, the Lebesgue Dominated Convergence Theorem holds in L1 (mT ), [8, Theorem 2.2], from which it follows that L1 (mT ) is a σ-order continuous B.f.s. based on (Ω, Σ, μ). In particular, sim Σ is dense in L1 (mT ). Define the integration operator ImT : L1 (mT ) → E associated with mT by f dmT , f ∈ L1 (mT ). ImT (f ) := Ω
Then ImT is linear and continuous with operator norm one, [13, p. 152]. Since ImT (f ) = Ω f dmT = T (f ) for f ∈ X(μ), it follows that X(μ) ⊆ L1 (mT ). Furthermore, this inclusion is continuous because f L1(mT ) ≤ 4 sup f dmT = 4 sup T (f χA ) E E A∈Σ A∈Σ A ≤ 4 T sup f χA X(μ) = 4 T · f X(μ) , f ∈ X(μ). A∈Σ
Consequently, ImT is an E-valued, continuous linear extension of T to the larger domain space L1 (mT ). The importance of ImT is manifested in the Curbera-Ricker Optimal Domain Theorem stating that the extension ImT of T to L1 (mT ) is maximal (see Section 1). Let us rephrase this. Lemma 2.1. Let Y (μ) be any σ-order continuous B.f.s. based on (Ω, Σ, μ) such that X(μ) ⊆ Y (μ) continuously and T admits a unique continuous linear extension TY (μ) : Y (μ) → E. Then Y (μ) is continuously embedded into L1 (mT ) and TY (μ) equals the restriction of ImT to Y (μ). In the notation of Lemma 2.1 above, observe that any continuous linear extension of T : X(μ) → E to a larger σ-order continuous B.f.s. Y (μ) is necessarily unique because such a continuous linear extension coincides with T on sim Σ which is dense in both Y (μ) and X(μ). Given 1 < p < ∞, we define Lp (mT ) as for scalar measures, i.e., Lp (mT ) := f ∈ L1 (mT ) : |f |p ∈ L1 (mT ) , which is an order ideal of L1 (mT ). Define 1/p f Lp(mT ) := |f |p L1 (m ) , T
f ∈ Lp (mT ).
From the fact that L1 (mT ) is a B.f.s. with σ-order continuous norm · L1 (mT ) , it follows that the functional · Lp (mT ) : Lp (mT ) → [0, ∞) is a lattice norm for which Lp (mT ) is also a σ-order continuous B.f.s. over (Ω, Σ, μ). This is in [13, Proposition 3.28] while the case for the space Lp (mT ) over R was given earlier in
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[16, Proposition 6]; a related statement for a general B.f.s. is in [5, Proposition 1.11]. The Lp -space (over R) of a vector measure was first defined in [16]. Later developments can be found in [3] and [13], for example. The adjoint index p/(p − 1) of p is denoted by p so that (1/p) + (1/p ) = 1. Given g ∈ Lp (mT ), the pointwise product gf belongs to L1 (mT ) for every f ∈ Lp (mT ), so that g · Lp (mT ) := gf : f ∈ Lp (mT ) ⊆ L1 (mT ). Moreover, the H¨ older type inequality gf L1(mT ) ≤ gLp(mT ) · f Lp(mT ) ,
g ∈ Lp (mT ),
f ∈ Lp (mT ),
(2.1)
holds. This was proved in the setting of general B.f.s.’ in [5, Proof of Proposition 1.10]; see also [13, Lemma 2.21(i) and (3.62)] with ν := mT for the setting of Lp (mT ) and Lp (mT ). If f ∈ L1 (mT ) is a function satisfying |f | ≤ |h| for some h ∈ g · Lp (mT ), then f (ω) = 0 whenever g(ω) = 0 and so f /g ∈ Lp (mT ) with the understanding that 0/0 = 0. It follows that g · Lp (mT ) is an order ideal of L1 (mT ) and the functional defined by hg·Lp(mT ) := h/gLp(mT ) , for h ∈ g · Lp (mT ), is a lattice norm on g · Lp (mT ). The further assumption that g ≥ c χΩ for some c > 0 guarantees the inclusion sim Σ ⊆ g · Lp (mT ). In this case, g · Lp (mT ) is a B.f.s. (over (Ω, Σ, μ) ) with σ-order continuous norm · g·Lp (mT ) because the linear isomorphism h → h/g, for h ∈ g · Lp (mT ), from g · Lp (mT ) onto Lp (mT ) preserves the norm and order. A subset W of L1 (mT ) is said to be uniformly μ-absolutely continuous if supf ∈W f χA L1 (mT ) → 0 as μ(A) → 0, [13, p. 56]. If, in addition, W is bounded in L1 (mT ), then W becomes relatively weakly compact. This can be found in [13, Proposition 2.39], for example. If L1 (mT ) is a real Banach lattice, then this follows from the fact that every bounded, uniformly μ-absolutely continuous set in a real Banach lattice is L-weakly compact and hence, is relatively weakly compact, [10, Proposition 3.6.5]. In the special case when mT = μ, uniform μ-absolute continuity is precisely uniform μ-integrability, [2, p. 74]. Recall that a linear operator between Banach spaces is said to be weakly compact if it maps the closed unit ball of the domain space to a relatively weakly compact subset of the codomain space.
Lemma 2.2. Let T : X(μ) → E be μ-determined. Given 1 < p < ∞, let g ∈ Lp (mT ) satisfy g ≥ c χΩ for some c > 0 so that g · Lp (mT ) is a σ-order continuous B.f.s. based on (Ω, Σ, μ). (g)
(i) The natural (identity) inclusion map αp from g · Lp (mT ) into L1 (mT ) is (g) weakly compact. In particular, αp is continuous. (g,p) (ii) Consider the restriction ImT : g · Lp (mT ) → E of the integration operator ImT : L1 (mT ) → E to g · Lp (mT ), that is, (g,p) Im = ImT ◦ α(g) p . T
(2.2)
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(ii-a) The restricted integration operator ImT is weakly compact. (g,p) (ii-b) The operator ImT is compact if and only if the range R(mT ) := mT (A) : A ∈ Σ = T (χA ) : A ∈ Σ of the vector measure mT : Σ → E is relatively compact. Proof. (i) Via σ-order continuity of · Lp(mT ) we have that g χA Lp(mT ) → 0 as μ(A) → 0. This and (2.1), with (g χA ) in place of g and (h/g) in place of f , imply (g) that αp B[g · Lp (mT )] is uniformly μ-absolutely continuous in L1 (mT ) because (g χ ) · (h/g) 1 α(g) ≤ g χA Lp(mT ) · h/gLp(mT ) p (h)χA L1 (mT ) = A L (mT ) = g χA Lp(mT ) · hg·Lp (mT ) ≤ g χA Lp(mT )
(2.3)
whenever h ∈ B[g · Lp (mT )] and A ∈ Σ. Moreover, (2.3) with A := Ω gives that (g) the subset αp B[g · Lp (mT )] is bounded and hence, is relatively weakly compact (g) in L1 (mT ); see the discussion prior to this lemma. Thus, αp is weakly compact. (ii) Since part (i) and (2.2) imply (ii-a), we only need to prove (ii-b). (g,p) Assume first that ImT is compact. Since χA g·Lp (mT ) = χA /gLp(mT ) ≤ c−1 χΩ Lp (mT ) , for A ∈ Σ, the subset {χA : A ∈ Σ} is bounded in g · Lp (mT ) (g,p)
and hence, its image under the compact operator ImT (which equals R(mT )) is a relatively compact set in E. Conversely, assume that R(mT ) is relatively compact in E. In the proof (p) of part (i), it was shown that αg (B[g · Lp (mT )] is bounded and uniformly μabsolutely continuous in L1 (mT ). Moreover, this set is mapped to a relatively compact set in E by ImT . This fact follows, for instance, from Proposition 2.41 of [13] with L1 (mT ) and ImT in place of X(μ) and T respectively, because ImT (χA ) = (g,p)
mT (A), for A ∈ Σ. So, ImT
is compact via (2.2).
Remark 2.3. In the notation of Lemma 2.2, the special case g := χΩ occurs in [13, Propositions 3.31(iii)and 3.56(I)]. Our current proof is an adaptation of the corresponding proofs given there. When the scalar field is real, see [3, Proposition 3.3, Corollary 3.4 and Theorem 3.6]. A construction : Let g · Lp (mT ) be as in Lemma 2.2. Suppose that there exists a σ-order continuous B.f.s. Y (μ) over (Ω, Σ, μ) such that Y (μ) ⊆ L1 (mT ) but g · Lp (mT ) is not contained in Y (μ). Then the linear span Z(μ) := Y (μ) + g · Lp (mT ) (2.4) of the order ideals Y (μ) and g ·Lp (mT ) in L1 (mT ) is also an order ideal of L1 (mT ). Indeed, if f ∈ L1 (mT ) is a function satisfying |f #| ≤ |ϕ # + ψ| for some # fψ ϕ # ∈ Y (μ) fϕ fψ fϕ # # ≤ |ψ|, it ≤ |ϕ| and # ϕ+ψ + ϕ+ψ . Since # ϕ+ψ and ψ ∈ g · Lp (mT ), then f = ϕ+ψ fϕ ∈ Y (μ) and follows that ϕ+ψ is an order ideal of L1 (mT ).
fψ ϕ+ψ
∈ g · Lp (mT ), so that f ∈ Z(μ), that is, Z(μ)
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f Z(μ) := inf ϕY (μ) + ψg·Lp (mT ) ,
(2.5)
Given f ∈ Z(μ), let
where the infimum is taken over all choices of ϕ ∈ Y (μ) and ψ ∈ g · Lp (mT ) satisfying f = ϕ + ψ. It is routine to prove that · Z(μ) is a lattice norm for which Z(μ) is a B.f.s. over (Ω, Σ, μ) and that the inclusion Y (μ) ⊆ Z(μ) is continuous. If Y (μ) and g · Lp (mT ) were over R, then our norm would be the same as that in [9, Definition 2.g.2]. Adapting the arguments in [13, p. 220], we can establish the fact that the lattice norm · Z(μ) is σ-order continuous because so are the lattice norms · Y (μ) and · g·Lp (mT ) . From the definition of · Z(μ) , it is clear that the natural embedding from Z(μ) into L1 (mT ) is continuous. Now we claim that 3 3 (g,p) B[g · Lp (mT )] . (2.6) ImT B[Z(μ)] ⊆ ImT B[Y (μ)] + Im T 2 2 In fact, let f ∈ B[Z(μ)]. There exist ϕ ∈ Y (μ) and ψ ∈ g · Lp (mT ) such that f = ϕ + ψ and ϕY (μ) + ψg·Lp (mT ) < f Z(μ) + 1/2 . So, ϕ Y (μ) < (3/2) and ψg·Lp (mT ) < (3/2), which implies that ImT (ϕ) ∈ (3/2) ImT B[Y (μ)] and (g,p) ImT (ψ) ∈ (3/2) ImT B[g ·Lp (mT )] . This establishes (2.6) because ImT is defined as the restriction of ImT to g · Lp (mT ) (see Lemma 2.2). The linear operator T : X(μ) → E has been so far assumed just to be continuous. We shall now prove Theorem 1.1 in which T is additionally assumed to be compact. Proof of Theorem 1.1. When the integration operator ImT : L1 (mT ) → E is compact, it necessarily equals the maximal compact linear extension of T because there is no further continuous linear extension of ImT to any larger σ-order continuous B.f.s. containing L1 (mT ). So, assume that ImT is not compact. Take any σ-order continuous B.f.s. Y (μ) over (Ω, Σ, μ) such that X(μ) ⊆ Y (μ) continuously and T admits a compact linear extension TY (μ) : Y (μ) → E. Such a space Y (μ) always exists because Y (μ) is allowed to be equal to X(μ). Since TY (μ) is continuous, we have the continuous inclusion Y (μ) ⊆ L1 (mT ) and ImT restricted to Y (μ) is equal to TY (μ) (see Lemma 2.1). Our aim is to find an E-valued, compact linear extension of TY (μ) to a σorder continuous B.f.s. strictly larger than Y (μ). To this end, fix 1 < p < ∞. Since ImT is not compact, the set L1 (mT ) \ Y (μ) is non-empty and hence, a contains function f0 . Let sgn f0 := f0 /|f0 | pointwise on Ω. The function g := |f0 |1/p + χΩ belongs to Lp (mT ) because both |f0 |1/p , χΩ ∈ Lp (mT ). Since |f0 |1/p ∈ Lp (mT ), it follows that f0
= =
|f0 |1/p · (sgn f0 ) |f0 |1/p
1/p sgn f0 1/p ∈ g · Lp (mT ). |f0 | + χΩ |f0 | |f0 |1/p + χΩ
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So, g · Lp (mT ) is not contained in Y (μ). Accordingly, we are in the situation of the above construction. That is, the σ-order continuous B.f.s. Z(μ) ⊆ L1 (mT ) as defined via (2.4) exists and is strictly larger than Y (μ). Moreover, X(μ) is continuously embedded in Z(μ) because of the continuous inclusions X(μ) ⊆ Y (μ) and Y (μ) ⊆ Z(μ). The restriction TZ(μ) of ImT to Z(μ) is an E-valued, continuous linear extension of T as Z(μ) is continuously embedded into L1 (mT ). To show that TZ(μ) is compact, observe that the range R(mT ), which equals the image of the bounded subset {χA : A ∈ Σ} of X(μ) under the compact operator (g,p)
T , is relatively compact in E. Thus, ImT is compact via Lemma 2.2(ii-b), that (g,p) is, ImT B[g · Lp (mT )] is relatively compact in E. Furthermore, TY (μ) B[Y (μ)] is also relatively compact in E via compactness of TY (μ) . Since ImT coincides with TY (μ) on Y (μ), it follows from (2.6) that TZ(μ) B[Z(μ)] = ImT B[Z(μ)] is relatively compact in E. So, TZ(μ) : Z(μ) → E is a proper compact linear extension of TY (μ) because Z(μ) is strictly larger than Y (μ). In other words, T does not admit a maximal continuous linear extension. It is known when the integration operator ImT is compact, [12, Theorems 1 and 4]. Let us present a compact linear operator whose maximal continuous linear extension is not even weakly compact. Example. The Volterra integral operator V : L1 ([0, 1]) → L1 ([0, 1]) (of order 1) is defined by t (V f )(t) := f (u) du, t ∈ [0, 1], f ∈ L1 ([0, 1]). (2.7) 0
That is, we have the setting: X(μ) := L1 ([0, 1]), E := L1 ([0, 1]) and T := V , with μ being Lebesgue measure on the Borel σ-algebra B([0, 1]) of the unit interval Ω := [0, 1]. Clearly L1 ([0, 1]) is a σ-order continuous B.f.s. The operator V is known to be compact. For this, see [6, Satz 11.6]; an alternative proof can be found in [13, pp. 151–152]. The operator V admits a natural extension V( to the space of all functions f ∈ L0 (μ) such that f is Lebesgue integrable over [0, u] u for every u ∈ [0, 1] and the function u → 0 f (t) dt, for u ∈ [0, 1], belongs to u E = L1 ([0, 1]), in which case V( (f )(u) = 0 f (t) du. Via Fubini’s Theorem, this extended domain can be shown to be the weighted Lebesgue space L1 ((1 − t)dt). It turns out that L1 ((1 − t)dt) is the maximal σ-order continuous domain L1 (mV ) and that V( = ImT . The maximal continuous linear extension ImV is not compact. In fact, it is not even weakly compact, [11, Example 2]. Given 1 < p < ∞, the Volterra integral operator from Lp ([0, 1]) into Lp ([0, 1]) can be defined also by (2.7) with Lp ([0, 1]) replacing L1 ([0, 1]). Its maximal continuous linear extension is also non-compact but, obviously weakly compact due to reflexivity of the codomain space Lp ([0, 1]), [14, §5].
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321
3. Weakly compact linear extension Proposition 3.1 below will present a “weakly compact analogue” of Theorem 1.1. Fix a positive, finite measure space (Ω, Σ, μ). Given a σ-order continuous B.f.s. X(μ) over (Ω, Σ, μ), we say that a Banach-space-valued, weakly compact operator T : X(μ) → E admits a maximal weakly compact linear extension if there exists a largest σ-order continuous B.f.s. Y (μ) (over (Ω, Σ, μ) ) into which X(μ) is continuously embedded and to which T admits an E-valued, weakly compact linear extension. Proposition 3.1. Let X(μ) be a σ-order continuous B.f.s. based on (Ω, Σ, μ) and E be a Banach space. A μ-determined weakly compact linear operator T : X(μ) → E admits a maximal weakly compact linear extension if and only if the integration operator ImT : L1 (mT ) → E is weakly compact. Proof. The proof is similar to that of Theorem 1.1. If the maximal continuous linear extension ImT : L1 (mT ) → E of T is weakly compact, then ImT is necessarily the maximal weakly compact extension of T . So, assume that ImT is not weakly compact and take any σ-order continuous B.f.s. Y (μ) (over (Ω, Σ, μ) ) into which X(μ) is continuously embedded and to which T admits a weakly compact linear extension TY (μ) : Y (μ) → E. Such a space Y (μ) can be equal to X(μ). Since TY (μ) is continuous, it follows from Lemma 2.1 that Y (μ) is continuously embedded into L1 (mT ) and that ImT and Y (μ) coincide on Y (μ). But, we have Y (μ) L1 (mT ) because ImT is not weakly compact whereas TY (μ) is. Let 1 < p < ∞. Select a function g ∈ Lp (mT ) such that g ≥ c χΩ for some c > 0 and g · Lp (mT ) is not contained in Y (μ); see the proof of Theorem 1.1. Consider the σ-order continuous B.f.s. Z(μ) equipped with the lattice norm · Z(μ) as constructed via (2.4) and (2.5). The subset TY (μ) B[Y (μ)] (resp. (g,p) (g,p) ImT (B[g·Lp (mT )]) ) of E is relatively weakly compact because TY (μ) (resp. ImT ) is weaklycompact by assumption (resp. by Lemma 2.2(ii-a)). This and (2.6) imply that ImT B[Z(μ)] is also relatively weakly compact in E because ImT coincides with TY (μ) on Y (μ). Hence, the restriction ImT to Z(μ) provides a proper weakly compact linear extension of TY (μ) to Z(μ) Y (μ). This implies that T does not admit a maximal weakly compact linear extension. Acknowledgment The author thanks O. Blasco and W.J. Ricker for useful discussions. Support from the Maximilian Bickhoff Universit¨ atsstiftung is gratefully acknowledged.
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References [1] G.P. Curbera and W.J. Ricker, Optimal domains for kernel operators via interpolation. Math. Nachr. 244 (2002), 47–63. [2] J. Diestel, and J.J. Uhl, Jr., Vector Measures. Math. Surveys 15, Amer. Math. Soc. Providence, 1977. [3] A. Fern´ andez, F. Mayoral, F. Naranjo, C. S´ aez and E.A. S´ anchez-P´erez, Spaces of p-integrable functions with respect to a vector measure. Positivity 10 (2006), 1–16. [4] A. Fern´ andez, F. Naranjo and W.J. Ricker, Completeness of L1 -spaces for measures with values in complex vector spaces. J. Math. Anal. Appl. 223 (1998), 76–87. [5] J. Garc´ıa Cuerva, Factorization of operators and weighted norm inequalities: in Nonlinear Analysis, Function Spaces and Applications (Proceedings of the Spring School held in Roudnice nad Labem), B.G. Teubner, Leipzig, 1990 (pp. 5–41). [6] K. J¨ orgens, Lineare Integraloperatoren. B.G. Teubner, Stuttgart, 1970. [7] I. Kluv´ anek and G. Knowles, Vector Measures and Control Systems. North-Holland Mathematics Studies 20, North-Holland, Amsterdam, 1975. [8] D.R. Lewis, Integration with respect to vector measures. Pacific J. Math. 33 (1970), 157–165. [9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II. Springer, Berlin, 1996. [10] P. Meyer-Nieberg, Banach Lattices. Springer, Berlin-New York, 1991. [11] S. Okada, W.J. Ricker, Non-weak compactness of the integration map for vector measures. J. Austral. Math. Soc. (Series A) 54 (1993), 287–303. [12] S. Okada and W.J. Ricker and L. Rodr´ıguez-Piazza, Compactness of the integration operator associated with a vector measure. Studia Math. 150 (2002), 133–149. [13] S. Okada, W.J. Ricker and E.A. S´anchez-P´erez, Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Birkh¨ auser, 2008. [14] W.J. Ricker, Compactness properties of extended Volterra operators in Lp ([0, 1]) for 1 ≤ p ≤ ∞. Arch. Math. (Basel) 66 (1996), 132–140. [15] W.J. Ricker, Rybakov’s theorem in Fr´echet spaces and completeness of L1 -spaces. J. Aust. Math. Soc. (Series A) 64(1998), 247–252. [16] E.A. S´ anchez P´erez, Compactness arguments for spaces of p-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces. Illinois J. Math. 45 (2001), 907–923. Susumu Okada 112 Marconi Crescent Kambah ACT 2902, Australia e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 323–325 c 2009 Birkh¨ auser Verlag Basel/Switzerland
A Note on R-boundedness in Bidual Spaces Ben de Pagter and Werner J. Ricker Abstract. We provide an alternate proof, via the Principle of Local Reflexivity, of the fact that a family of bounded linear operators T in a Banach space X is R-bounded iff T ∗∗ is R-bounded in the bidual X ∗∗ . Mathematics Subject Classification (2000). 46B07, 46E40, 47A05. Keywords. Bidual, R-boundedness.
The notion of R-boundedness for families of operators is due to E. Berkson and T.A. Gillespie, [1], (where it is called the R-property); it is also implicit in earlier work of J. Bourgain, [2]. Denote by X ∗∗ the bidual of a Banach space X and by L(X) the space of all bounded linear operators from X into itself. Theorem 1. Let X be a Banach space and T ⊆ L(X). Then T is R-bounded if and only if T ∗∗ := {T ∗∗ : T ∈ T } is R-bounded in X ∗∗ . This result, [5, Lemma 2.4], has recent applications to semigroups of operators [5], functional calculi [6], and the theory of spectral operators in bidual spaces, [7], [8]. The proof in [5] is based on properties of the subspace Rad(X) ⊆ L2 ([0, 1], X), which is generated by functions of the kind rj ⊗ xj , where xj ∈ X and {rj }∞ j=1 are the Rademacher functions on [0, 1]. We present an alternate proof of Theorem 1 based on the Principle of Local Reflexivity, [4]. We require the following formulation of this result which follows directly from the version in [4, p. 178]. Principle of Local Reflexivity. Let X be a Banach space. Given a finite-dimensional subspace E ⊆ X ∗∗ , a finite-dimensional subspace F ⊆ X ∗ and 0 < δ ∈ R, there exists an injective linear mapping u : E → X such that, by considering u−1 : u(E) → E, we have: 1. u ≤ 1 + δ and u−1 ≤ 1 + δ; 2. ux = x for all x ∈ E ∩ X, interpreting X as canonically embedded in X ∗∗ ; 3. ux∗∗ , x∗ = x∗∗ , x∗ , for all x∗∗ ∈ E and x∗ ∈ F . Recall that a non-empty collection T ⊆ L(X) is called R-bounded if n n 2 1/2 2 1/2 1 1 rj (t)Tj xj dt ≤M rj (t)xj dt 0
j=1
0
j=1
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for a constant M ≥ 0 and all {Tj }nj=1 ⊆ T , {xj }nj=1 ⊆ X, n ∈ N. We also write n n rj Tj xj ≤ M rj xj . 2
j=1
(1)
2
j=1
Proof of Theorem 1. Let (Ω, F , μ) be a positive, finite measure space and Y a Banach space. By [3, pp. 97–98], the Bochner space L2 (μ, Y ∗ ) can be identified isometrically with a closed subspace of L2 (μ, Y )∗ via the duality pairing f ∈ L2 (μ, Y ), g ∈ L2 (μ, Y ∗ ).
f, g := f (ω), g(ω) dμ(ω), Ω
Suppose that T is R-bounded in L(X) with M ≥ 0 satisfying (1). Take ∗∗ ∗∗ T1 , . . . , Tn ∈ T and x∗∗ 1 , . . . , xn ∈ X . Let Fn be the σ-algebra in [0, 1] generated by r1 , . . . , rn (i.e., Fn is the σ-algebra generated by all dyadic intervals in [0, 1] of length 2−n ). Since the X ∗∗ -valued function nj=1 rj Tj∗∗ x∗∗ j is Fn -measurable, by the previous paragraph we have n n #, -# # # ∗∗ ∗∗ 2 ∗ rj Tj xj = sup # rj Tj∗∗ x∗∗ j , f # : f ∈ L ([0, 1], Fn ; X ), f 2 ≤ 1 , 2
j=1
j=1
where L ([0, 1], Fn ; X ) denotes the subspace of L2 ([0, 1]; X ∗) consisting of all functions which are Fn -measurable. Take f ∈ L2 ([0, 1], Fn ; X ∗ ) with f 2 ≤ 1. Note that {f (t) : t ∈ [0, 1]} consists of at most 2n elements of X ∗ . Let F be the (finite-dimensional) linear subspace of X ∗ generated by {Tj∗f (t) : t ∈ [0, 1], j = ∗∗ 1, . . . , n} and let E be the subspace of X ∗∗ generated by {x∗∗ 1 , . . . , xn }. It follows from the Principle of Local Reflexivity that, for any given δ > 0, there exists a linear map u : E → X such that ux∗∗ , x∗ = x∗∗ , x∗ for all x∗∗ ∈ E and x∗ ∈ F , and ux∗∗ ≤ (1 + δ)x∗∗ for all x∗∗ ∈ E. Since + * + * ∗∗ ∗ + * + * ∗∗ ∗∗ ∗ ∗∗ Tj xj , f (t) = x∗∗ j , Tj f (t) = uxj , Tj f (t) = Tj (uxj ), f (t) 2
∗
for all j = 1, . . . , n and all t ∈ [0, 1], we find that n n n #, -# #, -# # # # # ∗∗ ∗∗ rj Tj∗∗ x∗∗ , f = r T (ux ), f ≤ r T (ux ) # # # # j j j j j j j f 2 j=1
j=1 n
≤ M
2
j=1
rj (ux∗∗ j ) = M u 2
j=1
n j=1
n ≤ (1 + δ)M rj x∗∗ j . j=1
2
Since this holds for all δ > 0, it follows that n n #, -# # # rj Tj∗∗ x∗∗ rj x∗∗ # j , f # ≤ M j . j=1
j=1
2
rj x∗∗ j
2
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Consequently, we have that n n ∗∗ ∗∗ rj Tj xj ≤ M rj x∗∗ j . j=1
2
j=1
2
This establishes the R-boundedness of T ∗∗ . The converse follows routinely from the definitions involved.
References [1] E. Berkson and T.A. Gillespie, Spectral decompositions and harmonic analysis in UMD spaces, Studia Math. 112 (1994), 13–49. [2] J. Bourgain, Some remarks on Banach spaces in which martingale differences are unconditional, Ark. Mat. 21 (1983), 163–168. [3] J. Diestel and J.J., Uhl, Jr. Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977. [4] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995. [5] M. Hoffmann, N.J. Kalton and Kucherenko, R-bounded approximating sequences and applications to semigroups, J. Math. Anal. Appl. 294 (2004), 373–386. [6] C. Kriegler and C. Le Merdy, Tensor extension properties of C(K)-representations and applications to unconditionality, Preprint 2009. [7] B. de Pagter and W.J. Ricker, R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry, Proc. Conf. “Positivity IV-Theory & Applications”, July 2005, Tech. Univ. Dresden, Eds: J. Voigt, M. Weber, (2006), 115–129. [8] B. de Pagter and W.J. Ricker, C(K)-representations and R-boundedness, J. London Math. Soc. (Ser. II), 76 (2007), 498–512. Ben de Pagter Department of Applied Mathematics, Faculty EEMCS Delft University of Technology, P.O. Box 5031 NL-2600 GA Delft, The Netherlands e-mail: [email protected] Werner J. Ricker Math.-Geogr. Fakult¨ at Katholische Universit¨ at Eichst¨ att-Ingolstadt D-85072 Eichst¨ att, Germany e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 327–338 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Salem Sets in the p-adics, the Fourier Restriction Phenomenon and Optimal Extension of the Hausdorff-Young Inequality Christos Papadimitropoulos Abstract. We prove the existence of Salem sets in the compact abelian group Zp of the p-adic integers and establish an L2 Fourier restriction estimate associated to such sets. This allows us to extend work of G.Mockenhaupt and W.J.Ricker about optimal extensions of the Hausdorff-Young inequality from the torus to the p-adic integers. Vector measures will play an important role. Mathematics Subject Classification (2000). 42B10, 46G10. Keywords. Fourier transform, p-adic numbers, vector measure, Salem set, Fourier restriction phenomenon.
1. Introduction The Hausdorff-Young inequality f1lq (Z) ≤ f Lq (T) where 1 ≤ q ≤ 2, 1q + q
q
1 q
=
1, asserts that the Fourier transform maps L (T) into l (Z) and is a bounded operator. Mockenhaupt and Ricker, in a recent work [10] extended the Fourier transform continuously and in an optimal way, keeping the range space lq (Z) fixed. More precisely, for a given 1 < q < 2, they construct a Banach function space Fq (T) such that Lq (T) ⊂ Fq (T) ⊂ L1 (T) with both inclusions proper and such that the Fourier transform F : Fq (T) → lq (Z) is bounded. Moreover f Fq (T) ≤ 4f Lq (T) and f L1(T) ≤ f Fq (T) and therefore Fq (T) contains continuously the space Lq (T) and is contained continuously in L1 (T). The norm Fq (T) is σ-order continuous; that is, if non-negative functions fn ∈ Fq (T) decrease to 0 as n → ∞, then fn → 0 in Fq (T) as n → ∞. Furthermore the space Fq (T) is maximal with these properties in the sense that if Z is a Banach function space over (T, B(T), dt) with σ-order continuous norm which contains continuously the space Lq (T) and the Fourier transform F : Lq (T) → lq (Z) has an extension to a continuous linear operator from Z into lq (Z), then Z is continuously included in Fq (T). We call the space q F (T) the optimal lattice domain for the Hausdorff-Young inequality on the torus
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T. It turns out that Fq (T) consists precisely of those functions f ∈ L1 (T) such that f · 1A ∈ lq (Z) for every Borel set A ⊆ T. It also turns out that for the endpoints q = 1, 2 the corresponding optimal lattice domains F1 (T) , F2 (T) are just L1 (T) and L2 (T) respectively. The work of Mockenhaupt and Ricker is based on the theory of Vector measures. This approach to optimal extensions for various operators via appropriate vector measures has been proved to be very effective (see [1], [2], [3], [4], [5], [11], [12], [13]). The aim of this paper is to reproduce the work of Mockenhaupt and Ricker to the p-adic setting, replacing the torus T by the compact abelian group of p-adic integers Zp . Salem sets in the p-adics will play an important role since together with the L2 -Fourier restriction phenomenon they enable us to prove the proper inclusion Lq (Zp ) Fq (Zp ) (1 < q < 2). However p-adic Salem sets are interesting in their own right.
2. p-adic Salem sets and the L2 -Fourier restriction phenomenon We begin with some well-known properties of p-adic numbers, (see [6],[7]). We fix a prime number p. Every rational number x can be written in a unique way as k x= m n p , where k ∈ Z and p does not divide m nor n. For such a rational number we define the norm |x|p := p−k . The set of the p-adic numbers Q p is defined as ∞ the completion of Q with respect to | |p . It turns out that Qp = { j=k cj pj : k ∈ Z, cj = 0, 1, . . . , p − 1} and the arithmetic operations are defined in the natural way to ensure Qp is a field anda locally compact group under addition. If the ∞ unique p-adic expansion of x is j=k cj pj with ck = 0, then |x|p = p−k . Hence if x, y ∈ Qp and |x|p > |y|p then |x + y|p = |x|p . Moreover |x + y|p ≤ max{|x|p , |y|p }. On account of this property, every two balls with the same radius either are disjoint or coincide. We will use this fact in the proof of the existence of p-adic Salem sets. We define jthe set of the p-adic integers as Zp := B 1 (0) = {x ∈ Qp : |x|p ≤ 1} = { ∞ j=0 cj p : cj = 0, 1, . . . , p − 1} and endow (Qp , +) with Haar measure dx which we normalize such that the measure of the compact abelian group Zp is equal to one. The characters of (Zp , +) are of the form χ(γm x), where (γm )∞ m=0 is some −1 j ∼ ordering of the set { j=−k cj p : k ∈ N, cj = 0, 1, . . . , p − 1} = Qp /Zp and χ is the fundamental character of Qp , i.e., χ( j cj pj ) = exp(2πi j≤−1 cj pj ). The other characters of (Qp , +) are of the form χξ (x) = χ(ξx), for some ξ ∈ Qp . After these preliminaries we proceed to prove the existence of Salem sets in Zp . Definition 2.1. A set E ⊆ Qp of measure zero is called Salem set if dimF E = dimH E, where dimF E, dimH E are the Fourier and Hausdorff dimensions of E respectively. We remind the reader that −β
μ(x)| ≤ C · |x|p 2 ∀x ∈ Qp }, dimF E := sup{β > 0 : ∃ μ = 0, supp μ ⊆ E s.t. |1 where μ 1(x) = Qp χ(xy)dμ(y).
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Theorem 2.2. For every 0 < α < 1 and > 0 there exist a set E ⊆ Zp of Hausdorff dimension α and a measure μ supported on E such that −α 2 +
|μ1 (x)| ≤ C · |x|p
∀x ∈ Qp .
To prove this theorem we follow Salem’s approach to establishing the existence of such sets in [0, 1] ⊆ R (see [14],[9]). We will make use of the following proposition. N Proposition 2.3. Let P (x) = N1 j=1 χ(aj x) with a1 , . . . , aN ∈ Qp , linearly independent over Q, and let m be an even natural number. Then ∃ T0 = T0 (N, aj , m) such that √ 1 m1 m m |P (x)| dx ≤ √ , ∀T ≥ T0 , ∀x0 ∈ Qp . T B T (x0 ) N Proof. Let m = 2k. Then 1 P (x)k = k N
k1 +···+kN =k
|P (x)|m = P (x)k · P (x)k =
k! χ((k1 a1 + · · · + kN aN )x), and k1 ! · · · kN !
1 Nm
k1 +···+kN =k
k! k1 ! · · · kN !
2 +
λj χ(bj x)
j
with integer coefficients not where λj ∈ N and bj is a linear combination of all zero. Hence bj = 0 ∀j. For ν ≥ 1, I = |x|p ≤pν χ(x)dx = 0 (because {x : |x|p ≤ ν p −1 ' ν B 1 (βi ) with βi ∈ cj pj : cj : 0, 1, . . . , p − 1 p } is the disjoint union (ai )N i=1
i=1 p
i=1
i=1
ν
and so I =
j=−ν
p
Therefore
ν
B 1 (βi ) χ(x)dx =
B T (x0 )
λj χ(bj x)dx =
j
χ(βi ) =
ν E
(1 + e
1 pj
+ ··· + e
2πi p−1 j p
) = 0).
j=1
λj χ(bj x0 )
j
=
2πi
λj χ(bj x0 )
j
|x|p ≤T
1 |bj |p
χ(bj x)dx
χ(x)dx, |x|p ≤T ·|bj |p
because the change of variables x = bj x implies that dx = |bj |p dx. Choosing T0 sufficiently big such that T0 |bj |p > 1 ∀j, we have that ∀T ≥ T0 the last integral vanishes for each j. Hence ∀ T ≥ T0 2 1 k! 1 1 m |P (x)| dx = dx T B T (x0 ) T Nm k1 ! · · · kN ! B T (x0 ) k1 +···+kN =k √ m k! m 1 1 = m k!N k ≤ √ m , ≤ m k! N k1 ! · · · kN ! N N k1 +···+kN =k since the measure of B T (x0 ) equals T .
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Ch. Papadimitropoulos We need also the following two lemmas.
Lemma 2.4. Let N < pk with N, k ∈ N. There exist a1 , . . . , aN ∈ Zp , linearly independent over Q such that |ai − aj |p > p−k ∀ i = j. Proof. We start with any b1 , . . . , bN ∈ Qp with {1, b1 , . . . , bN } linearly independent over Q (Qp is an infinite field extension over Q). From each bj we subtract that part of its p-adic expansion, say wj , which consists of the negative powers of p. So bj = bj − wj ∈ Zp . Since each wj is a rational number, we have that 1, b1 , . . . , bN are linearly independent over Q. Next we consider r1 , . . . , rN (different) p-adic integers whose p-adic expansions consist of zero after the pk -coefficient, k−1 i.e., ri = j=0 cj (i)pj . Last we set ai = ri + pk bi , i = 1, . . . , N . One can verify easily that a1 , . . . , aN have the desired properties. log N Lemma 2.5. Let 0 < α < 1 , N ∈ N and k = [ log pα ]. Then there is a sequence n
N kn ∈ {k, k + 1} such that lim log plog k1 +···+kn = α. n
Proof. Let θ =
log N log pα .
It is sufficient to find kn ∈ {k, k + 1} such that lim n
k1 + · · · + kn = θ. n
1 −1 We take k1 = k. We consider the smallest n1 ∈ N such that θ < n1 k+n . n1 Set k2 = · · · = kn1 = k + 1. Then we pick the smallest n2 ∈ N such that (n1 +n2 )k+n1 −1 < θ and set kn1 +1 = · · · = kn1 +n2 = k. We continue inductively n1 +n2 and we construct a sequence (nλ )λ such that
k
2λ j=1
nj +
2λ−1 j=1, j odd 2λ j=1
nj
nj − 1
k <θ<
2λ+1 j=1
2λ+1
nj +
nj − 1
j=1, j odd 2λ+1
nj
j=1
where n2λ , n2λ+1 are the smallest natural numbers for which the above inequalities hold. We claim that both sequences above converge to θ as λ → ∞. To see this, we suppose that this is not true for the LHS sequence which we will denote by θλ for the sake of brevity. Then ∃ > 0 such that for infinitely many λ ∈ N we have θλ < θ − and 2λ−1 k 2λ j=1 nj − k + j=1, j odd nj − 1 >θ 2λ j=1 nj − 1 where the last inequality arises from the construction of nλ . From these two 2λ inequalities we easily get · j=1 nj < θ − k for infinitely many λ which cannot be true. Similarly we prove the convergence of the RHS sequence. Proof of Theorem 2.2. Let 0 < α < 1, M be an even natural number and N = log N M M . Let k = [ log pα ]. From Lemma 2.5 we pick a sequence kn ∈ {k, k + 1}
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n
N such that k1 = k and lim log plog k1 +···+kn = α. For M sufficiently large we have n
N < pk1 (since α < 1). From Lemma 2.4 we consider a1 , . . . , aN ∈ Zp linearly independent over Q and |ai − aj |p > p−k1 ∀ i = j. We fix a sequence of p-adic integers (ξi )i with |ξi |p = p−ki ∀ i, and we carry out the following Cantor type construction. First we begin with the N balls B p−k1 (aj ), j = 1, . . . , N which are mutually disjoint since |ai − aj |p > p−k1 . At the second step we consider the N 2 balls B p−k1 −k2 (aj + ai ξ1 ) ⊆ B p−k1 (aj ), i, j = 1, . . . , N . These balls are also pairwise disjoint since |ai ξ1 − aj ξ1 |p = |ai − aj |p |ξ1 |p > p−k1 −k1 ≥ p−k1 −k2 . After n operations we get N n disjoint balls B p−k1 −···−kn (a0 + a1 ξ1 + · · · + an−2 ξ1 ξ2 · · · ξn−2 + an−1 ξ1 ξ2 · · · ξn−1 ) ⊆ B p−k1 −···−kn−1 (a0 + a1 ξ1 + · · · + an−2 ξ1 ξ2 · · · ξn−2 ) where j = 1, . . . , N . The limit of this procedure gives us a set E whose elements are given by the formula x = a0 + a1 ξ1 + · · · + an ξ1 ξ2 · · · ξn + · · · where aj ∈ {a1 , . . . , aN }. One can verify that for each M , the set E has Hausdorff dimension α. Next we consider the measures μ0 =
N N 1 1 δaj , μn = δa ξ ···ξ ∀ n ∈ N, N j=1 N j=1 j 1 n
where δ is the Dirac measure. Then the infinite convolution μ = μ0 ∗ μ1 ∗ · · · (which depends on M and the sequence (ξi )) is a probability measure supported √3 −α 2 (1− M )
on E. Our goal is to find a choice of ξi s.t. |1 μ(x)| ≤ CM |x|p ∞ E transform of μ is given by μ 1(x) = μ Fn (x) where
. The Fourier
n=0
μ F0 (x) =
N N 1 1 χ(aj x), μ Fn (x) = χ(aj ξ1 · · · ξn x) N j=1 N j=1
and χ is the fundamental character of Qp . Setting P (x) =
we get that μ 1(x) = P (x)
∞ E n=1
N 1 χ(aj x) N j=1
P (ξ1 · · · ξn x). So far (ξi )i was an arbitrary sequence
with |ξi |p = p−ki . We now consider ξi = pki + pki +1 ζi with ζi ∈ Zp . Hence the set E and the measure μ depend on (ζi )i ∈ Zp N . We endow the hypercube Zp × Zp × · · · with the measure product dζ1 dζ2 . . . of Haar measures on Zp .
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Since |P (x)| ≤ 1 and the measure of Zp equals one we have n > |1 μ(x)|M dζ1 dζ2 · · · ≤ |P (ξ1 · · · ξi x)|M dζ1 · · · dζn Zp N
Zp n i=1
n−1 >
=
Zp n−1 i=1
|P (ξ1 · · · ξi x)|M
|P (ξ1 · · · ξn x)|M dζn · · · dζ1 Zp
We deal first with the last integral with respect to the variable ζn . 1 |P (ξ1 · · · ξn−1 pkn x + ξ1 · · · ξn−1 pkn +1 ζn x)|M dζn = |P (ζn )|M dζn T B T (x0 ) |ζn |p ≤1 where T = |ξ1 · · · ξn−1 pkn +1 x|p = p−k1 −···−kn−1 −kn p−1 |x|p
and x0 = ξ1 · · · ξn−1 pkn x.
Hence T ≥ T0 ⇔ log |x|p − (k1 + · · · + kn ) log p − log p ≥ log T0 . We fix c such that 1 < c < (1 − √1M )−1 . Then there is a natural number n0 such that k1 + · · · + kn < N c nαlog log p for every n ≥ n0 . We consider 3 2 1 α log |x|p +1 . n = (1 − √ ) M log N
For |x|p sufficiently large we have log |x|p − (k1 + · · · + kn ) log p − log p 1 c log N − log p . > 1 − c(1 − √ ) log |x|p − α M The last quantity is bigger than log T0 if |x|p is big enough, say |x|p > L. Having chosen the above n = n(|x|p ), the inequality log |x|p − (k1 + · · ·+ ki ) log p− log p ≥ log T0 is also true for every i = 1, . . . , n. Hence, by Proposition 2.3, successive integrations give us √ Mn log |x| 1 M (1−M)(1− √1 )α log Mp M M |1 μ(x)| dζ1 dζ2 . . . ≤ √ ≤M2 N Zp N √1 −α 2 (M−1)(1− M )
= |x|p Therefore Zp N
α √3 2 (1− M
Zp N (|x|p
|x|p >L
)
√2 −α 2 M(1− M )
≤ |x|p
−α 2
|1 μ(x)|)M dζ1 dζ2 · · · ≤ |x|p
α M √3 2 (1− M ) |x|p |1 μ(x)| dxdζ1 dζ2 · · · ≤
√
M
, for |x|p > L. Hence
|x|p >L
α √3 2 (1−
)
, for |x|p > L .
−α 2
|x|p
√
M
dx ≤ C < ∞
M (|x|p |1 μ(x)|)M dx < ∞ for a.e (ζi )i ∈ Zp N . |x|p >L ' ' B 1 (γm ), where F ⊆ N {0} is finite and the balls Since {x : |x|p > L} =
for M big enough. So
m∈F /
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333
are mutually disjoint, we get α M √3 2 (1− M ) |x|p |1 μ(x)| dx < ∞ for a.e (ζi )i ∈ Zp N . m∈F /
B 1 (γm )
But for every x ∈ B 1 (γm ) = γm + B 1 (0) we have |x|p = |γm |p and μ 1(x) = μ 1(γm ). Hence, since the measure of B 1 (γm ) equals one, we get M α √3 2 (1− M ) |γm |p |1 μ(γm )| <∞ m∈F / √3 ) −α 2 (1−
M ∀m∈ / F and for a.e. (ζi )i ∈ Zp N . The which gives us |1 μ(γm )| ≤ CM |γm |p ' B 1 (γm ), used earlier, and the fact that μ 1 ∈ L∞ (Qp ) fact {x : |x|p > L} =
m∈F /
√3 −α 2 (1− M )
allow us to obtain |1 μ(x)| ≤ CM |x|p given > 0 we choose M such that α2 · proof of the theorem.
√3 M
∀x ∈ Qp , for a.e. (ζi )i ∈ Zp N . For a < and we set μ = μ concluding the
Next we prove a regularity property for μ which together with the decay of its Fourier transform will enable us to get an L2 - Fourier restriction phenomenon on E by using the Stein-Tomas argument (see [8],[9]). n We set θ = lim k1 +···+k . Then N = pθα . Let δ > 0. Then ∃ n0 = n0 (δ) n n
nθ n s.t. | k1 +···+k − 1| < δ ∀ n ≥ n0 . Also ∃ Λ > 0 s.t. | k1 +···+k − θ| < Λ ∀ n. n n Let now B p−k1 −···−kn (x0 ) one of the balls in the construction of E. We assume n ≥ n0 . Then α 1 1 ≤ (p−k1 −···−kn )α(1−δ) . μ (B p−k1 −···−kn (x0 )) = n = N pnθ
Let now n < n0 . Then 1 μ (B p−k1 −···−kn (x0 )) = n = N
1 pnθ
α
≤ pnΛα (p−k1 −···−kn )α .
Therefore μ (B r (x0 )) ≤ Cδ rα(1−δ) for every ball in the construction of E. Con sider now any ball B p−ν (x0 ). We assume that B p−ν (x0 ) E = ∅ otherwise we have nothing to do. We consider the smallest n ∈ N s.t. p−k1 −···−kn < p−ν . Since any two balls in Qp of different radius either are disjoint or the one of smaller radius is contained in the other, we have that B p−ν (x0 ) is contained in just one ball of the construction of E with radius p−k1 −···−kn−1 . This implies that B p−ν (x0 ) can contain at most N balls of the construction of E with radius p−k1 −···−kn . Hence μ (B p−ν (x0 )) ≤ N Cδ (p−k1 −···−kn )α(1−δ) ≤ N Cδ (p−ν )α(1−δ) . ∗ More
careful consideration shows that N can be equal to 1.
∗
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Having proved the above regularity property of μ and the decay of μ1 , the Stein-Tomas machine shows that for every 1 ≤ q < q(α) = 4−2α 4−3α there is an > 0 such that |f1(x)|2 dμ (x) ≤ C f 2Lq (Qp ) .
3. Optimal extension of the Hausdorff-Young inequality in Zp As it is well known the Hausdorff-Young inequality holds for abstract compact abelian groups. Hence f1lq ≤ f Lq (Zp ) for 1 ≤ q ≤ 2, 1q + q1 = 1, where and f1(γm ) = f (x)χ(γm x)dx. The aim of this section is to f1 = (f1(γm ))∞ m=0
Zp
extend the Hausdorff-Young inequality in an optimal way keeping the range space lq fixed. We follow Mockenhaupt’s and Ricker’s approach and hence we repeatedly refer to their work [10]. We define the vector measure mq : B(Zp ) → lq , mq (A) = 1F A , where B(Zp ) stands for the Borel subsets of Zp . We consider the Banach lattice L1 (mq ) of the mq -integrable functions f , that is, Zp |f |d| mq , (bn ) | < ∞ for all (bn ) ∈ lq and
is a unique element in lq , denoted by set A ⊆ Zp there moreover for every Borel q A f dmq , such that A f dmq , (bn ) = A f d mq , (bn ) for every (bn ) ∈ l , where
mq , (bn ) denotes the complex measure A → mq (A), (bn ) . The norm L1 (mq ) is defined as f L1(mq ) := sup Zp |f |d| mq , (bn ) | and is σ-order continuous. bn lq =1 We also consider the integration map I : L1 (mq ) → lq , I(f ) = Zp f dmq and we note that it is a bounded operator with I ≤ 1 (see [10]). The next theorem ensures that this operator is precisely the continuous extension of the Fourier transform. Theorem 3.1. (i) For 1 < q < 2 the following are valid: Lq (Zp ) L1 (mq ) L1 (Zp ), f dmq = f6 1A for all f ∈ L1 (mq ) and A ∈ B(Zp ), f L1 (mq ) ≤ A 4f Lq (Zp ) for every f ∈ Lq (Zp ), f L1 (Zp ) ≤ f L1 (mq ) for all f ∈ L1 (mq ). Moreover for q = 1, 2 we have L1 (m1 ) = L1 (Zp ) and L1 (m2 ) = L2 (Zp ). (ii) If Z is any Banach function space over (Zp , B(Zp ), dx) with σ-order continuous norm which contains the space Lq (Zp ) continuously and if also the Fourier transform F : Lq (Zp ) → lq has a continuous linear extension T : Z → lq , then Z is contained continuously in L1 (mq ). Proof. We only focus on the proper inclusions Lq (Zp ) L1 (mq ) L1 (Zp ) for 1 < q < 2 since all the other parts of the theorem can be easily proved following the proof of Theorem 1.1 in [10]. From now on we write Fq (Zp ) instead of L1 (mq ).
Salem Sets in the p-adics and Fourier Restriction
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Fq (Zp ) L1 (Zp ), 1 < q ≤ 2: So far the ordering (γm )∞ m=0 of the elements of the set −1 cj pj : k ∈ N, cj = 0, 1, . . . , p − 1 j=−k
was completely arbitrary. We now need to consider a specific one. We take mν m1 m0 γmν pν +···+m1 p+m0 = ν+1 + · · · + 2 + p p p where mj = 0, 1, . . . , p − 1. Let f (x) = |x|1 p · log21 1 · 1| |x|p
|p <1
, x = 0. It is easy to see that f ∈ L1 (Zp ).
Next we compute the Fourier coefficients f1(γm ) ⎧ j ⎨ p − pj−1 , −pj−1 , 1| |p =pj (x) = ⎩ 0 ,
for m = 0. Using the formula |x|p ≤ p−j |x|p = p1−j |x|p ≥ p2−j
we get f1(γm ) =
∞
1 j 1 p log2 p j=1 j 2
=−
|x|p =p−j
χ(γm x)dx
log2 p 1 p−1 + log2 p log2 (|γm |p p−1 ) log2 p pj ≥|γ
m |p
≥
−1
pj
1 −j p (1 − p−1 ) j2
−1
1 1−p log p −p +C ≥ C log |γm |p log2 (|γm |p p−1 ) log2 p log |γm |p
where C is a positive number and |γm |p is big enough, say |γm |p ≥ pn0 . So ∞ (p−1)pn−1 f1q ≥ C n q = ∞. lq
n=n0
(log p )
Before we pass to the proper inclusion Lq (Zp ) Fq (Zp ), 1 < q < 2, we state a theorem in which the space Fq (Zp ) is described in a much more concrete way. For the proof see [10]. Theorem 3.2. Fq (Zp ) = {f ∈ L1 (Zp ) : |f ||h|dt < ∞ ∀h ∈ Lq (Zp ) with 1 h ∈ lq } = 1 q 1 q ∞ 6 F {f ∈ L (Zp ) : f 1A ∈ l ∀A ∈ B(Zp )} = {f ∈ L (Zp ) : f g ∈ l ∀g ∈ L (Zp )}. We need also the following lemma. Lemma 3.3. (i) If rβ (x) =
· 1Zp , 0 < β < 1, then ⎧ 1−β (1−p−1 ) ⎨ p 1−β p −1 1−β r1β (ξ) = (1−p−β ) 1 ⎩ p p1−β · |ξ|1−β −1
1 |x|β p
p
, |ξ|p ≤ 1 , |ξ|p > 1
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(ii) If μ is a finite compactly supported measure in 1Qp , and if also Et (μ) denotes the t- energy of μ defined as Et (μ) := |x−y|tp dμ(x)dμ(y), then Et (μ) = |1μ(ξ)|2 C · Qp |ξ|1−t dξ. p
(iii) For a probability compactly supported measure μ in Qp , if Et (μ) < ∞ then dimH (supp μ) ≥ t. Proof. The proof of (i) is a direct computation. For (ii) note |1 μ(ξ)|2 1 lim 1−t dξ = k→∞ 1−t χ((x − y)ξ)dξdμ(y)dμ(x) k Qp |ξ|p |ξ| ≤p |ξ|p p 1 1 dμ(y)dμ(x) = C · Et (μ) = C · lim −kt k→∞ p |(x − y)p−k |tp
where in the last equality we made the change of variables ξ = pk ξ and we used (i) and the fact that for k big enough |(x − y)p−k |p > 1 since μ is compactly supported. We note that the corresponding proof in the Euclidean case is much more difficult. For (iii), we can imitate the corresponding proof in the Euclidean case. We omit the details. Lq (Zp ) Fq (Zp ), 1 < q < 2: From the end of Section 2 we have that for every 1 ≤ q < q(α) = 4−2α 4−3α there is an > 0 such that |f1(x)|2 dμ (x) ≤ C f 2Lq (Qp ) . (3.1) Let rβ (x) =
1 |x|β p
· 1Zp (x), 0 < β < 1. We set Iβ, (x) = rβ ∗ μ (x) =
1 |x − y|βp
dμ (y).
Since (3.1) is translation invariant, following [10], we get |f1(x)|2 dμt (x) ≤ C · f 2Lq (Qp ) where μt is the translation of μ by t ∈ Qp . Multiplying by rβ and then integrating with respect to t we get (3.2) |f1(x)|2 Iβ, (x)dx ≤ C f 2Lq (Qp ) . q Let (fm )∞ m=0 ∈ l and f =
∞ m=0
fm · 1B1 (γm ) . Then
q Qp |f (x)| dx =
∞ m=0
|fm |q . We
also have 1 B 1 (γm ) (x) = χ(γm x) · 1Zp (x), and since suppIβ, ⊆ Zp (3.2) gives |
∞ m=0
fm χ(γm x)| Iβ, (x)dx ≤ C 2
∞ m=0
2/q |fm |
q
.
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Hence by duality we get ∞ q q 2 q /2 |gIβ, (γm )| ≤ C ( |g(x)| Iβ, (x)dx) ≤ C gL∞ (Zp ) ( Iβ, (x)dx)q /2 m=0
for every g ∈ L∞ (Zp ). Since Iβ, ∈ L1 (Zp ), Theorem 3.2 implies that for every 1 ≤ q < q(α) there is an > 0 such that Iβ, ∈ Fq (Zp ) for every 0 < β < 1. The regularity property μ (B r (x0 )) ≤ Cδ, ·rα(1−δ) implies that Iβ, ∈ L∞ (Zp ) ∀ β < α. Since Iβ, ∈ L1 (Zp ) ∀ 0 < β < 1, by convexity we get, for β > α, that 1−α . (3.3) Iβ, ∈ Lq (Zp ) for q < β−α From Plancherel and Lemma 3.3 (i) we have |Iβ, (x)|2 = |r1β (ξ)|2 |μ1 (ξ)|2 dξ = C +
|ξ|p >1
Qp
For
1 2
< β < 1 we get
|μ1 (ξ)|2 1−(2β−1)
|ξ|p
dξ.
|F μ (ξ)|2 1−(2β−1) dξ. So from Lemma 3.3 Qp |ξ|p Hence β ≤ 1+α 2 (by Lemma 3.3 (iii)).
|Iβ, (x)|2 dx = C +
(ii), if Iβ, ∈ L2 (Zp ) then E2β−1 (μ ) < ∞. Next we prove that condition (3.3) is sharp: Let us suppose (for contradiction) 1−α+γ
that there is β0 , β0 > α such that Iβ0 , ∈ L β0 −α (Zp ) for some γ > 0. Since . From Iβ, ∈ L∞ (Zp ) ∀ β < α, by convexity we have Iβ, ∈ L2 (Zp ) ∀ β < 1+α+γ 2 the above argument about the energy of μ we get γ = 0. We are now able to conclude the proof of Theorem 3.1 : Let q0 ∈ (1, 2). For α0 sufficiently close to 1 we get q0 < q(α0 ). Therefore there is an 0 such that Iβ,0 ∈ Fq0 (Zp ) ∀ 0 < β < 1. Choosing β0 such that β0 > α0 and sufficiently close 0 < q0 and hence Iβ0 ,0 ∈ / Lq0 (Zp ). to 1 we have that β1−α 0 −α0 Acknowledgment The author is grateful to his supervisor professor Jim Wright. Many thanks also to professor Gerd Mockenhaupt for bringing the problem to our attention.
References [1] G.P. Curbera and W.J. Ricker, Optimal domains for kernel operators via interpolation. Math. Nachr. 244 (2002), 47–67. [2] G.P. Curbera and W.J. Ricker, Optimal domains for the kernel operator associated with Sobolev’s inequality. Studia Math. 158 (2003), 131–152 and 170 (2005), 217–218. [3] G.P. Curbera and W.J. Ricker, Compactness properties of Sobolev imbeddings for rearrangement invariant norms. Trans. Amer. Math. Soc. 359 (2007), 1471–1484. [4] G.P. Curbera and W.J. Ricker, Banach lattices with the Fatou property and optimal domains of kernel operators. Indag. Math. (N.S.) 17 (2006), 187–204. [5] O. Delgado and J. Soria, Optimal extension of the Hardy Operator. J. Funct. Anal. 244 (2007), 119–133. [6] G.B. Folland, A course in Abstract Harmonic Analysis. CRC Press, 1995.
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[7] N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions. Springer-Verlag, 1977. [8] G. Mockenhaupt, Salem sets and restriction properties of Fourier transforms. Geom. Funct. Anal. 10 (2000), no. 6, 1579–1587. [9] G. Mockenhaupt, Bounds in Lebesgue Spaces of Oscillatory Integral Operators. Habilitationsschrift, Siegen 1996. [10] G. Mockenhaupt and W.J. Ricker, Optimal extension of the Hausdorff-Young inequality. Journal f¨ ur die reine und angewandte Mathematik. 620 (2008), 195–211. [11] S. Okada and W.J. Ricker, Optimal domains and integral representations of convolution operators in Lp (G). Integral Equations Operator Theory, 48 (2004), no. 4, 525–546. [12] S. Okada and W.J. Ricker, Optimal domains and integral representations of Lp (G)valued convolution operators via measures. Math. Nachr. 280 (2007), 423–436. [13] S. Okada, W.J. Ricker and E.A. Sanchez Perez, Optimal Domains and Integral Extensions of Operators. Birkh¨ auser, 2008. [14] R. Salem, On Singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1 (1950), 353–365. Christos Papadimitropoulos School of Mathematics University of Edinburgh JCMB, King’s Buildings Mayfield Road Edinburgh EH9 3JZ, Scotland e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 339–344 c 2009 Birkh¨ auser Verlag Basel/Switzerland
L-embedded Banach Spaces and a Weak Version of Phillips Lemma Hermann Pfitzner Abstract. After a short survey on L-embedded Banach spaces – for which Bochner spaces of the form L1 (X) with reflexive X serve as examples – we prove that these spaces satisfy a weak form of Phillips’ lemma: It is proved that the L-projection of an L-embedded Banach spaces sends relatively w∗ sequentially compact sets to relatively weakly sequentially compact sets. Mathematics Subject Classification (2000). 46B20. Keywords. L-embedded, L-projection, Phillips lemma, Bochner spaces.
L-embedded Banach spaces. A projection P on a Banach space Z is called an L-projection if P z + z − P z = z for all z ∈ Z. A Banach space X is called L-embedded (or an L-summand in its bidual) if it is the image of an L-projection on its bidual. In this case we write X ∗∗ = X ⊕1 Xs . The standard reference for L-embedded Banach spaces is the monograph [3] which contains everything of this introduction except for [7]. For general Banach space theory and undefined notation we refer to [4], [6], or [1]. A special class of L-embedded spaces consists of the duals of M-embedded Banach spaces: A Banach space Y is called M-embedded (or an M-Ideal in its bidual) if its annihilator Y ⊥ in Y ∗∗∗ is the range of an L-projection on Y ∗∗∗ in which case the dual Y ∗ identifies easily with the kernel of the L-projection and is therefore L-embedded. Examples of such L-embedded spaces include l1 (Γ) = (c0 (Γ))∗ (Γ any set), N(H) = (K(H))∗ (the nuclear operators on a Hilbert space H), (K(lp ,lq ))∗ where 1 < p ≤ q < ∞, the Hardy space H01 = (C(T)/A)∗ where A is the disk algebra. L1 -spaces are L-embedded and so are, more generally, the preduals of von Neumann algebras, even the preduals of JBW∗ -triples. As to a generalization towards Bochner spaces of the form L1 (X) not much seems to be known: From [3, p. 199] we resume the following cases. Since the projective tensor product of L1 and the predual of von Neumann algebra is again the predual of a von Neumann algebra the Bochner space L1 (X) is L-embedded if X is a von Neumann predual. The support of ANR-06-BLAN-0015 is gratefully acknowledged.
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H. Pfitzner
Further, if the underlying measure space is finite and if X ⊂ L1 is (in Godefroy’s terminology) nicely placed, i.e., L-embedded, for example H01 ⊂ L1 (T), then L1 (X) is L-embedded, too. Finally, if X is reflexive (hence trivially L-embedded) then L1 (X) is L-embedded. The class of L-embedded spaces is much larger than the one of duals of Membedded spaces because among L1 -spaces only l1 (Γ) (Γ a set) is the dual of an M-embedded space, and, analogously, the only von Neumann predual which is the dual of an M-embedded space is the direct sum in the l1 -sense of N(H)spaces which in turn is equivalent to the von Neumann predual having RNP [3, Prop. III.2.9, Prop. IV.2.9]. In particular, an L-embedded space need not have RNP whereas the dual of an M-embedded space does. There are dual L-embedded spaces with RNP that are not duals of M-embedded spaces [3, Ex. IV.4.12]. The interest of L-embeddedness lies in the fact that it unifies some Banach space properties which before were known only as special cases. A kind of breakthrough for this general point of view was Godefroy’s discovery that L-embedded Banach spaces are weakly sequentially complete. Another example of this unifying point of view is the fact [7] that (at least) separable L-embedded spaces are unique preduals which comprises for example the corresponding result of Dixmier-Sakai for von Neumann preduals. Talagrand [10] has shown that the Bochner space L1 (X) is weakly sequentially complete if and only if X is and Randrianantoanina [9] has shown that L1 (X) has Pe lczy´ nski’s property (V∗ ) if and only if X does (see also [5]). Since L-embedded spaces are weakly sequentially complete and have (V∗ ) [3, IV.2.7] it seems plausible that L1 (X) is L-embedded for more Banach spaces X than the ones mentioned above. A weak version of Phillips’ lemma. Phillips’ lemma [8] states that the canonical projection from the second dual of l1 onto l1 is w∗ -norm-sequentially continuous which if one takes l1 ’s Schur property for granted is equivalent to the projection being w∗ -weak-sequentially continuous. (Cf., for example, [1, Ch. VII].) Therefore the fact that the canonical projection from the third onto the first dual of a Banach space is w∗ -weak sequentially continuous if the space in consideration has Pe lczy´ nski’s property (V) (cf. [3, Prop. III.3.6]) can be considered a generalization of Phillips’ lemma. In this note we give a modest – see the remark after the proof – generalization in a similar direction by looking at l1 as an L-embedded Banach space. Although the result is new the technique of its proof is not. In fact it is a modification of the proof in [7] but holds, contrary to the latter one, for all L-embedded Banach spaces, not only separable ones. Theorem The L-projection of an L-embedded Banach space sends relatively w∗ sequentially compact sets into relatively weakly sequentially compact sets. a Banach space Z is called weakly unconProof. We recall that a series zj in ditionally Cauchy (wuC for short) if |z ∗ (zj )| converges for each z ∗ ∈ Z ∗ or,
Banach Spaces and a Weak Version of Phillips Lemma
341
n equivalently, if there is a number M such that j=1 αj zj ≤ M max1≤j≤n |αj | for all n ∈ N and all scalars αj . Let X be an L-embedded Banach space with L-projection P . We have the decomposition X ∗∗ = X ⊕1 Xs where Xs denotes the range of the projection Q = idX ∗∗ − P . Let (xn ) be a sequence in X and let (tn ) be a sequence in Xs . Further, consider a cluster point x + xs of the xn in the w∗ -topology of X ∗∗ (with x ∈ X, xs ∈ Xs ). ∗ Claim. Given x∗ ∈ X ∗ of norm one there are a wuC-series xk in X ∗ and an increasing sequence (nk ) in N such that tnk (x∗k ) = lim x∗k (xnk ) = k
0
∀k ∈ N
(1)
xs (x∗ ).
(2)
Proof of the claim. Let 1 > ε > 0 and E let (εj ) be a sequence of numbers decreasing to zero such that 0 < εj < 1 and ∞ j=1 (1 + εj ) < 1 + ε. By induction over k ∈ N0 = N∪{0} we shall construct two sequences (x∗k )k∈N0 and (yk∗ )k∈N0 in X ∗ (of which the first members x∗0 and y0∗ are auxiliary elements used only for the induction) and an increasing sequence (nk ) of indices such that, for all (real or complex) scalars αj and with β = xs (x∗ ), the following hold: x∗0 = 0, k αj x∗j α0 yk∗ +
≤
j=1
y0∗ = 1, k > (1 + εj ) max |αj |, 0≤j≤k
j=1
tnk (x∗k ) yk∗ (x) |x∗k (xnk )
=
(3) if k ≥ 1,
0, xs (yk∗ )
= 0, and − β| < εk
(4) (5)
= β, if k ≥ 1.
(6) (7)
We set n0 = 1, x∗0 = 0 and y0∗ = x∗ . For the following it is useful to recall some properties of P : The restriction of P ∗ to X ∗ is an isometric isomorphism from X ∗ onto Xs⊥ with (P ∗ y ∗ )|X = y ∗ for all y ∗ ∈ X ∗ , Q is a contractive projection and X ∗∗∗ = Xs⊥ ⊕∞ X ⊥ (where X ⊥ is the annihilator of X in X ∗∗∗ ). For the induction step suppose now that x∗0 , . . . , x∗k , y0∗ , . . . , yk∗ and n0 , . . . , nk have been constructed and satisfy conditions (3)–(7). Since x + xs is a w∗ -cluster point of the xn there is an index nk+1 such that |xs (yk∗ ) − yk∗ (xnk+1 − x)|
<
εk+1 .
Put E
=
lin({x∗ , x∗0 , . . . , x∗k , yk∗ , P ∗ x∗0 , . . . , P ∗ x∗k , P ∗ yk∗ }) ⊂ X ∗∗∗ ,
F
=
lin({xnk+1 , tnk+1 , x, xs }) ⊂ X ∗∗ .
(8)
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Clearly Q∗ x∗j , Q∗ yk∗ ∈ E for 0 ≤ j ≤ k. By the principle of local reflexivity there is an operator R : E → X ∗ such that (1 − εk+1 )e∗∗∗ ∗∗
∗∗∗
f (Re ) R|E∩X ∗
≤ Re∗∗∗ ≤ (1 + εk+1 )e∗∗∗ , ∗∗∗
∗∗
= e (f ), = idE∩X ∗
(9) (10) (11)
for all e∗∗∗ ∈ E and f ∗∗ ∈ F . We define x∗k+1 = RP ∗ yk∗
∗ and yk+1 = RQ∗ yk∗ . 0 In the following we use the convention j=1 (· · · ) = 0. Then we have that ∗ α0 yk+1 +
k+1
k k αj x∗j = R Q∗ (α0 yk∗ + αj x∗j ) + P ∗ (αk+1 yk∗ + αj x∗j ) .
j=1
j=1
j=1
Now (4) (for k + 1 instead of k) can be seen as follows: k+1 ∗ + αj x∗j α0 yk+1 j=1
≤
k k (1 + εk+1 ) Q∗ (α0 yk∗ + αj x∗j ) + P ∗ (αk+1 yk∗ + αj x∗j )
=
k k αj x∗j ) , P ∗ (αk+1 yk∗ + αj x∗j ) (1 + εk+1 ) max Q∗ (α0 yk∗ +
≤
k k αj x∗j , αk+1 yk∗ + αj x∗j (1 + εk+1 ) max α0 yk∗ +
≤
k+1 > (1 + εj ) max{ max |αj |, max |αj |}
(9)
j=1
j=1
j=1
j=1
j=1
=
0≤j≤k
j=1
j=1
1≤j≤k+1
k+1 > (1 + εj ) max |αj | j=1
0≤j≤k+1
where the last inequality comes from (3) if k = 0, and from (4), if k ≥ 1. The conditions (5) and (6) (for k + 1 instead of k) are easy to verify because P tnk+1 = 0, Qx = 0 and Qxs = 0 thus, by (10) tnk+1 (x∗k+1 ) = yk∗ (P tnk+1 ) = 0, ∗ (x) = yk∗ (Qx) = 0 yk+1
and ∗ ) = Q∗ yk∗ (xs ) = xs (yk∗ ) = β. xs (yk+1
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Finally, we have x∗k+1 (xnk+1 ) − β = yk∗ (xnk+1 ) − β = yk∗ (xnk+1 − x) − xs (yk∗ ) by (6) whence (7) for k + 1 by (8). This ends the induction and proves the claim. ∗ ∗∗ Let (x∗∗ n ) be an arbitrary w -null sequence. We write xn = xn + tn where the xn and tn are as before the statement of the claim, with x + xs a w∗ -cluster point of he xn . For an arbitrary x∗ ∈ X ∗ let (x∗j ) and (nk ) be as given by the on the subsets of N claim. We define a sequence (μn ) of finitely additive measures ∗ ∗ by μn (A) = x∗∗ ( x ) for all A ⊂ N where x is to be understood in n k∈A k k∈A k the w∗ -topology of X ∗ . Then μn (A) → 0 for all A ⊂ N and by (1) and Phillips’ original lemma ([8], [1, p. 83]) we get ∗ ∗ |x∗k (xnk )| = |x∗∗ |x∗∗ |μnk ({j})| → 0. nk (xk )| ≤ nk (xj )| = j ∗
j
∗
Thus xs (x ) = 0 by (2). Since x was an arbitrary normalized element this means that xs = 0. It follows that the w∗ -closure in X ∗∗ of the set consisting of the xn lies in X. Hence this set is relatively weakly compact (or, equivalently, relatively weakly sequentially compact). This proves the theorem. It is not clear whether the L-projection is actually w∗ -weak-sequentially continuous. In two cases this happens trivially: firstly if the dual X ∗ is a Grothendick space (which by definition means that in X ∗∗ w∗ -convergent sequences converge weakly) and secondly if X is the dual of an M-embedded Banach space Y (see [3, Ch. III]) because in this case P is the canonical restriction projection from Y ∗∗∗ onto Y ∗ ([3, Prop. III.2.4]) and the xn tend to 0 on the elements of Y and thus any weak cluster point x of (xn ) is necessarily zero. Alternatively, in the latter case the result [3, Prop. III.3.6], which was mentioned above, applies because M-embedded Banach spaces have property (V) ([2] or [3, Th. III.3.4]).
References [1] J. Diestel. Sequences and Series in Banach Spaces. Springer, Berlin-Heidelberg-New York, 1984. [2] G. Godefroy and P. Saab. Weakly unconditionally convergent series in M -ideals. Math. Scand., 64:307–318, 1989. [3] P. Harmand, D. Werner, and W. Werner. M -ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics 1547. Springer, 1993. [4] W.B. Johnson and J. Lindenstrauss. Handbook of the Geometry of Banach Spaces, Volumes 1 and 2. North Holland, 2001, 2003. [5] P.K. Lin. K¨ othe-Bochner function spaces. Birkh¨ auser, Boston, 2004. [6] J. Lindenstrauss and L. Tzafriri. Classical Banach Spaces I and II. Springer, BerlinHeidelberg-New York, 1977, 1979.
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[7] H. Pfitzner. Separable L-embedded Banach spaces are unique preduals. Bull. Lond. Math. Soc., 39:1039–1044, 2007. [8] R.S. Phillips. On linear transformations. Trans. Amer. Math. Soc., 48:516–541, 1940. [9] N. Randrianantoanina. Complemented copies of 1 and Peczyski’s property (V ∗ ) in Bochner function spaces. Canadian J. Math., 48:625–640, 1996. [10] M. Talagrand. Weak Cauchy sequences in L1 (E). Amer. J. Math., 106:703–724, 1984. ¨ [11] A. Ulger. The weak Phillips property. Colloq. Math., 87:147–158, 2001. Hermann Pfitzner Universit´e d’Orl´eans BP 6759 F-45067 Orl´eans Cedex 2, France e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 345–359 c 2009 Birkh¨ auser Verlag Basel/Switzerland
When is the Space of Compact Range Measures Complemented in the Space of All Vector-valued Measures? Luis Rodr´ıguez-Piazza Abstract. Let X be a Banach space and denote by M(X) the Banach space of all X-valued measures, defined on the σ-algebra of Borel subsets of [0, 1], and equipped with the semivariation norm. Denote by M0 (X) the (closed) subspace of M(X) consisting of the measures with relatively compact range. The aim of this paper is to prove that M0 (X) is complemented in M(X) if and only if M0 (X) = M(X). Mathematics Subject Classification (2000). Primary 28B05, 46G10; Secondary 46B20, 47L05. Keywords. Banach spaces, spaces of vector measures, compact range, complementability.
1. Introduction In the last part of this section, once the notation is fixed, we will present the main result of the paper, to whose proof we will devote the rest of the paper. But, let us begin with some motivation concerning the meaning of complementability. 1.1. Complementability In the analysis of mathematical objects one sometimes tries to decompose them into simpler parts or into different parts, each one carrying the different properties of the object. For instance one can decompose a finite measure μ as the sum μ = μpa + μc
(1.1)
of the purely atomic part μpa and the non-atomic (continuous) part μc . Another example is the Lebesgue decomposition, where μ can be decomposed with respect Partially supported by Spanish project MTM2006-05622.
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to another finite measure σ as the sum μ = μa + μs
(1.2)
of its absolutely continuous part μa / σ and its singular part μs ⊥ σ. The decompositions (1.1) and (1.2) can also be done for complex measures, and even for vector (Banach space valued) measures, [2, pp. 30–31]. In the vector case we may even consider further properties to decompose the measure. For instance, we can ask about the variation or about the compactness of the range. For instance, if μ is a Banach space valued measure, we can write μ = μ1 + μ2 ,
(1.3)
where μ1 has σ-finite variation, and μ2 has everywhere infinite variation, that is every measurable set has μ2 variation either 0 or ∞. This can also be done for compactness of the range. We can write μ = μcr + μncr
(1.4)
with μcr having relatively compact range and μncr having the property that the range of its restriction to every measurable set is either {0} or a non relatively compact set (see [5] and [13] for these facts). We can also ask about the way in which these decompositions act in the space of vector measures. One can see that the maps μ → μc in (1.1) and μ → μa in (1.2) are both linear and continuous. These maps are projections and, thanks to this, we can say (for X a Banach space) that the spaces of continuous measures, of purely atomic measures, and of absolutely continuous measures are complemented in the space of X-valued measures. Thus, complementability can be viewed as a “fair way”(with good properties) to decompose objects. Unfortunately, the decompositions (1.3) and (1.4) do not behave well. First, in general, they are not unique. In order to get uniqueness we need to impose orthogonality, i.e., μ1 ⊥ μ2 in (1.3) and μcr ⊥ μncr in (1.4). But, even with this requirement they do not produce complementability. In general, the map μ → μcr in (1.4) is neither linear nor continuous, and the same is true for the map μ → μ1 in (1.3). Nevertheless, we can still ask for a “fair way” to decompose vector measures into a compact range part and another part. That is, when is the space of relatively compact range X-valued measures complemented in the space of X-valued measures? The purpose of this paper is to provide the answer to this question: we will see that there is no complementability unless these two spaces coincide (that is, unless every X-valued measure has relatively compact range). We remark, for the case of the variation, that the answer is even worse because, in general, the space of σ-finite variation measures is not closed!
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1.2. Notation Vector Measures. The book [2] is the basic reference for all general facts we will need about vector measures. We will consider only the case of Banach spaces over the field of real numbers. So, let X be a Banach space and (Ω, Σ) be a measurable space. Then M(Σ, X) denotes the space of all X-valued vector measures defined on (Ω, Σ). That is, M(Σ, X) = {μ : Σ → X | μ is countably additive }, which is a Banach space for the semivariation norm. The range of a measure μ is μ(Σ) = {μ(A) : A ∈ Σ}. This is always a relatively weakly compact subset of X. We will also require M0 (Σ, X) = {μ ∈ M(Σ, X) : μ(Σ) is relatively compact }, which is a closed subspace in M(Σ, X). Suppose now that σ is a finite (positive) measure defined on (Ω, Σ). We will say that μ ∈ M(Σ, X) is absolutely continuous with respect to σ if μ(A) = 0, for every A ∈ Σ with σ(A) = 0. This is equivalent to limσ(A)→0 μ(A) = 0 and, as usual, is denoted by μ / σ. We can then consider the spaces Mσ (Σ, X) = {μ ∈ M(Σ, X) : μ / σ}, M0σ (Σ, X) = Mσ (Σ, X) ∩ M0 (Σ, X). Both are closed subspaces of M(Σ, X). The Lebesgue decomposition (1.2) provides us with a projection from M(Σ, X) onto Mσ (Σ, X). It is easy to see that this projection sends M0 (Σ, X) onto M0σ (Σ, X). When Ω is the interval [0, 1] and Σ is the σ-algebra B of its Borel subsets, we will simply write M(X) and M0 (X) instead of M(B, X) and M0 (B, X). The Lebesgue measure on B is denoted by λ. We will then use the notation Mλ (X) instead of Mλ (B, X); analogously we will use M0λ (X). Each space Lp (λ), 1 ≤ p ≤ ∞, is denoted simply by Lp . Operators. If X and Y are Banach spaces, then L(X, Y ) denotes the space of bounded linear operators from X to Y . The subspace of compact operators is denoted by K(X, Y ). If X is the dual of a Banach space Z (i.e., X = Z ∗ ), then we also consider the subspace of L(X, Y ) consisting of the operators that are continuous from the weak* topology in X (as the dual of Z) to the weak topology of Y . These are characterized by the inclusion T ∗ (Y ∗ ) ⊂ Z, where T ∗ denotes the adjoint operator of T . We also adopt the notations Lw∗ ,w (X, Y ) = T ∈ L(X, Y ) : T ∗ (Y ∗ ) ⊂ Z , Kw∗ ,w (X, Y ) = Lw∗ ,w (X, Y ) ∩ K(X, Y ).
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1.3. Main result As announced above, the aim of this paper is to establish the following theorem. Theorem 1.1. Let X be a Banach space. Then M0 (X) is complemented in M(X) if and only if M(X) = M0 (X). Observe that, besides finite-dimensional Banach spaces, there also exist other Banach spaces X for which every X-valued measure has relatively compact range. This is the case for all Banach spaces X with the Schur property (e.g., 1 ), because every weakly compact set is norm compact. There are also spaces X without the Schur property satisfying M(X) = M0 (X); for instance, X = p , for 1 < p < 2, [17, Remark 2, p. 211]. Remark 1.2. The first point concerning the proof of this theorem is that it suffices to prove that M0λ (X) is not complemented in Mλ (X) whenever M(X) = M0 (X). Indeed, M0λ (X) is complemented in M0 (X) by the Lebesgue decomposition. So, were M0 (X) complemented in M(X), then M0λ (X) would be complemented in M(X) and, consequently, also in Mλ (X). The question solved by Theorem 1.1 was proposed to the author by L. Drewnowski. In fact, he had already proved some results giving uncomplementability of M0 (X) in M(X) for many Banach spaces X, and he asked if these cases covered all the possibilities of when M0 (X) = M(X). Concretely, Drewnowski (see [3] and [4]) proved the following result. Theorem 1.3. Let X be a Banach space. The following assertions are equivalent. (a) Mλ (X) contains an isomorphic copy of c0 . (b) There exists a non compact operator T : 2 → X. (c) There exists an isomorphic embedding j : ∞ → Mλ (X), such that j(c0 ) = j(∞ ) ∩ M0λ (X), and j(c0 ) is complemented in M0λ (X). The fact that (c) produces uncomplementability of M0 (X) in M(X) is an easy consequence of the well-known fact that c0 is not complemented in ∞ . Then M0 (X) is not complemented in M(X) as soon as L(2 , X) = K(2 , X).
(1.5)
It can be proved (see [4]) that M (X) = M(X) is equivalent to 0
L(∞ , X) = K(∞ , X).
(1.6)
So, if we knew that (1.6) implies (1.5), then we would have a proof of Theorem 1.1 via Theorem 1.3. Unfortunately, this implication is not true in general, although it is true under some additional hypothesis, for instance, if X has finite cotype. The fact that (1.6) does not always imply (1.5) is a consequence of the following result which will appear in a forthcoming paper (still in preparation). Theorem 1.4. There exists a rearrangement invariant Banach lattice R of functions on [0, 1] such that L∞ [0, 1] ⊂ R ⊂ L1 [0, 1] ,
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and, for all 1 < p < +∞, we have L(p , R) = K(p , R) . Observe in the case of the lattice R in Theorem 1.4 that the map A → χA is an R-valued measure whose range is not relatively compact. So we need a different approach to prove Theorem 1.1. In Section 2 we will establish the result when the Banach space X is separable. The non-separable case will need some modifications which are explained in Section 3.
2. Proof of Theorem 1.1: the separable case 2.1. Recalling why c0 is not complemented in ∞ It is a very well-known fact in the geometry of Banach spaces that c0 is not complemented in ∞ . This result is implicit in Phillips’ paper [15], and first stated by Sobczyk in [18]. In [19], Whitley gave a different proof whose ideas will be useful for the proof of our main result. Let us recall his arguments. Suppose that X is a Banach space, and Y is a subspace of X. If Y is complemented in X, then there exists a closed subspace E in X such that X = Y ⊕ E. Therefore the quotient space X/Y is isomorphic to E. Whitley proved that the quotient ∞ /c0 is not isomorphic to any subspace of ∞ , thereby yielding that c0 is not complemented in ∞ . The following result, whose easy proof is left to the reader, provides a useful criterion to determine when a Banach space is not embeddable in ∞ . Lemma 2.1. Let Z be a Banach space. Suppose there exists in Z a family {zt : t ∈ R} satisfying zt ≥ 1, for every t ∈ R, and such that the set {t ∈ R : z ∗ (zt ) = 0} is at most countable, for every z ∗ ∈ Z ∗ . Then Z is not isomorphic to a subspace of ∞ . In fact, every bounded operator from Z to ∞ is not injective. Denote by P∞ (N) the family of all infinite subsets of N. The other ingredient in Whitley’s argument is the following result whose proof can be found in [19] or [1, p. 45]. Lemma 2.2. There exists a family {Et : t ∈ R} ⊂ P∞ (N) which is almost disjoint, that is, every Et is an infinite subset of N and Et ∩ Et is finite whenever t = t . Let {Et : t ∈ R} be as in Lemma 2.2. For every t ∈ R, consider the characteristic function χEt of the set Et as an element of ∞ and let zt = χEt + c0 be its coset in the quotient space Z = ∞ /c0 . Then this family {zt : t ∈ R} satisfies the hypothesis of Lemma 2.1. We conclude that ∞ /c0 is not embeddable in ∞ (see [19] for the details).
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2.2. Identifying measures and operators In the case that X is a separable Banach space, it is not difficult to see that the space Mλ (X) is isomorphic to a subspace of ∞ . So, we can still use Lemma 2.1 to see that, whenever M0 (X) = M(X), the quotient space Mλ (X)/M0λ (X) is not embeddable in Mλ (X), because it is not embeddable in ∞ . It will be more convenient for us to work with spaces of operators than with spaces of vector measures. We recall an identification between some of these spaces. Let σ be a finite measure defined on a measurable space (Ω, Σ). Consider L∞ (σ) as the dual of L1 (σ). If T : L∞ (σ) → X is any weak* to weak continuous linear map, i.e., T ∈ Lw∗ ,w L∞ (σ), X , we define Φ(T ) as the X-valued set function Φ(T )(A) = T (χA ),
A ∈ Σ.
It easy to see that Φ(T ) is weakly countably additive and, thanks to the Orlicz– Pettis Theorem, that Φ(T ) ∈ M(Σ, X). Actually, Φ(T ) is absolutely continuous with respect to σ and we have the following result. above is an Proposition 2.3. The map Φ : Lw∗ ,w L∞ (σ), X → Mσ (Σ, X) defined ∞ ∗ isomorphism between these two Banach spaces. The subspace Kw ,w L (σ), X is sent by Φ onto the space M0σ (Σ, X). The proof of this proposition can be found amongst different parts of [2]; see, for instance, Theorem I.1.13 and Lemma IX.1.3. The inverse of Φ is the map sending every measure μ to the operator defined by the Bartle integral with respect to μ. It is in terms of this inverse operation Φ−1 that these matters are explained in [2]. Let us now use this identification to explain why Mλ (X) is embeddable in ∞ whenever X is separable: it suffices to see that this is so for Lw∗ ,w L∞ , X . Lemma 2.4. Let X be a separable Banach space. Then Lw∗ ,w L∞ , X is isomorphic to a subspace of ∞ . Proof. Observe that the unit ball B of L∞ (λ) is weak* separable (metrizable and compact), so we can select a dense sequence {fn } in B. Separability of X can then be used to choose a norming sequence {x∗m } in the unit ball of X ∗ . In fact, it is the existence of such a norming sequence which is the only requirement we will need throughout this section. Since we have T = sup{T fnX : n ∈ N} = sup{| x∗m , T fn | : n, m ∈ N}, for every T ∈ Lw∗ ,w L∞ , X , the map T → x∗m , T fn n,m is an isometric embedding of Lw∗ ,w L∞ , X into ∞ (N × N) ∼ = ∞ .
Remark 2.5. Using Proposition 2.3 we have reduced 1.1 to the proof of Theorem proving the uncomplementability of Kw∗ ,w L∞ , X in Lw∗ ,w L∞ , X , whenever M0 (X) = M(X).
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Let us mention that there is a long series of papers devoted to the study of the uncomplementability between different spaces of operators: bounded, compact, weakly compact, . . . . This property has usually been related to the presence of copies of c0 . The interested reader can take a look, for instance, at the papers of Kalton [14], Feder [9], Emmanuele [6, 7], Emmanuele and John [8], Ghenciu [10], . . . . 2.3. Rademacher functions Let us recall some facts about the sequence of Rademacher functions rn (x) = sign(sin(2n πx)),
for x ∈ [0, 1] and n ∈ N.
Defined on the probability space ([0, 1], B, λ), from the probabilistic point of view {rn }n≥1 is a sequence of independent variables all having the same distribution: essentially each rn takes only the values 1 and −1 and each one in a set of measure 1/2. As every rn has mean 0, the independence yields that the Rademacher sequence is an orthonormal system in L2 = L2 [0, 1]. But it is not a complete orthonormal system. In L∞ the sequence {rn }n≥1 is weak* null. So, if X is a Banach space, and T ∈ Kw∗ ,w L∞ , X , then limn→∞ T (rn ) = 0. We will need the following proposition. Its proof follows the pattern of a construction by Girardi and Johnson, [11], used to prove the existence of an universal non completely continuous operator on L1 spaces. Proposition 2.6. Let X be a Banach space. If there exists a measurable space (Ω, Σ) such that M0 (Σ, X) = M(Σ, X), then there exists an operator β : L∞ → X which is weak* to weak continuous and satisfies β(rn ) ≥ 1,
for all n ∈ N.
Proof. Let μ ∈ M(Σ, X) \ M0 (Σ, X). Purely atomic vector measures always have relatively compact ranges. So, using the decomposition (1.1) we may assume that μ has no atoms. There exists then a probability measure σ without atoms such that μ / σ. For instance, one can take σ = |x∗0 ◦ μ|, for a suitable Rybakov functional x∗0 [2, p. 268]. Thanks to Proposition 2.3 there exists a non compact operator T0 ∈ Lw∗ ,w L∞ (σ), X . Since T0 is not compact, there exists δ > 0, such that, for every subspace Y of L∞ (σ) with finite codimension the restriction of T0 to Y has norm T0 |Y > δ, [16, Proposition 2.4.10]. We may assume that δ = 1 and now proceed to construct a “Rademacherlike” sequence {fn } in L∞ (σ) such that T0 (fn ) > 1, for every n ∈ N. We first require the following observation: if Y is a weak* closed subspace of L∞ (σ) with finite codimension, and f is an extreme point of the unit ball of Y , then |f | = 1 almost everywhere. Indeed, if this were not true, then there should exist ε > 0 and A ∈ Σ with σ(A) > 0 such that |f (ω)| ≤ 1 − ε, for every ω ∈ A.
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Since σ has no atom, the space Z of functions g ∈ L∞ (σ), which are 0 almost everywhere in Ω \ A is infinite dimensional. Now, Y has finite codimension and so there exists g ∈ Z ∩ Y with 0 < g∞ < ε. It is easy to check that both f + g and f − g are in the unit ball of Y . This contradicts the fact that f is an extreme point. To begin the construction, consider the subspace Y1 = f ∈ L∞ (σ) : f dσ = 0 , which is weak* closed and finite codimensional. Using the Krein–Milman theorem for the unit ball of Y1 , and the facts that T0 is weak* to weak continuous with T0 |Y1 > 1, we see that there exists an extreme point f1 of this unit ball such that T0 (f1 ) > 1. We know, by the above observation, that |f1 | = 1 almost everywhere which we can assume actually happens everywhere. So, f1 takes only values 1 and −1. Moreover, σ({f1 = 1}) = σ({f1 = −1}) = 1/2 because f1 dσ = 0. Suppose we have found f1 , f2 , . . . , fn , all taking only the values 1 and −1 in such a way that 7 n σ {fj = sj } = 2−n , for every (s1 , s2 , . . . , sn ) ∈ {1, −1}n, (2.1) j=1
and j = 1, 2, . . . , n. (2.2) T0 (fj ) > 1, n Write Cn for nthe family of 2 pairwise disjoint sets obtained by forming all intersections j=1 {fj = sj }, for the different choices of (s1 , s2 , . . . , sn ) ∈ {1, −1}n. Define Yn+1 = f ∈ L∞ (σ) : D f dσ = 0, for every D ∈ Cn . Observe that Yn+1 is finite codimensional and weak* closed. Choose, as before, any ±1-valued function fn+1 which is an extreme point of the unit ball of Yn+1 such that T0 (fn+1 ) > 1. Then, for every D ∈ Cn , we have σ D ∩ {fn+1 = 1} = σ D ∩ {fn+1 = 1} = 2−n−1 , which is an analog of (2.1) for n + 1. The probabilistic interpretation of (2.1) is that the sequence {fn } consists of independent variables. Define now a measurable function φ : Ω → [0, 1] by φ(ω) =
∞ 1 − fn (ω) , 2n+1 n=1
ω ∈ Ω.
(2.3)
G H k n −n Let I = k−1 . 2n , 2n , with 1 ≤ k ≤ 2 be a dyadic subinterval of [0, 1] of length 2 −1 It is not difficult to check that φ (I) is, up to a σ-null set, an element of Cn and hence, σ φ−1 (I) = 2−n = λ(I). This is true for every n, from which we deduce that the image of σ by φ is the Lebesgue measure λ. The reader can check that also fn = rn ◦ φ σ-a.e., for every n. (2.4)
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Finally define the operator β : L∞ → X via β(h) = T0 (h ◦ φ),
h ∈ L∞ .
Thanks to (2.2) and (2.4) we have β(rn ) > 1, for every n. To see that β is weak* to weak continuous we can again use Proposition 2.3. We simply need to observe that B → β(χB ) is both countably additive on B and absolutely continuous with respect to λ. The first claim is an easy consequence of the facts that A → T0 (χA ) is countably additive on Σ and that β(χB ) = T0 χφ−1 (B) , for every B ∈ B. For the second claim, let B ∈ B be a λ-null set. Then σ φ−1 (B) = 0, χB ◦ φ = 0 σ-a.e., and β(χB ) = T0 (χB ◦ φ) = 0. Remark 2.7. A consequence of the last proposition is that M0λ (X) = Mλ (X) whenever there exists a measurable space (Ω, Σ) such that M0 (Σ, X) = M(Σ, X). This is part of a result of L. Drewnowski [4, Theorem 3.2]. We also have seen, at the beginning of the proof of Proposition 2.6, that whenever M0 (Σ, X) = M(Σ, X), then the σ-algebra Σ admits an atomless probability measure σ. The σ-algebras with this property are called σ-algebras of type (NA). Among other equivalences, Drewnowski proved M0λ (X) = Mλ (X) if and only if M0 (Σ, X) = M(Σ, X) for every σ-algebra Σ of type (NA), [4, Theorem 3.2]. 2.4. Walsh system The Rademacher sequence {rn } is an orthonormal sequence in L2 which is not complete. However, it is part of a complete orthonormal system, the so-called Walsh system {wD : D ∈ Pf (N)}, defined as follows, where Pf (N) is the family of all finite subsets of N. Namely, for each D ∈ Pf (N), we have > wD (x) = rn (x), x ∈ [0, 1]. n∈D
Observe that w∅ ≡ 1, and w{n} = rn , for every n ∈ N. Let us introduce some new operators. For every subset E (finite or infinite) of N, let ΣE denotes the σ-algebra in [0, 1] generated by the functions {rn : n ∈ E}. Thus, Σ∅ = {∅, [0, 1]} and ΣN is essentially B (every Borel set is equal almost everywhere to a set in ΣN ). We then denote by SE the conditional expectation over ΣE , namely SE (f ) = E(f | ΣE ), f ∈ L1 . p Then, for every p ∈ [1, +∞], the operator SE : L → Lp has norm one. Moreover, for p = ∞, SE is weak* to weak* continuous since it is the adjoint of SE : L1 → L1 . Considering the Walsh system we see that $ wD , if D ⊂ E; SE (wD ) = (2.5) 0, if D ⊂ E. That is, SE is a diagonal operator on L2 with respect to the orthonormal basis consisting of the Walsh system.
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Proof of Theorem 1.1 for X separable. Recall Remark 2.5 and fix a separable Ba0 briefly Lw∗ ,w in place of nach space ∞ X such that M (X) = M(X). Let us write Lw∗ ,w L , X , and Kw∗ ,w in place of Kw∗ ,w L∞ , X . To be proved is that Kw∗ ,w is not complemented in Lw∗ ,w . By Lemma 2.4, Lw∗ ,w is embeddable in ∞ . So, we will be done if we show that the quotient Lw∗ ,w /Kw∗ ,w is not isomorphic to a subspace of ∞ . In order to use Lemma 2.1 we consider the following operators. Let β be an operator as specified by Proposition 2.6 and let {Et : t ∈ R} be an almost disjoint family in P∞ (N) as in Lemma 2.2. For every t ∈ R, consider the operator Tt = β ◦ SEt , that is SE
β
t −→ L∞ −−−−→ X. Tt : L∞ −−−−−
(2.6)
Observe that Tt ∈ Lw∗ ,w , because SEt is weak* to weak* continuous. Moreover, thanks to (2.5), we also have Tt (rn ) = β(rn ), for n ∈ Et . Denote by T(t the class of Tt in the quotient space Lw∗ ,w /Kw∗ ,w , that is, the coset T(t = Tt + Kw∗ ,w .
(2.7)
Since {rn } is a weak* null sequence, given any K ∈ Kw∗ ,w , we have that {K(rn )} is weakly null in X. But, this latter sequence is contained in a compact subset of X since K is a compact operator. Accordingly, limn→∞ K(rn ) = 0. Then Tt + K ≥ sup Tt (rn ) + K(rn ) ≥ lim sup β(rn ) ≥ 1. n∈Et
n∈Et ,n→∞
This shows that T(t ≥ 1, for every t ∈ R. We then apply Lemma 2.1 to the family {T(t : t ∈ R}, provided that the hypotheses of this lemma hold: this is established in (b) in the next proposition. Granted this, we see that the quotient Lw∗ ,w /Kw∗ ,w is not embeddable into ∞ which finishes the proof in the case that X is separable. Proposition 2.8. Let {T(t : t ∈ R} be given by (2.7). (a) For every ε > 0, there exists Nε ∈ N such that, for every n ≥ Nε , and for every family of n distinct real numbers t1 , t2 , . . . , tn , we have n @ ≤ εn. T tj j=1
Lw∗ ,w /Kw∗ ,w
∗
(b) For every φ in the dual of Lw∗ ,w /Kw∗ ,w the set {t ∈ R : φ∗ T(t = 0} is at most countable.
Proof. First let us see why (b) follows from (a). If {t ∈ R : φ∗ T(t = 0} were uncountable, then there would exist ε > 0 such that either {t ∈ R : φ∗ T(t ≥ 2εφ∗ } or {t ∈ R : φ∗ T(t ≤ −2εφ∗ } is infinite. Then, for every n, we can select distinct numbers t1 , t2 , . . . , tn in one of these sets, namely, the one that is
Complementability of Compact Range Measures infinite, and we have n ∗ T@ φ tj j=1
Lw∗ ,w /Kw∗ ,w
355
# ## # ∗ n # @ ≥#φ Ttj ## ≥ 2εnφ∗ , j=1
thereby giving a contradiction to (a). So, let us prove (a). Fix ε > 0. Since β ∈ Lw∗ ,w L∞ , X , using the identification given in Proposition 2.3 and the absolute continuity of the associated vector measure A → β(χA ), we get that there exists δ > 0 such that for all f ∈ L∞ and all A ∈ B with λ(A) < δ. (2.8) 2 Select Nε large enough so that 2β/nε < δ, for every n ≥ Nε . Fix n ≥ Nε and pick any family of n distinct real numbers t1 , t2 , . . . , tn . If l = j, then the set Etl ∩ Etj is finite. Let Δ be any finite set in N containing all of these intersections Etl ∩ Etj , j = l, 1 ≤ j, l ≤ n, and define β(f χA ) ≤ (ε/4)f ∞ ,
Rj = SEtj − SEtj ∩Δ ,
1 ≤ j ≤ n,
and R =
n
Rj .
(2.9)
j=1
Observe that SEtj ∩Δ is a finite rank operator and so, it is compact. Therefore β ◦ Rj is in the same equivalence class as that of Ttj for j = 1, 2, . . . , n, and n @ ≤ β ◦ R L∞ →X . Ttj j=1
Lw∗ ,w /Kw∗ ,w
We now show that β ◦ RL∞ →X ≤ nε and (a) will thereby be proved. We have Rj L∞ →L∞ ≤ 2, for every 1 ≤ j ≤ n and so RL∞ →L∞ ≤ 2n. By (2.5) every conditional expectation operator SE is a diagonal operator on L2 with respect to the Walsh system. By (2.9) the same is true for the operators R and Rj , 1 ≤ j ≤ n. Additionally observe that either Rj (wD ) = 0 or Rj (wD ) = wD and that the last case happens if and only if D ⊂ Etj and D ⊂ Δ. By the property of Δ it cannot simultaneously happen Rj (wD ) = wD and Rl (wD ) = wD for l = j. Thus, we also have that either R(wD ) = 0 or R(wD ) = wD , for every D ∈ Pf (N) and we deduce that RL2 →L2 ≤ 1. Take now f in the unit ball of L∞ , and let g = Rf . We know that g∞ ≤ 2n and g2 ≤ 1. Consider the set A = {x ∈ [0, 1] : |g(x)| > (nε/2β)}. Using Chebyshev’s inequality we have 2β 2 g22 λ(A) ≤ ≤ < δ. 2 (nε/2β) nε Then by (2.8) we have β(gχA ) ≤ (ε/4)g∞ ≤ nε/2. On the other hand, with B = [0, 1] \ A, we have gχB ∞ ≤ nε/2β. We get nε + βgχB ∞ ≤ nε. (β ◦ R)f ≤ β(gχA ) + β(gχB ) ≤ 2 This yields β ◦ RL∞ →X ≤ nε and the proof is finished.
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3. Proof in the non separable case All the constructions we have made in Section 2 can also be carried out in the non separable case. In fact the only thing that is not at our disposal for every Banach space is Lemma 2.4. So, let X be a Banach such that M0 (X) = M(X). ∞ space We need that Kw∗ ,w = Kw∗ ,w L , X is not complemented in Lw∗ ,w = to prove Lw∗ ,w L∞ , X . Consider again β as in Proposition 2.6 and {Tt : t ∈ R} as defined in (2.6). We will require the following result. Proposition 3.1. Given the above setting, there exists a closed subspace Y of Lw∗ ,w , and an operator j : Y → ∞ so that: (a) Kw∗ ,w ⊂ Y . (b) Tt ∈ Y , for every t ∈ R. (c) j(T ) = 0, whenever T ∈ Y and T ∈ / Kw∗ ,w . Before giving the proof let us see how Proposition 3.1 allows us to complete the non separable case. If Kw∗ ,w were complemented in Lw∗ ,w , then it would also be complemented in Y and so Y = Z ⊕ Kw∗ ,w , for a closed subspace Z. Thanks to condition (c) in Proposition 3.1, we see that the restriction of j to Z is an injective operator from Z into ∞ . But, Z is isomorphic to the quotient space Y /Kw∗ ,w . We can still apply Proposition 2.8 to this quotient because Tt ∈ Y , for every t ∈ R. Again using Lemma 2.1 we see that there cannot exist an injective operator from Y /Kw∗ ,w into ∞ . This is a contradiction to the fact that Y /Kw∗ ,w is isomorphic to Z. Proof of Proposition 3.1. The operator β : L∞ → X is weak* to weak continuous, and L∞ is weak* separable. So, β(L∞ ) is weakly (and then norm) separable in X. Thus, we may consider a separable closed subspace X0 of X such that β(L∞ ) ⊂ X0 . Define Y as the set consisting of all T ∈ Lw∗ ,w for which there exist a compact set K in X and a positive number c > 0 so that T (BL∞ ) ⊂ K + cBX0 , where BL∞ and BX0 are the closed unit balls of L∞ and X0 , respectively. It is then trivial to see that Y is a linear subspace with Kw∗ ,w ⊂ Y . Moreover, as Tt (L∞ ) ⊂ β(L∞ ) ⊂ X0 , we also see that Tt ∈ Y , for every t ∈ R. To see that Y is closed, consider the quotient map Q : X → X/X0. Compact sets in the quotient can always be lifted, that is, if C is a compact subset of X/X0 , then there exists a compact set F in X with C = Q(F ). From this observation it is easy to see that a set A ⊂ X has the property that there exist a compact set K ⊂ X and positive number c > 0 such that A ⊂ K + cBX0 if and only if A is bounded and Q(A) is relatively compact in X/X0 . So, given T ∈ Lw∗ ,w , we see that T ∈ Y if and only if Q ◦ T is a compact operator. Thus, Y is closed because ∞ it is the inverse image of Kw∗ ,w L , X/X 0 under the map T → Q ◦ T which is continuous from Lw∗ ,w L∞ , X to Lw∗ ,w L∞ , X/X0 .
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It remains to construct j : Y → ∞ satisfying (c). Now, X0 is separable and so, after an application of the Hahn–Banach theorem, we see that there exists a sequence {x∗m } in the unit ball of X ∗ which is norming for X0 , that is x = sup{ | x∗m , x | : m ∈ N },
for every x ∈ X0 .
Choose any sequence {gn } dense in the unit ball of L and define j : Y → ∞ (N × N) by j(T ) = x∗m , T gn n,m .
∞
(3.1)
for the weak* topology
In order to prove (c), suppose that j(T ) = 0 for some T ∈ Y . Then we need to show that T ∈ Kw∗ ,w in which case we are finished. So, let D be the image by T of the unit ball, i.e., D = T (BL∞ ). By the density of {gn } and the condition j(T ) = 0, we have
x∗m , d = 0,
for all m ∈ N and all d ∈ D.
(3.2)
To see that D is totally bounded, fix ε > 0. We need to cover D with a finite family of sets of diameter less than ε. As T ∈ Y , we know that there exist a compact set K in X and c > 0 such that D ⊂ K + cBX0 .
(3.3)
Select a finite family K1 , K2 , . . . ,Kn of sets of diameter less than ε/2 covering the compact set K. Then writing Dj = D ∩ (Kj + cBX0 ), we have (by (3.3)) that D=
n )
j = 1, 2, . . . , n;
Dj .
j=1
To see that every Dj has diameter no larger than ε, fix j and take any two points d and d in Dj . There exist k, k ∈ Kj and x, x ∈ BX0 such that d = k + cx,
and
d = k + cx .
Since d and d belong to D, (3.2) implies that
x∗m , k − k = x∗m , cx − cx ,
for all m ∈ N,
and then, as k − k ≤ ε/2, we have | x∗m , cx − cx | ≤ x∗m k − k ≤ ε/2,
for all m ∈ N.
Using (3.1) we get cx − cx ≤ ε/2. Accordingly, d − d ≤ cx − cx + k − k ≤ ε/2 + ε/2, and so the diameter of Dj is indeed no larger than ε.
In the main result, we have focused our attention on the σ-algebra of Borel subsets of [0, 1]. Now we are going to extend it to other measurable spaces. Remember that a σ-algebra is called of type (NA) if it admits an atomless probability. For the proof of the following result we use ideas ideas similar to those in [4].
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Theorem 3.2. Let X be a Banach space. If there exists a measurable space (Δ, N ) such that M0 (N , X) = M(N , X) then, for every σ-algebra Σ of type (NA), the space M0 (Σ, X) is not complemented in M(Σ, X). In particular, given any measurable space (Ω, Σ), we have that M0 (Σ, X) is complemented in M(Σ, X) if and only if M(Σ, X) = M0 (Σ, X). Proof. We merely sketch the idea of the proof. Observe that the second part follows from the first part since M(Σ, X) = M0 (Σ, X) implies that Σ is of type (NA); see Remark 2.7. Suppose now that M0 (N , X) = M(N , X) for a certain measurable space (Δ, N ). This is the same hypothesis as in Proposition 2.6. This proposition allows us to continue the proof we have already given, both in the separable and the non separable case, that M0λ (X) is not complemented in Mλ (X). So, we have Kw∗ ,w L∞ , X is not complemented in Lw∗ ,w L∞ , X . (3.4) If σ is an atomless probability measure defined on (Ω, Σ), then there exists a countably generated σ-algebra Σ0 ⊂ Σ such that σ is still atomless on Σ0 (consider the σ-algebra generated by a Rademacher-like sequence defined on (Ω, Σ, σ)). Let us denote by σ0 the restriction σ|Σ0 . Then the measure space (Σ0 , σ0 ) is isomorphic to (B, λ) [12, 41.C]. This yields, using (3.4), that Kw∗ ,w L∞ (σ0 ), X is not complemented in Lw∗ ,w L∞ (σ0 ), X . (3.5) Observe that L∞ (σ0 ) can be considered as a weak* closed subspace of L∞ (σ) which is complemented by a weak* to weak* continuous projection: namely, the conditional expectation. ∞ ∗ Thus, it is not difficult to see that ∞ Lw ,w L (σ0 ), X can be embedded as sends a complemented subspace of Lw∗ ,w L (σ), X and that this embedding ∞ ∞ Kw∗ ,w L (σ0 ), X to a complemented subspace of Kw∗ ,w L (σ), X . ∞ ∗ ,w L (σ), X is not compleTherefore, using (3.5), we deduce that K w mented in Lw∗ ,w L∞ (σ), X , or equivalently, that M0σ (Σ, X) is not complemented in Mσ (Σ, X). Applying the same reasoning as in Remark 1.2, it follows that M0 (Σ, X) is not complemented in M(Σ, X). Acknowledgment I wish to thank to the referee for valuable suggestions improving the paper.
References [1] F. Albiac and N. Kalton, Topics in Banach space theory. Graduate Texts in Mathematics 233, Springer, New York, 2006. [2] J. Diestel and J.J. Uhl, Vector Measures. Surveys Amer. Math. Soc. 15, 1977. [3] L. Drewnowski, When does ca(Σ, X) contain a copy of l∞ or c0 ?, Proc. Amer. Math. Soc. 109, (1990), 747–752. [4] L. Drewnowski, Another note on copies of l∞ and c0 in ca(Σ, X), and the equality ca(Σ, X) = cca(Σ, X), Funct. Approx. Comment. Math. 26, (1998), 155–164.
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[5] L. Drewnowski and Z. Lipecki, On vector measures which have everywhere infinite variation or noncompact range, Dissertationes Math. (Rozprawy Mat.) 339, (1995), 39 pp. [6] G. Emmanuele, A remark on the containment of c0 in spaces of compact operators, Math. Proc. Cambridge Philos. Soc. 111, (1992), 331–335. [7] G. Emmanuele, About the position of Kw∗ (E ∗ , F ) inside Lw∗ (E ∗ , F ), Atti Sem. Mat. Fis. Univ. Modena 42, (1994), 123–133. [8] G. Emmanuele and K. John, Uncomplementability of spaces of compact operators in larger spaces of operators, Czechoslovak Math. J. 47(122), (1997), 19–32. [9] M. Feder, On the nonexistence of a projection onto the space of compact operators, Canad. Math. Bull. 25, (1982), 78–81. [10] I. Ghenciu, Complemented spaces of operators, Proc. Amer. Math. Soc. 133, (2005), 2621–2623. [11] M. Girardi and W.B. Johnson, Universal non-completely-continuous operators, Israel J. Math. 99, (1997), 207–219. [12] P.R. Halmos, Measure Theory. Graduate texts in Mathematics 19, Springer-Verlag, New York, 1974. [13] L. Janicka and N. Kalton, Vector measures of infinite variation, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 25, (1977), 239–241. [14] N. Kalton, Spaces of compact operators, Math. Ann. 208, (1974), 267–278. [15] R.S. Phillips, On linear transformations, Trans. Amer. Math. Soc. 48, (1940), 516– 541. [16] A. Pietsch, Eigenvalues and s-numbers. Cambridge Studies in Advanced Mathematics 13, Cambridge University Press, Cambridge, 1987. [17] H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp (μ) to Lr (ν), J. Functional Analysis 4, (1969), 176–214. [18] A. Sobczyk, Projection of the space (m) on its subspace (c0 ), Bull. Amer. Math. Soc. 47, (1941), 938–947. [19] R.J. Whitley, Projecting m onto c0 , Amer. Math. Mon. 73, (1966), 285–286. Luis Rodr´ıguez-Piazza Facultad de Matem´ aticas Universidad de Sevilla Aptdo. de correos 1160 E-41080 Sevilla, Spain e-mail: [email protected]
Operator Theory: Advances and Applications, Vol. 201, 361–369 c 2009 Birkh¨ auser Verlag Basel/Switzerland
When is the Optimal Domain of a Positive Linear Operator a Weighted L1-space? Anton R. Schep Abstract. Let E be a reflexive Banach function space. Let T be a positive order continuous operator with values in E. Then the optimal domain [T, E] is (isomorphic to) a weighted L1 -space if and only if the operator T is an integral operator with kernel T (x, y), the adjoint operator T is a Carleman integral operator and there exists 0 ≤ g ∈ E such that φ(y) = Ty (·)E ≤ T g(y) a.e. on Y . In this case [T, E] = L1 (Y, φ dν). Mathematics Subject Classification (2000). Primary: 47B38; Secondary: 47B65, 47B34, 47G10. Keywords. Optimal domains, integral operators, Carleman operator, weighted L1 -spaces.
1. Introduction Let (Y, ν) and (X, μ) be σ-finite measure spaces and let L be an ideal of measurable functions on Y , i.e., a linear subspace of L0 (Y, ν) such that if f ∈ L and |g| ≤ |f | in L0 , then g ∈ L. Let E be a Banach function space on X (for terminology involving Banach function spaces we follow [14]). For a positive order continuous operator T : L → E one can define the optimal (or maximal) domain [T, E] for which T |f | ∈ E. This optimal domain can again be made into a Banach function space. In many papers (see, e.g., [4], [5], [9], [10]) this optimal domain was related to a space L1 (λ) of a vector measure λ. Curbera had already raised in [2] the question when L1 of a vector measure is an AL-space, so that in many of these papers this question was discussed for the operators at hand. A close reading of these papers revealed that in case the optimal domain was a weighted L1 -space that the weight was in every case equal to T (·, y)E , where T (x, y) was the kernel of the integral operator T . This lead us to consider whether this would always be the case, and in Theorem 3.3 we actually show this. In fact, in Theorem 3.3 we This work was generously supported by the Alexander von Humboldt Foundation.
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give a necessary condition on the operator T in order that the optimal domain is a weighted L1 -space, where the weight can then always be taken to be T (·, y)E . The paper is organized as follows. In Section 2 we briefly discuss the optimal domain for operators with values in L0 . Here our results are closely related to the paper [8]. The main difference is that we will deal with arbitrary positive order continuous operators, while in [8] only integral operators were considered. In Section 3 we then consider the optimal domain for positive order continuous operators with values in a Banach function space. Here the main result is Theorem 3.3, which states if E is a reflexive Banach function space and T is positive order continuous operator with values in E, then the optimal domain [T, E] is a weighted L1 -space if and only if the operator T is an integral operator with kernel T (x, y), the adjoint operator T is a Carleman integral operator and there exists 0 ≤ g ∈ E such that φ(y) = Ty (·)E ≤ T g(y) a.e. on Y . Moreover [T, E] = L1 (Y, φ dν) in this situation. In case T is already given to be an integral operator we can weaken the hypothesis on E to the condition that E has order continuous norm and the Fatou property. As [T, E] inherits a norm from T and E, we can also ask the question when [T, E] is isometric to a weighted L1 -space with respect to this norm. We will show that for E with strictly convex norm this happens if and only if T is a rank one operator. Finally we show then how our results can be used to give short proofs for many of the examples occurring in the above-mentioned literature.
2. The L0 case In this section we briefly deal with the case L0 . We use L0 (X, μ) to denote the space of a.e. finite measurable functions on X and let M (X, μ) denote the space of extended real-valued measurable functions on X. Assume that T is defined on an ideal L of measurable functions. By L+ we denote the collection of nonnegative functions in L. A positive linear operator T : L → L0 (X, μ) is called order continuous if 0 ≤ fn ↑ f a.e. and fn , f ∈ L imply that T fn ↑ T f a.e. We first recall, see [7], that such operators have “adjoints”. Theorem 2.1. Let L be an ideal of measurable functions on (Y, ν) and let T be a positive order continuous operator from L into L0 (X, μ). Then there exists an operator T t : L0 (X, μ)+ → M (Y, ν)+ such that for all f ∈ L+ and all g ∈ L0 (X, μ)+ we have f (T t g)dν.
(T f )gdμ = X
Y
We denote by T the positive linear operator obtained by restricting T t to f for which T t (|f |) < ∞ a.e. Example. Let T (x, y) ≥ 0 be μ × ν-measurable function on X × Y . Let L = {f ∈ L0 (Y, ν) such that T (x, y)|f (y)|dν(y) < ∞ a.e.} and define T as the integral operator T f (x) = Y T (x, y)f (y)dν(y) on L. Then one can check (using Tonelli’s theorem) that the operator T as defined above is the integral operator
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T (x, y)g(x)dμ(x). Therefore we will use T (y, x) = T (x, y) to denote the kernel of T . X
Let now T : L → L0 (X, μ) be a positive order continuous linear operator. As T is order continuous the absolute kernel {f ∈ [T, L0 ] : T (|f |) = 0} is a band in L0 (Y, ν), so that by removing the “support” of this band from Y , we can assume that the absolute kernel of T equals {0}, i.e., that T is strictly positive on L. From now on we will make this assumption on ideals L. There exists a largest order ideal [T, L0 ] containing L to which T can be extended as a positive order continuous linear operator. Here f ∈ [T, L0 ] if there exist fn ∈ L, 0 ≤ fn ↑ |f | such that sup T fn < ∞ a.e. and then T is extended to [T, L0 ] by T f = sup T fn for positive f satisfying this condition. We shall denote this extension again by T . One can verify that T : [T, L0 ] → L0 is again order continuous. Theorem 2.2. Let L be an ideal of measurable functions on (Y, ν) and let T be a positive order continuous operator from L into L0 (X, μ). Then [T, L0 ] contains a strictly positive function. Proof. There exist disjoint fn in L+ (Y, ν) such that ∪n supp(fn ) = Y (recall that we assume that the support of L is Y ). Now find k n such that μ(x ∈ X : T fn (x) > kn 2−n ) < 2−n . Now define gn = k1n fn and g = n gn . Clearly g(y) > 0 for all y ∈ Y , so it remains to show that g ∈ [T, L0 ]. Let > 0 and then choose N such that 2−N < . Then μ{x ∈ X : T gn (x) > 2−n } < 2−n for all n ≥ 1 implies that for all n ≥ N + 1 we have that $ % $ ∞ % ∞ ) −n μ x∈X: T gn (x) > 1 ≤ μ {x ∈ X : T gn (x) > 2 } n=N +1
≤ ∞
n=N +1 ∞ −n
2
= 2−N < .
n=N +1
Hence μ{x X : n=1 T gn (x) = ∞} < . As this holds for all > 0 we get that ∈ ∞ T g(x) = n=1 T gn (x) < ∞ a.e., so g ∈ [T, L0 ]. We now indicate when [T, L0 ] is a weighted L1 -space. Theorem 2.3. Let L be an ideal of measurable functions on (Y, ν) and let T be a positive order continuous operator from L into L0 (X, μ). Then [T, L0 ] is a weighted L1 -space if and only if there exists a strictly positive g ∈ L0 such that the range T ([T, L0]) of T is contained in L1 (X, gdμ). In this case [T, L0 ] = L1 (Y, T g dν). Proof. Assume first that there exists a strictly positive g ∈ L0 such that the range T ([T, L0]) of T is contained in L1 (X, gdμ). Then 0 ≤ T : [T, L0 ] → L1 (X, gdμ). As T is order continuous the absolute kernel {f ∈ [T, L0 ] : T (|f |) = 0} is a band in L0 (Y, ν), so that by removing the “support” of this band from Y , we can assume that the absolute {0}, i.e., that T is strictly positive on [T, L0 ]. kernel of T equals Then |f | = X T (|f |)g dμ = Y |f |T g dν defines an AL-norm on [T, L0 ] and the
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maximality of [T, L0 ] implies that [T, L0 ] is complete with respect to this norm. Now it is clear that [T, L0 ] = L1 (Y, T g dν). Conversely, if [T, L0 ] is a weighted L1 -space, then T maps the positive part, B + , of the unit ball of this weighted L1 space to a convex set bounded in measure. By the Maurey-Nikisin theorem, see, e.g., [7] or [13], there exist a strictly positive g ∈ L0 such that g1 T (B + ) is contained in the unit ball of L1 (X, μ). Hence the range of T is contained in L1 (X, gdμ). Remark 2.4. Note the above g, or corresponding weighted L1 -space, is not unique. In fact, if 0 / h ≤ g and the range T ([T, L0]) of T is contained in L1 (X, gdμ), then the range T ([T, L0]) of T is also contained in L1 (X, hdμ). This implies that L1 (X, T gdν) = L1 (X, T hdν) in this case, which requires that T g and T h satisfy T h ≤ T g ≤ cT h for some constant c. An example of an operator T for which the range T ([T, L0]) of T is contained in some L1 (X, gdμ) is given by T f (x) = 1 xf (x)+ 0 f (x) dx. One can easily verify that T g and T h satisfy T h ≤ T g ≤ cT h for some constant c if we choose g identically equal to one and 0 / h ≤ g as above. Corollary 2.5. Let L be an ideal of measurable functions on (Y, ν) and let T be a positive order continuous operator from L into L0 (X, μ). Then [T, L0 ] is a weighted L1 -space if and only if [T, L0 ] is a Banach function space. Remark 2.6. The above corollary was proved for positive integral operators by Labuda and Szeptycki in [8].
3. The Banach function space case Let E denote a Banach function space on (X, μ) with the Fatou property, i.e., if 0 ≤ fn ↑ in E with sup fn E < ∞, then f = sup fn ∈ E and f E = sup fn E . Let L be an ideal of measurable functions on (Y, ν) and let T be a positive order continuous operator from L into E. By removing a measurable set from Y we can and will assume that T is strictly positive. Then the optimal domain [T, E] is a Banach function space with the Fatou property with respect to the norm f = T (|f |)E . The following theorem shows that the maximality of the optimal domain [T, E] is reflected by minimality of the associate space [T, E] with respect to the range of T . The proof of this theorem is modelled after a proof of G. Bennett, who considered in [1] the special case E = p and counting measures. Theorem 3.1. Let T and E be as above and assume that E has order continuous norm. Then [T, E] equals the ideal generated by T (E ) in L0 . Proof. It suffices to prove that for all 0 ≤ h0 ∈ [T, E] there exists g0 ∈ E such that h0 ≤ T g0 . Let 0 ≤ h0 ∈ [T, E] and assume h0 = 0. Let V = {g ∈ E : g = T (f ), f h0 dν = 1, 0 ≤ f ∈ [T, E]}. Then V = ∅ and V is convex. Let g ∈ V and let 0 ≤ f ∈ [T, L0 ] such that T (f ) = g and f h0 dν = 1. Then 1 = f h0 dν ≤ f [T,E] h0 [T,E] = T (f )E h0 [T,E] = gE h0 [T,E] .
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−1 Hence gE ≥ h0 −1 [T,E] for all g ∈ V . Let W = {g ∈ E : gE < h0 [T,E] }. Then W is an open ball such that V ∩ W = ∅. By the Hahn-Banach theorem there exists 0 = g0 ∈ E and C such that hg0 dμ < C ≤ gg0 dμ X
X
for all h ∈ W and all g ∈ V . Since W is a ball, it follows that C > 0. By absorbing C into g0 we can assume then that C = 1. Then we have 1≤ T (f )g0 dμ = f T (g0 ) dν X
Y
for all 0 ≤ f ∈ [T, E] such that Y f h0 dν = 1. By homogeneity this implies that f h0 dν ≤ f T (g0 ) dν Y
Y
for all 0 ≤ f ∈ [T, E]. This implies immediately that h0 ≤ T g0 , which completes the proof of the theorem. To characterize the operators for which [T, L0 ] equals a weighted L1 -space, we need the notion of a Carleman integral operator. Recall, see [12], that a positive linear operator T from a Banach function space E into L0 is called a Carleman operator, if there exists a μ × ν-measurable T (x, y) ≥ 0 on X × Y such that 1. T f (x) = Y T (x, y)f (y) dν(y) a.e. for all f ∈ E, i.e., T is an integral operator on E with kernel T (x, y), and 2. for almost every x ∈ X we have Tx (y) = T (x, y) ∈ E , i.e., Tx (·)E < ∞. We recall from [12] the following result. Theorem 3.2. Let E denote a Banach function space on (X, μ) with order continuous norm and let T : E → L0 be a linear operator. Then the following are equivalent. 1. T is a Carleman operator. 2. There exists 0 ≤ g ∈ L0 such that |T (f )(x)| ≤ f E g(x) a.e. on X. Moreover Tx (·)E = inf{g ∈ L0 : |T (f )(x)| ≤ f E g(x)a.e.} We remark that the above theorem still holds without the order continuity hypothesis, if we assume that T is already an integral operator. Recall from [14] that a Banach function space E is reflexive if and only if E has the Fatou property and the norms on E and E are order continuous. For this reason we list reflexivity instead of order continuity properties as a hypothesis in the following theorems. Theorem 3.3. Let L be an ideal of measurable functions on (Y, ν), let E denote a reflexive Banach function space on (X, μ) and let T be a positive order continuous operator from L into E. Then the following are equivalent. 1. The optimal domain [T, E] is a weighted L1 -space.
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2. The operator T is an integral operator with kernel T (x, y), the adjoint operator T is a Carleman integral operator and there exists 0 ≤ g ∈ E such that φ(y) = Ty (·)E ≤ T g(y) a.e. on Y . Moreover in either case [T, E] = L1 (Y, φ dν). Proof. Assume first that the optimal domain [T, E] is a weighted L1 -space, say L1 (Y, ψ dν). Then T : [T, E] → E has an adjoint T : E → Aψ , where Aψ = {g : |g| ≤ Cψ for some constant C} denotes the order ideal generated by ψ. It follows now from Theorem 3.2 that T is a Carleman operator, which implies by Tonelli’s theorem that T is an integral operator. The fact that T is bounded implies that |T (g)(y)| ≤ Cψ(y)gE for all g ∈ E and some constant C (the norm of T ). This implies by the moreover part of Theorem 3.2 that φ(y) = Ty (·)E ≤ Cψ(y) a.e. on Y , i.e., φ ∈ Aψ . Now by Theorem 3.1 there exists g ∈ E such that φ ≤ T g. This completes the proof in one direction. Assume now that (2) holds. Then the inequality |T h| ≤ φhE implies that the range of T is contained in the ideal Aφ and thus by Theorem 3.1 we have that [T, E] ⊂ Aφ . On the other hand the condition φ(y) ≤ T g for some g ∈ E implies that Aφ ⊂ [T, E] . Hence [T, E] = Aφ , which implies that [T, E] = [T, E] = Aφ = L1 (Y, φ dν) and the proof of the theorem is complete. With the same proof we can weaken the hypothesis that E is reflexive, if we assume that T is already an integral operator (by the remark following Theorem 3.2). Theorem 3.4. Let L be an ideal of measurable functions on (Y, ν), let E denote an order continuous Banach function space on (X, μ) with the Fatou property and let T be a positive integral operator from L into E. Then the following are equivalent. 1. The optimal domain [T, E] is a weighted L1 -space. 2. The adjoint operator T is a Carleman integral operator and there exists 0 ≤ g ∈ E such that φ(y) = Ty (·)E ≤ T g a.e. on Y . Moreover in either case [T, E] = L1 (Y, φ dν). Remark 3.5. If in the above theorem we have in part (2) only that T is a Carleman operator, then we still have that L1 (Y, φ dν) ⊂ [T, E], but the inclusion will be strict if the other condition of (2) fails. The following proposition is elementary, but we want to record the statement. Proposition 3.6. Let L be an ideal of measurable functions on (Y, ν) and let T be a strictly positive operator from a non-trivial ideal L into L1 (X, μ). Then [T, L1 (X, μ)] = L1 (Y, T 1dν). We will now indicate briefly the question when [T, E] with the norm f = T (|f |)E is isometric to a weighted L1 -space. In case E = L1 , then it is clear from the proof of the above proposition that [T, L1 (X, μ)] is isometric to L1 (Y, T 1dν) for any positive linear operator T . Therefore we need an additional hypothesis on
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E to rule out this situation. Recall that a Banach space is called strictly convex, if x, y ∈ E with x = y and x = y = 1 implies x + y < 2. Proposition 3.7. Let E be a strictly convex Banach function space, let L be an ideal of measurable functions on (Y, ν) and let T be a positive order continuous operator from L into E. Then the optimal domain [T, E] is isometric to a weighted L1 -space if and only if T is a rank one operator. Proof. If T = φ ⊗ ψ, where we can assume that ψE = 1, then it is clear that T f E = f L1(Y,φdν) , so [T, E] is isometric to a weighted L1 -space. Conversely, assume [T, E] is isometric to a weighted L1 -space. Let 0 < f0 ∈ [T, E] 1 with T f0 E = 1. Let 0 ≤ g ∈ [T, E] with T g = 0. Let g0 = T g g. The E T f0 + T g0 E = T f0E + T g0 E = 2 implies that T f0 = T g0 , i.e., T g = T gE T f0 . Hence T has rank one.
4. Examples We now present as application some examples from the literature. Example (1). Let 1 < p ≤ ∞ and T : Lp ([0, 1]) → Lp ([0, 1]) denote the Volterra opx 1 erator T f (x) = 0 f (y) dy. Then T g(y) = y f (x) dx, so that φ(y) = Ty (x)p = 1
(1 − y) p . This shows that T is a Carleman operator. On the other hand we will show that there exists no g ∈ Lp such that φ ≤ T g, so that [T, Lp ] is not a weighted L1 -space, a result which was shown by W. Ricker in [11]. Assume that φ ≤ T g for some g ∈ Lp . Then 1 p1 1 p1 1 1 p g(x)dx ≤ |g| dx 1 dx . (1 − y) p ≤ 1
y
y
y
This implies that 1 ≤ y |g(x)|p dx → 0 as y ↑ 1, which is a contradiction. Note that if E is a rearrangement invariant Banach function space on [0, 1], then by interpolation T : E → E. Assume in addition that E is reflexive, then by the same argument we have that [T, E] is not a weighted L1 -space. On the other hand for p = 1 we get that [T, L1 ] = L1 ([0, 1], (1 − y) dy). This follows immediately from the above corollary, since now T 1(y) = (1 −y). Curbera in [3] considered Volterra x convolution operators of the form T f (x) = 0 φ(x − y)f (y) dy, where φ is strictly positive, non-decreasing integrable function. By the same methods as above one can show that in this case again [T, Lp ] is not a weighted L1 -space for 1 < p < ∞. Curbera showed in [3] that [T, Lp ] is a real interpolation space of weighted L1 spaces, but even in case of the ordinary Volterra operator that does not seem to give an explicit description of [T, Lp ]. The following example uses similar arguments. Example (2). Let 1 < p < ∞ and T : Lp ([0, ∞)) → Lp ([0, ∞)) denote the x ∞ Hardy operator T f (x) = x1 0 f (y) dy. Then T g(y) = y f (x) x dx, so that φ(y) =
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Ty (x)p =
1 1 p−1 y p−1
p1
. Hence T is again a Carleman integral operator. On the
other hand we will show that there exists no g ∈ Lp such that φ ≤ T g, so that again [T, Lp ] is not a weighted L1 -space, a special case of a result of O. Delgado and J. Soria in [6]. Assume that φ ≤ T g for some g ∈ Lp . Then
1 1 p−1 p−1y
p1
≤ y
∞
g(x) dx ≤ x
y
∞
1 p |g| dx p
y
∞
p1 1 dx . xp
∞ Hence 1 ≤ y |g|p dx → 0 as y → ∞, which is a contradiction. Along the same lines on can prove that if E is a reflexive Banach function space and F is a Banach function space such that T : F → E, then [T, E] is not a weighted L1 -space. To see this, note that if we have φ(y) = Ty (x)E ≤
y
∞
g(x) dx ≤ gχ(y,∞) E φ(y), x
so that either φ(y) = ∞ on a set of positive measure, or 1 ≤ gχ(y,∞)E a.e. which leads to contradiction as · E is order continuous. Example (3). Let G denote a (Hausdorff) compact Abelian group and let λ denote the normalized Haar measure on G. Let 1 < p < ∞ and 0 ≤ μ ∈ M (G) a bounded positive non-zero Borel measure on G. Then Tμ (f ) = f ∗ μ defines a positive order continuous operator from Lp (G, λ) into Lp (G, λ). Now it is easy to see that T is a Carleman operator on Lp (G, λ) if and only if μ = g dλ for some 0 ≤ g ∈ Lp . Moreover φ(y) = g(· − y)p = gp . Moreover φ(y) ≤ T h holds trivially for g h = g p1 1. Hence [Tμ , Lp ] is a weighted L1 -space if and only if μ = g λ for some 0 ≤ g ∈ Lp . Moreover in this case [Tμ , Lp ] = L1 (G, λ). This recovers results of Okada and Ricker from [9] and [10]. Note that essentially the same arguments extend to the case where we replace Lp (G) by a rearrangement invariant reflexive Banach function space on G. Example (4). Let now G denote a (Hausdorff) locally compact, non-compact Abelian group and let λ denote Haar measure on G (which is unique up to a positive constant). Let 1 < p < ∞ and let 0 ≤ g ∈ L1 (G). Then Tg (f ) = f ∗ g defines a positive operator from Lp (G, λ) into Lp (G, λ). We will show that [Tg , Lp ] is never equal to a weighted L1 -space. To see this note first that T is a Carleman operator if and only if φ(y) = g(· − y)p = gp < ∞. Therefore assume now that g ∈ L1 ∩ Lp . Denoting with g˜ the function g˜(x) = g(−x), we have that g ∗ h)(y) g˜ ∗ h ∈ C0 (G) for all h ∈ Lp . Assume now that φ(y) = gp ≤ T h(y) = (˜ p for some h ∈ L . Then the constant function is in C0 (G) which is a contradiction, as G is non-compact. Hence [Tg , Lp ] is never equal to a weighted L1 -space.
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References [1] Grahame Bennett, Factorizing the classical inequalities, Mem. Amer. Math. Soc. 120 (1996), no. 576, viii+130. [2] Guillermo P. Curbera, When L1 of a vector measure is an AL-space, Pacific J. Math. 162 (1994), no. 2, 287–303. [3] , Volterra convolution operators with values in rearrangement invariant spaces, J. London Math. Soc. (2) 60 (1999), no. 1, 258–268. [4] Guillermo P. Curbera and Werner J. Ricker, Optimal domains for kernel operators via interpolation, Math. Nachr. 244 (2002), 47–63. [5] , Banach lattices with the Fatou property and optimal domains of kernel operators, Indag. Math. (N.S.) 17 (2006), no. 2, 187–204. [6] Olvido Delgado and Javier Soria, Optimal domain for the Hardy operator, J. Funct. Anal. 244 (2007), no. 1, 119–133. [7] Ralph Howard and Anton R. Schep, Norms of positive operators on Lp -spaces, Proc. Amer. Math. Soc. 109 (1990), no. 1, 135–146. [8] Iwo Labuda and Pawel Szeptycki, Domains of integral operators, Studia Math. 111 (1994), no. 1, 53–68. [9] S. Okada and W.J. Ricker, Optimal domains and integral representations of convolution operators in Lp (G), Integral Equations Operator Theory 48 (2004), no. 4, 525–546. [10] , Optimal domains and integral representations of Lp (G)-valued convolution operators via measures, Math. Nachr. 280 (2007), no. 4, 423–436. [11] W.J. Ricker, Compactness properties of extended Volterra operators in Lp ([0, 1]) for 1 ≤ p ≤ ∞, Arch. Math. (Basel) 66 (1996), no. 2, 132–140. [12] A.R. Schep, Generalized Carleman operators, Indagationes Mathematicae (Proceedings) 83 (1980), no. 1, 49–59. [13] , Convex solid subsets of L0 (X, μ), Positivity 9 (2005), no. 3, 491–499. [14] A.C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, NorthHolland Publishing Co., Amsterdam, 1983. Anton R. Schep Department of Mathematics University of South Carolina Columbia, SC 29212, USA e-mail: [email protected] URL: http://www.math.sc.edu/~schep/
Operator Theory: Advances and Applications, Vol. 201, 371–380 c 2009 Birkh¨ auser Verlag Basel/Switzerland
Liapounoff Convexity-type Theorems Rudolf G. Venter Abstract. We investigate Liapounoff convexity-type theorems for the case of Fr´echet space-valued measures defined on Fields of sets. We extend theorems by I. Kluv´ anek [10], J.J. Uhl [21] and S. Ohba [15] to this case. Results relating bounded finitely additive measures with strongly additive measures is also of interest. Mathematics Subject Classification (2000). 46G10; 46A04. Keywords. Vector measure; Liapounoff convexity theorem; strong continuity; Fr´echet spaces.
1. Introduction A. Liapounoff [12] showed that if m is a σ-additive measure defined on a σ-field, Σ, taking values in a finite-dimensional vector space, E1 , we say m ∈ ca(Σ, E1 ), then the range of m denoted by Rm is compact and if m is non-atomic then Rm is convex, see also J. Lindenstrauss [13]. Various well-known related theorems for infinite-dimensional vector spaces exists. Some of these theorems are listed below in terms of Fr´echet space-valued measures. Let F be a Fr´echet space and m ∈ ca(Σ, F ) a non-atomic measure: • (I. Kluv´ anek [8, Theorem 1, Corollary 3.1]). The weak closure of Rm coincides with the closure of co(Rm). • (S. Ohba [15] see also I. Kluv´ anek and G. Knowles [11, Theorem IV.6.1] and [18]). If Rm is relatively compact then the closure of Rm is convex. • (J.J. Uhl [21] generalized by S. Ohba [15]). If F has the Radon-Nikod´ ym property and if m is also of bounded variation then the closure of Rm is compact and convex. In this paper, these theorems are investigated for the case of finitely additive, bounded finitely additive and strongly additive Fr´echet space-valued measures defined on fields (of sets) and fields of sets with the interpolation property (I) of G.L. Seever [19].
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In Section 3, we show that non of the above-mentioned theorems will hold if the σ-field is replaced by a field. Here a property stronger than non-atomicity must be considered. A. Sobczyk and P.C. Hammer [20] utilized the concept of “continuous” set function. To avoid confusion, we call this concept strongly continuous as done in [2]. In Section 4, we investigate the relationship between non-atomicity, strong continuity and Darboux properties for the case of non-negative finite measures defined on a fields of sets and fields of sets with property (I). The strong continuity property is introduced for the case of Fr´echet space-valued measures in Section 5. Finally in Section 6 we give the mentioned Liapounoff theorems and discuss conditions under which bounded finitely additive measures are strongly additive.
2. Definitions and notation A field F of subsets of a set Ω has the interpolation property (I) if and only if for any two sequences (An ) and (Bn ) in F satisfying the condition that An ⊆ Bm for all n, m there exist a set C ∈ F such that An ⊆ C ⊆ Bm . Let ca+ (F) (resp. ba+ (F)), indicate the space of real-valued non-negative bounded σ-additive (resp. finitely additive) measures. The notion measure is used for any set function. In this paper F denotes a Fr´echet space (locally convex, metrizable by a complete metric) over C topologized by an increasing sequence of seminorms, denoted by P . Let ca(F, F ) (resp. f a(F, F )), indicate the space of all F -valued σ-additive (resp. finitely additive) measures defined on F. All of the bounded elements in f a(F, F ) is denoted by ba(F, F ). Let sa(F, F ) indicate the space of all strongly additive measures, that is, all F -valued measures m defined on F, with the prop erty that ∞ m(A n ) converges and the sum belongs to F , for any collection of n=1 pairwise disjoint elements {An }∞ n=1 ⊂ F. For any x ∈ F , let m, x : F → C denote the complex measure A → m(A), x for all A ∈ F. Let wca(F, F ) indicate the space of all weakly σ-additive measures, that is all measures m : F → F with the property that m, x ∈ ca(F, C). If m ∈ sa(F, F ) and is also weakly σ-additive then m has a unique extension to a σ-additive F -valued measure on σ(F), see [8]. Let p ∈ P , the p-semivariation of a measure m : F → F is denoted by the function p(m) : F → [0, ∞) defined by p(m)(A) = sup{| m, x |(A) : x ∈ U ◦ } for all A ∈ F and where U ◦ denotes the polar of U = {x ∈ F : p(x) ≤ 1} and | m, x | denotes the total variation of m, x . Let m : F → F be a vector measure and μ ∈ ba+ (F). The following notions take the place of “absolute continuity” for measures defined on a field: m is μ-null if m(A) = 0 whenever μ(A) = 0; μ is m-null if μ(A) = 0 whenever m(A) = 0; μ is m-continuous if μ(A) → 0 whenever p(m)(A) → 0 for all p ∈ P (equiv. m(A) → 0 in F ) and m is μ-continuous if p(m)(A) → 0 for all p ∈ P whenever μ(A) → 0.
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Vector measure m and μ ∈ ba+ (F) are said to be equivalent if m is μ-continuous and μ is m-continuous. There exists a Boolean isomorphism i from F onto F1 the field of all clopen sets of a totally disconnected compact Hausdorff space Ω1 . Under a Boolean isomorphism unions, intersections and complements are continuous. There exists an isomorphism denoted by B from sa(F, F ) onto ca(σ(F1 ), F ) where for each m ∈ sa(F, F ), the vector measure Bm(iE) = m(E) for all E ∈ F. We call the triple [Ω1 , σ(F1 ), Bm] the Stone Representation of [Ω, F, m]. See for instance [6, Section I.12] and [4, Theorem I.5.7]. Let Σ be a σ-field of subsets of Ω. For every m ∈ ca(Σ, F ) there exists a μ ∈ ca+ (Σ) such that m and μ are equivalent, see [11, Corollary II.1.2]. As a result of the Stone Representation theorem there exists a μ ∈ ba+ (F) for every m ∈ sa(F, F ) such that m and μ are equivalent. The proof is along the lines of [4, Corollary I.5.3]. The range of a measure m over a set A ∈ F is denoted by (Rm)(A) := {m(B) : B ⊂ A, B ∈ F} and let Rm := (Rm)(Ω). The collection of all finite partitions of Ω is denoted by P(Ω). For any p ∈ P let Πp : F → Fp := F/p−1 (0) be the canonical quotient map, i.e., Πp (x) = x + p−1 (0) for all x ∈ F . If we let Πp (x)p := p(x) for all @p indicate the Banach space x ∈ F . Then · p is a well-defined norm on Fp . Let F completion of Fp with respect to · p . The norm on Fp is also denoted by · p .
3. Example The following example is la raison d’ˆetre for the structures studied in this paper. This example shows that the classical Liapounoff Convexity theorem and the mentioned theorems by I. Kluv´ anek, J.J. Uhl and S. Ohba can’t be extended to the case of a non-atomic vector measure on a field. In fact these theorems can’t even be extended to a non-atomic σ-additive vector measure of bounded variation on a field. Let Ω = [0, 1] and let F be the field generated by all sets of the form [a, b) where a < b and are rational numbers in Ω. Let α be any number in Ω. It is important to note that, since a σ-field isn’t under consideration, {α} ∈ / F. Let μ be the “indicator” measure on F for the point α, i.e., for any set A ∈ F if α ∈ A then μ(A) = 1 otherwise μ(A) = 0. Clearly, μ is an atomic measure. Let λ be the restriction of the Lebesgue measure on Ω to F. The non-negative measure λ is non-atomic, since for every set A ∈ F such that λ(A) > 0 there exists a subset B of A in F such that 0 < λ(B) < λ(A). The vector measure m : F → R2 defined by 3 2 λ m= μ
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is σ-additive since λ and μ are both σ-additive measures on F. For any π ∈ P(Ω), only one set in π, say set A, can contain the point α. Under the sup-norm of R2 m(E)∞ = m(A)∞ + m(E)∞ < 2 E∈π
E∈π,A=E
Hence, the measure m is of bounded variation and thus also strongly additive, see [4, Proposition I.1.9]. Now, since λ is non-atomic, m is also non-atomic. It is obvious that Rm is neither compact, nor convex. Since the rational numbers are dense in the real numbers the closure of Rm denoted by Rm is compact but non-convex. Let m ˜ denote the extension of m to σ(F). Since Rm is dense in Rm, ˜ it’s worth studying the relationship between m and m, ˜ specifically the non-atomicity relationship. Although m on F is non-atomic, m ˜ on σ(F) is atomic, since 3 2 0 = 0 m({α}) ˜ = 1 but {α} does not contain any non-empty subset. We call {α} an embedded atom of F in terms of m. That is, an embedded atom of a field F in terms of a vector measure m is a set in σ(F) which is an atom of m, ˜ the extension of m to σ(F).
4. Non-negative scalar measures We consider the relationships between the following properties of scalar measures: Definition 4.1. Let F be a field of subset of a set Ω and μ ∈ ba+ (F), then (i) μ is non-atomic if for any A ∈ F such that μ(A) = 0, there exists a B ⊂ A in F such that μ(B) = 0 and μ(B) = μ(A); (ii) μ is strongly continuous if for every > 0 there exists a π ∈ P(Ω) ⊂ F such that μ(E) < for every E ∈ π, i.e., inf π maxE∈π μ(E) = 0 where the infimum is taken over all π ∈ P(Ω); (iii) μ has the Darboux property if for any A ∈ F and β ∈ (0, μ(A)) there exist a set B ⊂ A in F such that μ(B) = β. If μ is non-negative then it is trivial to show that (iii)⇒(ii)⇒(i). For a full treatment of these concepts in the setting of non-zero scalar measures, see [2]. N. Dinculeanu [5, p. 26] showed that if a σ-additive non-negative measure on a δ-ring is non-atomic, then it has the Darboux property. Here we use a construction from this proof to prove this result for the case of a field with the interpolation property (I). Theorem 4.2. Let F be a field of subsets of some set Ω and μ ∈ ca+ (σ(F)). Then the following statements are equivalent: (i) μ has the Darboux property on σ(F); (ii) μ is strongly continuous on σ(F);
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(iii) μ is non-atomic on σ(F); (iv) μ|F is strongly continuous on F; If F has property (I) these results are also equivalent to (v) μ|F has the Darboux property on F; (vi) μ|F is non-atomic on F. Proof. The following are easy to show: (i)⇒(ii)⇒(iii); (v)⇒(iv); (iv)⇒(vi). (ii)⇔(iv) is from [2, Proposition 5.3.7]. We now prove that (vi)⇒(v) and since a σ-field has the interpolation property it also follows that (iii)⇒(i). Let E ∈ F be of positive measure and α ∈ (0, μ(E)). Dinculeanu [5, Theorem I.2.7] constructed sequences (An ) and (Bm ) in F with the following properties: (Note that this construction only depends on the non-atomiticity and finite additivity of μ.) (a) A0 ⊂ A1 ⊂ A2 ⊂ · · · ⊂ B2 ⊂ B1 ⊂ E (b) If we put an = sup{μ(A) : An−1 ⊂ A ⊂ Bn−1 , μ(A) ≤ α} and bn = sup{μ(B) : An ⊂ B ⊂ Bn−1 , μ(B) ≥ α} then the sequence (an ) is monotone decreasing and (bn ) is monotone increasing and we have an ≤ α ≤ bn for all n ∈ N. (c) There exist sequences (kn ) and (ln ) which tend to zero such that an − kn < μ(An ) ≤ an and bn ≤ μ(Bn ) < bn + ln . Taking the limits of sequences (an ) and (bn ) to a and b respectively, we have lim μ(An ) = a ≤ α ≤ b = lim μ(Bn )
n→∞
n→∞
From property (I) of F there exists a set C ∈ F such that A0 ⊂ A1 ⊂ A2 ⊂ · · · ⊂ C ⊂ · · · ⊂ B2 ⊂ B1 and a ≤ μ(C) ≤ b. If μ(C) ≤ α from Conditions (b) and (c) we deduce that an − kn ≤ μ(C) ≤ an+1 for every n. Consequently, μ(C) = a. We can show in a similar manner that μ(C) = b. Hence μ(C) = α. The converse is trivial.
5. Vector measures Definition 5.1. Let F be a field of subsets of a set Ω, F a Fr´echet space topologized by a countable collection of seminorms P , F the dual space of F and m : F → F an F -valued measure. Then (i) m is non-atomic, if for every A ∈ F such that m(A) = 0 there exists a B ⊂ A in F such that m(B) = 0 and m(A − B) = 0; (ii) m is strongly continuous if there exists a sequence {πn }n∈N in P(Ω) ⊂ F such that for every > 0 and every p ∈ P there exists Np ∈ N such that p(m)(E) < for every E ∈ πn and n ≥ Np , i.e., limn→∞ maxE∈πn p(m)(E) = 0;
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(ii ) m is strongly continuous if there exists a sequence {πn }n∈N in P(Ω) such that for every > 0 and every p ∈ P there exists Np ∈ N such that sup{p(x) : x ∈ (Rm)(E)} < for every E ∈ πn and n ≥ Np ; (iii) m is w-strongly continuous if m, x is strongly continuous for all x ∈ F . Condition (ii ) is an alternative, but equivalent, version of Condition (ii) since for each p ∈ P , sup{p(x) : x ∈ (Rm)(A)} ≤ p(m)(A) ≤ 4 sup{p(x) : x ∈ (Rm)(A)} for all A ∈ F, see [11, Chapter II.1]. U.K. Bandyopadhyay [1] studied a Darboux-type property of Banach spacevalued measures, that is, a Banach space valued measure m on a ring R has this property if for any set A ∈ R and α ∈ (0, 1) there exist a set B ⊂ A in R such m(B) = αm(A). This property implies the non-atomicity of m in fact it implies that the range of m over R is convex. However, unlike the scalar case, a non-atomic σ-additive Banach space valued measure defined on a σ-field need not have this Darboux property. See [21] for an example of a non-atomic Banach space valued measure on a σ-field with a nonconvex range, thus not possessing this Darboux property. It is obvious that if m is strongly continuous then | m, x | is strongly continuous for all x ∈ F . That is, the strong continuity of m, implies its w-strong continuity. A Fr´echet space has the Rybokov property if for every F -valued, σ-additive measure m there exists an x ∈ F such that m is | m, x |-continuous. A Fr´echet space F has the Rybakov property if and only if F does not contain a linear homeomorphic copy of CN , see [7] and [16]. A list of conditions implying the Rybakov property for real Fr´echet spaces is stated in [11, Theorem VI.3.1]. All Banach spaces have the Rybakov property, see [4, Chapter IX.2]. Lemma 5.2. Let F be a field of subsets of a set Ω and F a Fr´echet space. If F has the Rybakov property then the strong continuity property and the w-strong continuity property are equivalent for all strongly additive F-valued measures. Proof. We only need to show that w-strong continuity implies strong continuity. Let F be a field and m ∈ sa(F, X). Let [Ω1 , σ(F1 ), m1 ] be the Stone Representation of [Ω, F, m]. There exists an x ∈ F such that m1 is | m1 , x |-continuous. This still holds if we restrict the domain of m1 to F1 . Since m1 (iA) = m(A) for all A ∈ F it follows that m is | m, x |-continuous. Let i indicate the Boolean isomorphism from F onto F1 . Let x ∈ F be such that m is | m, x |-continuous. For every > 0 there exists a π ∈ P(Ω) such that | m, x |(A) < for every A ∈ π. The strong continuity of m follows since m is | m, x |-continuous. Lemma 5.3. Let F be a field of subsets of a set Ω and F a Fr´echet space. Let μ ∈ ba+ (F) and m ∈ sa(F, F ). Then (i) m is non-atomic if m is μ-null and μ is non-atomic; (ii) μ is non-atomic if μ is m-null and m is non-atomic;
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(iii) m is strongly continuous if m is μ-continuous and μ is strongly continuous; (iv) μ is strongly continuous if μ is m-continuous and m is strongly continuous. Theorem 5.4 is a consequence of Lemma 5.3 and Theorem 4.2. Theorem 5.4. Let F be a field of subsets of some set Ω, F a Fr´echet space and m ∈ ca(σ(F), F ). The following statements are equivalent: (i) m is strongly continuous on σ(F); (ii) m is non-atomic on σ(F); (iii) m|F is strongly continuous on F; (iv) m|F is non-atomic on F if F has property (I); (v) any of the equivalent statements of Theorem 4.2 for a scalar measure are equivalent for the vector measure m; (vi) m0 ∈ sa(F0 , F ) is strongly continuous on a field F0 of subsets of a set Ω0 , this is if [Ω, σ(F), m] is the Stone Representation of [Ω0 , F0 , m0 ]. Proof. There exists a μ ∈ ca+ (σ(F)) equivalent to m, as discussed in Section 2. The equivalence of properties (i) to (v) follows from the relationship between m and μ stated in Lemma 5.3 and the properties of μ in Theorem 4.2. We now prove that (vi)⇔(iii). For every > 0 there exists a partition π ∈ P(Ω0 ) ⊂ F0 such that p(m|F )(iA) = p(m0 )(A) < for every A ∈ F0 . It is trivial to verify that iπ := {iA : A ∈ π} is a finite partition of Ω consisting of elements of F. The converse is proved in the same way. Lemma 5.5. Let F be a field, F a Fr´echet space and m ∈ sa(F, F ). Let [Ω1 , F1 , m1 ] be the Stone Representation of [Ω, F, m]. Then the closures in F of the sets Rm and Rm1 are equal, i.e., Rm = Rm1 .
6. Liapounoff convexity-type theorems Lemma 6.1. If m : F → F is of bounded variation or Rm is compact then m is strongly continuous. This result must be well known. For each p ∈ P let mp := Πp ◦ m. If Rm is precompact, then Rmp is also precompact since Πp is a continuous map or if m is of bounded variation then it is easy to verify that mp is also of bounded variation. In both cases it follows from the Banach space-valued case that mp is strongly continuous for all p ∈ P and hence m is strongly continuous. Theorem 6.2. Let F be a field of subsets of a set Ω, F a Fr´echet space and m ∈ f a(F, F ) strongly continuous. Then (i) if m ∈ sa(F, F ) then the weak closure of Rm coincides with its closed convex hull and is weakly compact; (ii) if Rm is compact then Rm is convex; (iii) if F has the Radon-Nikod´ym property and m is of bounded variation then the closure of Rm is compact and convex.
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If F has the interpolation property (I) then the strong continuity condition can be replaced by non-atomicity. Proof. If m has a precompact range or is of bounded variation then m is strongly continuous, see Lemma 6.1. Let [Ω1 , σ(F1 ), m1 ] be the Stone Representation of [Ω, F, m]. From the definition of Stone Representation, m1 is σ-additive on σ(F1 ). Also, m1 is non-atomic since m is strongly continuous or alternatively if F has property (I) and m is non-atomic, see Theorem 5.4. From Lemma 5.5 we know that Rm = Rm1 . To complete each proof: (i) since Rm = Rm1 , we know that the weak closure and closed convex hulls of Rm and Rm1 are equal. An appeal to [9] completes the proof; (ii) the proof follows immediately from [9] since Rm = Rm1 ; (iii) since m is of bounded variation, m1 is also of bounded variation. An appeal to [15] completes the proof. We now consider a case where the strong additivity assumption in Theorem 6.2(i) can be relaxed. Definition 6.3. A field F of subsets of a set Ω has the Vitali-Hahn-Saks property, if every sequence {μn } ⊂ ba+ (F), where {μn (A)} converges for every A ⊂ F, is uniformly strongly additive. Lemma 6.4. Let X be a Banach space and let F be a field of subsets of a set Ω. Let m ∈ ba(F, F ) if m takes values in a finite-dimensional subspace of X then m is of bounded variation and hence strongly additive. The proof of this lemma follows easily from the case of signed measures. Lemma 6.5. The space C(−∞, ∞), of all continuous functions on the reals equipped with the topology of uniform convergence on compact sets is a Fr´echet space with a Schauder basis. Proof. C(−∞, ∞) equipped with the topology of uniform convergence on compact sets is a Fr´echet space, see [17, Example 1.44]. C(−∞, ∞) is isomorphic to C([0, 1])N , a countable product of copies of C([0, 1]), this follows easily from [22, Theorem 3.3.6.2, p. 496]. It is well known that C([0, 1]) has a Schauder basis [23, II.B.12], hence C([0, 1])N and thus C(−∞, ∞) also have Schauder bases. The following theorem contains some ideas in [3] applied to a Fr´echet space setting. Theorem 6.6. Let F be a separable Fr´echet space generated by an increasing family of seminorms denoted by P and let F be a field of subsets of a set Ω with the VitaliHahn-Saks property. Let m ∈ ba(F, F ) then m is strongly additive. Proof. Every separable Fr´echet space is linearly homeomorphic to a subspace of C(−∞, ∞) equipped with the topology of uniform convergence on compact sets, see [14, p. 144].
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Denote the Schauder basis of C(−∞, ∞) by (en ) and let (fn ) denote the associated sequence of coefficient functionals. Each fn ∈ F and each x ∈ F can be uniquely represented in the form x = Σn fn (x)en , hence m(A) = Σn fn (m(A))en . The vector measure mk (A) := Σkn=1 fn (m(A))en takes its values in a finite-dimensional subspace of C(−∞, ∞). @p , · p ) be the Banach space defined in terms the quotient Let p ∈ P and (F map Πp : F → F/p−1 (0). Since m is bounded, mk is also bounded and it follows that Πp ◦ mk is a bounded finitely additive measure hence Πp ◦ mk is strongly additive. There exists a measure μk ∈ ba+ (F) such that mk is μk -continuous, see [4, Corollary I.5.3]. Hence Πp ◦ mk p (E) → 0 as μk (E) → 0 where Πp ◦ mk p (E) indicates the semivariation of Πp ◦ mk over a set E ∈ F, see [4, p. 2]. Since F has the Vitali-Hahn-Saks property, μ 1p := sup{μk : k ∈ N} is strongly additive, hence bounded. If μ 1p (A) → 0 then Πp ◦ mk p (A) → 0 for each k ∈ N which implies that p(m(A)) = Πp ◦ m(A) → 0, see [4, Corollary I.5.4]. μp }p∈P as done We can construct a single measure μ 1 ∈ ba+ (F) from the set {1 in [11, Corollary II.2.2], with the property that μ 1(A) → 0 implies that μ 1p (A) → 0 for all p ∈ P . If {Am } ⊂ F is a mutually disjoint sequence of sets then μ 1(Am ) → 0 which implies that p(m(Am )) = Πp ◦ m(Am )p → 0 for each p ∈ P . Acknowledgment The author thanks Prof. J. Diestel for many helpful comments and discussions relating to this work. The author also thanks Prof. J. Bonet for the short proof of Lemma 6.5.
References [1] Utpal K. Bandyopadhyay. On vector measures with the Darboux property. Quart. J. Math. Oxford Ser. (2), 25:57–60, 1974. [2] K.P.S. Bhaskara Rao and M. Bhaskara Rao. Theory of charges, volume 109 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. [3] J. Diestel. Applications of weak compactness and bases to vector measures and vectorial integration. Rev. Roumaine Math. Pures Appl., 18:211–224, 1973. [4] J. Diestel and J.J. Uhl, Jr. Vector measures. American Mathematical Society, Providence, R.I., 1977. [5] N. Dinculeanu. Vector measures. International Series of Monographs in Pure and Applied Mathematics, Vol. 95. Pergamon Press, Oxford, 1967. [6] Nelson Dunford and Jacob T. Schwartz. Linear Operators. I. General Theory. With the assistance of W.G. Bade and R.G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York, 1958. [7] A. Fern´ andez and F. Naranjo. Rybakov’s theorem for vector measures in Fr´echet spaces. Indag. Math. (N.S.), 8(1):33–42, 1997.
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[8] Igor Kluv´ anek. The extension and closure of vector measure. In Vector- and operatorvalued measures and applications (Proc. Sympos., Alta, Utah, 1972), pages 175–190. Academic Press, New York, 1973. [9] Igor Kluv´ anek. The range of a vector-valued measure. Math. Systems Theory, 7:44– 54, 1973. [10] Igor Kluv´ anek. The range of a vector measure. Bull. Amer. Math. Soc., 81:609–611, 1975. [11] Igor Kluv´ anek and Greg Knowles. Vector measures and control systems. NorthHolland Publishing Co., Amsterdam, 1976. North-Holland Mathematics Studies, Vol. 20, Notas de Matem´ atica, No. 58. [Notes on Mathematics, No. 58]. [12] A. Liapounoff. Sur les fonctions-vecteurs compl`etement additives. Bull. Acad. Sci. URSS. S´er. Math. [Izvestia Akad. Nauk SSSR], 4:465–478, 1940. [13] Joram Lindenstrauss. A short proof of Liapounoff’s convexity theorem. J. Math. Mech., 15:971–972, 1966. [14] S. Mazur and W. Orlicz. Sur les espaces m´etriques lin´eaires. II. Studia Math., 13:137– 179, 1953. [15] Sachio Ohba. The range of vector measure and Radon-Nikod´ ym property. Rep. Fac. Engrg. Kanagawa Univ., (16):4–5, 1978. [16] W.J. Ricker. Rybakov’s theorem in Fr´echet spaces and completeness of L1 -spaces. J. Austral. Math. Soc. Ser. A, 64(2):247–252, 1998. [17] Walter Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York, second edition, 1991. [18] E. Saab and P. Saab. Lyapunov convexity type theorems for non-atomic vector measures. Quaest. Math., 26(3):371–383, 2003. [19] G.L. Seever. Measures on F -spaces. Trans. Amer. Math. Soc., 133:267–280, 1968. [20] A. Sobczyk and P.C. Hammer. A decomposition of additive set functions. Duke Math. J., 11:839–846, 1944. [21] J.J. Uhl, Jr. The range of a vector-valued measure. Proc. Amer. Math. Soc., 23:158– 163, 1969. [22] Manuel Valdivia. Topics in locally convex spaces, volume 67 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1982. Notas de Matem´ atica [Mathematical Notes], 85. [23] P. Wojtaszczyk. Banach spaces for analysts, volume 25 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1991. Rudolf G. Venter Department of Mathematics and Applied Mathematics University of Pretoria, 0002 South Africa e-mail: [email protected]
List of Participants Albrecht, E., Univ. des Saarlandes, Germany Arendt, W., Univ. Ulm, Germany Atanasiu, D., Univ. College of Boras, Sweden Barcenas, D., Univ. de Los Andes, Venezuela Becker, R., CNRS Paris VI, France Biegert, M., Univ. Ulm, Germany Bierstedt, K., Univ. Paderborn, Germany Blasco, O., Univ. de Valencia, Spain Boccuto, A., Univ. di Perugia, Italy Bonet, J., Univ. Polit´ecnica de Valencia, Spain Bongiorno, B., Univ. di Palermo, Italy Bongiorno, D., Univ. di Palermo, Italy Calabuig, J.M., Univ. Polit´ecnica de Valencia, Spain del Campo, R., Univ. de Sevilla, Spain Cascales, B., Univ. de Murcia, Spain Cicho´ n, K., Poznan Univ. of Technology, Poland Cicho´ n, M., A. Mickiewicz Univ., Poznan, Poland Corduneanu, S.-O., Technical Univ. of Iasi, Romania Croitoru, A., Univ. Alexandru Ioan Cuza, Romania Curbera, G.P., Univ. de Sevilla, Spain Delgado, O., Univ. Polit´ecnica de Valencia, Spain Di Piazza, L., Univ. di Palermo, Italy Diestel, J., Kent State Univ., USA Dodds, P.G., Flinders Univ. of South Australia Dodds, T., Flinders Univ. of South Australia Doust, I.R., Univ. of New South Wales, Australia
Fechner, W., Silesian Univ., Poland Fechner, Z., Silesian Univ., Poland Fern´ andez, A., Univ. de Sevilla, Spain Ferrando, I., Univ. Polit´ecnica de Valencia, Spain Florescu, L., Univ. Alexandru Ioan Cuza, Romania Fourie, J.H., North-West Univ., South Africa Galaz Fontes, F., Centro de Investigacion en Matematicas, Mexico Garc´ıa Raffi, L.M., Univ. Polit´ecnica de Valencia, Spain Gillespie, T.A., Univ. of Edinburgh, UK Girardi, M., Univ. of South Carolina, USA Haluska, J., Slovak Acad. of Sciences, Slovakia Hilger, S., Katholische Univ. Eichst¨ attIngolstadt, Germany Hyt¨onen, T.P., Univ. of Helsinki, Finland Jarchow, H., Univ. Z¨ urich, Switzerland Jefferies, B., Univ. of New South Wales, Australia Juan, M.A., Univ. Polit´ecnica de Valencia, Spain Kawabe, J., Shinshu Univ., Japan Knowles, G., Flinders Univ. of South Australia K¨onig, H., Univ. des Saarlandes, Germany Kunze, M., Univ. Ulm, Germany Labuda, I., Univ. of Mississippi, USA Leinert, M., Univ. Heidelberg, Germany Lipecki, Z., Polish Acad. of Sciences, Wroclaw, Poland Lusky, W., Univ. Paderborn, Germany Macansantos, P., Univ. of the Philippines Baguio, Philippines Maepa, S.M., Univ. of Pretoria, South Africa
382
List of Participants
Marraffa, V., Univ. di Palermo, Italy Mayoral, F., Univ. de Sevilla, Spain Miana, P.J., Univ. de Zaragoza, Spain Mockenhaupt, G., Univ. Siegen, Germany Musial, K., Univ. Wroclaw, Poland Naranjo, F., Univ. de Sevilla, Spain Okada, S., Univ. of Wollongong, Australia de Pagter, B., Delft Univ. of Technology, The Netherlands Papadimitropoulos, C., Univ. of Edinburgh, UK Pfitzner, H., Univ. d’Orl´eans, France Popa, E., Univ. Alexandru Ioan Cuza, Romania Ressel, P., Katholische Univ. Eichst¨ attIngolstadt, Germany Ricker, W.J., Katholische Univ. Eichst¨ attIngolstadt, Germany Rodr´ıguez-Piazza, L., Univ. de Sevilla, Spain S´ anchez P´erez, E.A., Univ. Polit´ecnica de Valencia, Spain Satco, B., Stefan cel Mare Univ., Romania Schep, A.R., Univ. of South Carolina, USA
Amin Sofi, M., Univ. of Kashmir, India Swart, J., Univ. of Pretoria, South Africa Sommer, M., Katholische Univ. Eichst¨ attIngolstadt, Germany Terauds, V., Univ. of Newcastle, Australia Tradacete, P., Univ. Complutense de Madrid, Spain Uglanov, A., St.-Petersburg State Polytechnical Univ., Russia Uhl, Jr., J.J., Univ. of Illinois (Urbana), USA van Neerven, J., Delft Univ. of Technology, The Netherlands V¨ ath, M., Univ. W¨ urzburg, Germany Venter, R.G., Univ. of Pretoria, South Africa Veraar, M.C., Univ. Karlsruhe, Germany Weis, L., Univ. Karlsruhe, Germany Wickstead, A.K., Queen’s Univ. Belfast, UK Wnuk, W., A. Mickiewicz Univ., Poznan, Poland Wright, J., Univ. of Edinburgh, UK Yost, D., Univ. Ballarat, Australia
