Collect. Math. https://doi.org/10.1007/s13348-018-0225-y

Extension of Pettis integration: Pettis operators and their integrals Oscar Blasco1

· Lech Drewnowski2

Received: 21 November 2017 / Accepted: 8 May 2018 © Universitat de Barcelona 2018

Abstract In this note, the authors discuss the concepts of a Pettis operator, by which they mean a weak∗ –weakly continuous linear operator F from a dual Banach space to an L 1 -space, and of its Pettis integral, understood simply as the dual operator F ∗ of F. Applications to radial limits in weak Hardy spaces of vector-valued harmonic and holomorphic functions are provided. Keywords Pettis integral · Pettis operator · Weak∗ –weakly continuous operator · Weak Hardy space Mathematics Subject Classification 46G10 · 28B05

1 Introduction Throughout, X is a Banach space with its dual space X ∗ whose closed unit ball is denoted B(X ∗ ). Furthermore, (S, , μ) is a σ -finite measure space, and the Banach spaces L 1 (μ) and L ∞ (μ) = L 1 (μ)∗ of scalar measurable functions are understood in the standard way, and their norms are denoted ·1 and ·∞ , respectively. The equalities and inequalities between the elements of these spaces are meant to hold μ-a.e. between any representatives of those elements. In some parts of this note, however, some sub-σ -algebras A of  and the restrictions μ|A of μ to A enter the scene, and as the latter need no longer be σ -finite, we

to the memory of Joe Diestel (1943–2017)

B

Oscar Blasco [email protected] Lech Drewnowski [email protected]

1

Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain

2

Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Pozna´n, Poland

123

O. Blasco, L. Drewnowski

are forced to assume that the measure μ is finite. In such cases, we state this requirement explicitly atthe beginning of relevant sections. For all ϕ ∈ L 1 (μ) and ψ ∈ L ∞ (μ) we shall  often write ψ ϕdμ in place of S ψϕdμ. (This notation is highly nonstandard, though worth testifying.) In the paper we deal, in varying contexts, with weak∗ –weakly continuous linear operators between Banach spaces X ∗ and Y . The collection of all such operators is clearly identical with the closed subspace L w∗ ,w (X ∗ , Y ) of the usual space L(X ∗ , Y ) of (norm) continuous operators from X ∗ to Y , consisting of those F ∈ L(X ∗ , Y ) for which the dual operator F ∗ acts from Y ∗ into X , where X is obviously viewed canonically as a subspace of X ∗∗ . Since, for each F ∈ L w∗ ,w (X ∗ , Y ), one has x ∗ , F ∗ (y ∗ ) = y ∗ , F(x ∗ ) for all x ∗ ∈ X ∗ ,

y∗ ∈ Y ∗,

the map F → F ∗ is a linear isometry between the spaces L w∗ ,w (X ∗ , Y ) and L w∗ ,w (Y ∗ , X ). For more detailed information, mostly in the context of general locally convex spaces, see e.g. [6,13]. In our generalization of Pettis integrable functions and of the Pettis integral, the particular case of the above with Y = L 1 (μ) will be of crucial importance. Let us recall that one of possible (and well known) approaches to the Pettis integral is the following: A weakly measurable function f : S → X is said to be Pettis (μ-)integrable if x ∗ f ∈ L 1 (μ) for every x ∗ ∈ X ∗ and the operator P f : X ∗ → L 1 (μ), given by x ∗ → x ∗ f , is weak∗ –weakly continuous. Of course, one should think here of x ∗ f as a genuine element of L 1 (μ); and notice that then the dual operator P ∗f acts from L ∞ (μ) into X ⊂ X ∗∗ . This, in turn, is equivalent to the apparently less restrictive condition that P ∗f (χ A ) ∈ X for every A ∈ . Here, χ A stands for the characteristic function of the set A. We denote by P1 (μ, X ) the space of Pettis μ-integrable functions f endowed with the operator norm  f  = P f . Now, by definition, the (indefinite) Pettis (μ-) integral of f is the (countably additive) vector measure m f :  → X given by the formula  m f (A) = P ∗f (χ A ) =: (P) f dμ for all A ∈ . A

Moreover, one then also has   P ∗f (ψ) = ψdm f = (P) ψ f dμ, for all ψ ∈ L ∞ (μ). S

S

Here, the first integral is the elementary Bartle-Dunford-Schwartz integral of ψ with respect to the vector measure m f , see e.g. [7], while the other one is the Pettis integral of the function  ψ f . The latter will sometimes be written as (P) ψ f dμ. This looks strange, but is possibly worth testifying in practice (except the situations when ψ has a complex form). For more precise formulations see, e.g., [18, Th. 4.3] and [9, (4.3)]. Basic information on the Pettis integral can be found in the monographs [7,21], and for a more comprehensive and up-to-date survey of what is known on this subject, including, in particular the weak Radon-Nikodym property (w-RNP) of Banach spaces, along with a vast literature, we refer the reader to [18,19]. Also [6, Sec. 4] is worth looking up. Unfortunately, the theory of Pettis integration has a number of “weak points” as compared to the classical case or even the case of the Bochner integration. Thus, for instance, in general • the normed spaces of Pettis integrable functions are incomplete, (see [10] for a thorough discussion). • Not every Pettis integrable function admits a conditional expectation with respect to a sub-σ -algebra (see [17,20]);

123

Extension of Pettis integration

• the Fubini theorem fails to hold in a dramatic way (see [16,22]). • the Fatou theorem for harmonic functions fails. In particular, whenever X is infinite dimensional, there exists a Pettis-integrable function f : T → X such that limr →∞ Pr ∗ F(t) = ∞ uniformly in t ∈ T (see [12]). The purpose of this paper is to propose a certain formalism that allows one, at least to some extent, to get rid of or bypass some of the disadvantages that the usual theory of Pettis integration has. Its basic idea is to go beyond the class of “usual” Pettis integrable functions f : S → X and the associated operators P f , and consider instead the space of all weak∗ –weakly continuous linear operators F : X ∗ → L 1 (μ), which we chose to call Pettis operators as objects which are subject to “integration”. Certainly, this approach does not annihilate the specific delicate problems of the proper Pettis integration, though it still might well encourage one to try to understand or interpret them within the proposed more general setting. We discuss various aspects of the proposed approach in the subsequent sections or subsections, with no intention, however, to develop a full-range theory. For example, among other things, we identify the space of Pettis operators with certain spaces of vector-valued measures and of vector-valued harmonic functions. In particular, we establish that it coincides with the space of μ-continuous countably additive X -valued measures, or that the space of holomorphic functions f from the unit disc D into X such that x ∗ , f  ∈ H 1 (D) for all x ∗ ∈ X ∗ can be identified, using radial limits, with the space of analytic Pettis operators from X ∗ into L 1 (T) (see Sect. 7 for the precise definitions and statement).

2 Pettis operators and their integrals Definition 2.1 We shall denote by P(X ∗ , L 1 (μ)) the space of all weak∗ –weakly continuous linear operators F : X ∗ → L 1 (μ). We shall call them Pettis operators. For each F ∈ P(X ∗ , L 1 (μ)) and A ∈  we define the Pettis μ-integral of F over A as the result of the action of F ∗ , the dual operator which belongs to L(L ∞ (μ)), X ), on χ A , that is  (P) Fdμ = F ∗ (χ A ). A ∗ ∗ More generally,  for any ψ ∈ L ∞ (μ), thought as a member of L 1 (μ) , F (ψ) will be denoted by (P) ψ Fdμ, and is a unique element of X such that     x ∗ , (P) Fdμ = ψ · F(x ∗ )dμ for all x ∗ ∈ X ∗ . ψ

S

Remark 2.2 More generally, one may consider as Pettis operators the weak∗ –weakly continuous operators F : X ∗ → L, where L is any Banach space of scalar μ-measurable functions, with its dual L ∗ also represented as a Banach space M of this type, and the duality form on  M × L being the natural one: ψ, ϕ = S ψϕdμ. An obvious example for this is of course L = L p (μ), M = L q (μ), where p, q is any pair of conjugate exponents. At least in this case most of the things to be discussed in the following sections should go through smoothly in a greater generality, with only minor and rather obvious modifications like replacing L 1 or 1 with L p or  p . Example 2.3 P1 (μ, X ) ⊆ P(X ∗ , L 1 (μ)) by means of the embedding f → F = P f . To produce more examples of Pettis operators we can use the following basic facts:

123

O. Blasco, L. Drewnowski

1. Any bounded linear operator in L(X ∗ , Y ) from reflexive Banach spaces X is always weak∗ –weakly continuous. 2. If X, Y, E and Z are Banach spaces, F ∈ L w∗ ,w (X ∗ , Y ), H ∈ L(X, E), and T ∈ L(Y, Z ), then TFH∗ ∈ L w∗ ,w (E ∗ , Z ). Example 2.4 Let X be a reflexive Banach space and let F : X ∗ → L 1 (μ) be a bounded linear operator. Then F ∈ P(X ∗ , L 1 (μ)). Example 2.5 Let μ and ν be σ -finite measures on some measurable spaces, and T : L 1 (μ) → L 1 (ν) be a continuous linear operator. Then if F : X ∗ → L 1 (μ) is a Pettis operator, so is TF : X ∗ → L 1 (ν). Furthermore,   (P) TFdν = (P) Fdμ for all γ ∈ L ∞ (ν). T ∗ (γ )

γ

Let us study the coincidence between P1 (μ, X ) and P(X ∗ , L 1 (μ)). Let us first analyse the case, where S = N,  = P(N) and μ is the counting measure γ . Theorem 2.6 P1 (γ , X ) is isometrically isomorphic to P(X ∗ , 1 ). Proof Recall first that  P1 (γ , X ) can be identified with the space of sequences x = (xn ) ∈ X such that the series n xn is unconditionally (or subseries) convergent in X . In particular, for x ∈ P1 (γ , X ),   (P) bxdγ = βn xn for all b = (βn ) ∈ ∞ A

n∈A

where the series on the right converges unconditionally (the bounded multiplier property). Clearly, the Pettis operator Px : X ∗ → 1 associated with x is given by the equality  Px (x ∗ ) = x ∗ (xn ) n∈N . Conversely, let F ∈ P(X ∗ , 1 ) and let (en ) be the sequence of unit vectors in ∞ . Then for each n ∈ N, xn =: (P) en Fdγ = F ∗ (en ) ∈ X . Further, for any A ⊂ N and x ∗ ∈ X ∗ ,   ∗ ∗ one has x A := (P)  n∈A x (x n ). Hence, by the Orlicz-Pettis A Fdγ ∈ X , and x (x A ) = theorem, the series n xn is subseries convergent, which shows that x ∈ P1 (γ , X ).  Remark 2.7 An analogous result holds also for the general spaces 1 (S). Note that when S is uncountable, the counting measure γ on S is not σ -finite. Remark 2.8 An analogue of Theorem 2.6 will state now that the general form of Pettis operators F : X ∗ →   p is exactly as in that theorem, but using sequences x = (x n ) in X such that the series n βn xn is unconditionally convergent in X for each sequence (βn ) ∈  p with 1/ p + 1/ p = 1. In general, e.g., when the measure μ is atomless and the space X is of infinite dimension, P(X ∗ , L 1 (μ)) is larger than P1 (μ, X ). While we are not ready yet to make this claim more precise, we are still able to strongly support it by the two examples below. First, it is rather easy to construct desired examples for some reflexive Banach spaces X . Example 2.9 Let X = L p ([0, 1]) for 1 < p < ∞ and (S, , μ) be the Lebesgue measure space [0, 1]. Let 1/ p + 1/ p = 1, and let F : L p ([0, 1]) → L 1 ([0, 1]) be the inclusion map. In view of Example 2.4, F ∈ P(L p ([0, 1]), L 1 ([0, 1]). We are going to show that

123

Extension of Pettis integration

F ∈ / P1 (μ, L p ([0, 1]). To this end, observe that g(t) = g, δt  for any 0  t  1 and any function g ∈ C([0, 1], where δt is the Dirac mass at the point t. It follows that the function t → f˜(t) = δt is a weak∗ -measurable function from [0, 1] to (C[0, 1])∗ and that F(g)(t) = g, f˜(t) for all g ∈ C([0, 1]) and t ∈ [0, 1]. Finally, if we assume that there exists a weakly measurable function f from [0, 1] to L p ([0, 1]) for which the function t → F(g)(t) = g, f (t) belongs to L 1 ([0, 1]) for all g ∈ L p ([0, 1]), in particular we get that g, f˜(t) = g, f (t) for any g ∈ C([0, 1]). Hence the Dirac mass δt should belong to L p ([0, 1]) for almost all t ∈ [0, 1] getting a contradiction.  Second, to achieve the same end in the case of non-reflexive Banach spaces we shall make use of the following well know example. Example 2.10 Let X = c0 and (S, , μ) the Lebesgue measure space [0, 1]. Let F : 1 → L 1 ([0, 1]) the operator defined by x ∗ → x ∗ , (rn (t)) where (rn ) stands for the sequence of Rademacher functions. Since (rn (t)) ∈ / c0 for t ∈ [0, 1] then F ∈ / P1 (μ, c0 ). However, taking into account that for x ∗ ∈ 1 and g ∈ L ∞ ([0, 1]) we have

 1 ∗ ∗ F(x ), g = x , g(t)rn (t) dt 0

and

1 0



g(t)rn (t) dt ∈ c0 one gets that F ∈ P(1 , L 1 ([0, 1])).



Let us now present natural extensions to Pettis operators of some operations that are routinely expected to be admissible in any potentially useful integration theory. Multiplication of Pettis operators by bounded measurable scalar functions Let ψ ∈ L ∞ (μ) and denote by Mψ the map ϕ → ψϕ. Now for each F ∈ P(X ∗ , L 1 (μ)) we have that ψ F ∈ P(X ∗ , L 1 (μ)), where ψ F stands for the operator Mψ F, that is (ψ F)(x ∗ ) := ψ F(x ∗ ) for each x ∗ ∈ X ∗ . Moreover   Fdμ (P) ψ Fdμ = (P) γ

γψ

for all γ ∈ L ∞ (μ). Convolutions of Pettis operators with scalar integrable functions Let (S, .) be a locally compact abelian group and let μ be its Haar measure. Then the convolution (ϕ1 , ϕ2 ) → ϕ1 ∗ ϕ2 , defined by  ϕ1 ∗ ϕ2 (t) = ϕ1 (ts −1 )ϕ2 (s)dμ(s) S

is a continuous bilinear map from L 1 (μ) × L 1 (μ) to L 1 (μ). In particular each ϕ ∈ L 1 (μ) defines a bounded linear operator Tϕ : L 1 (μ) → L 1 (μ) given by Tϕ (ψ) = ψ ∗ ϕ. Therefore, for every operator F in P(X ∗ , L 1 (μ)) and every function ϕ in L 1 (μ) one may define their convolution ϕ ∗ F to be the operator Tϕ F ∈ P(X ∗ , L 1 (μ)), that is (ϕ ∗ F)(x ∗ ) = ϕ ∗ F(x ∗ ).

123

O. Blasco, L. Drewnowski

Recall that for ϕ1 , ϕ2 ∈ L 1 (μ) and ϕ3 ∈ L ∞ (μ) one has, writing ϕ(t) ˜ = ϕ(t −1 ),   (ϕ1 ∗ ϕ2 )(t)ϕ3 (t)dμ(t) = ϕ1 (s)(ϕ˜2 ∗ ϕ3 )(s)dμ(s). S

S

This shows that

  (P) ϕ ∗ Fdμ = (P) γ

γ ∗ϕ˜

Fdμ

for all γ ∈ L ∞ (μ). Similarly, whenever Y is a Banach space and S ∈ L(L 1 (μ), Y ) we can define S ∗ ϕ ∈ L(L 1 (μ), Y ) as STϕ , that is, (S ∗ ϕ)(ψ) = S(ψ ∗ ϕ). It is also immediate to see that if F ∈ P(X ∗ , L 1 (μ)) and ϕ ∈ L 1 (μ), then (ϕ ∗ F)∗ = F ∗ ∗ ϕ. ˜ The reader is referred to [1,2] for the use of S ∗ ϕ in several settings. Conditional expectation Here we assume that μ is a finite measure. Let A be a sub-σ -algebra of . Whenever convenient, we shall consider the space L 1 (μ|A) = L 1 (S, A, μ|A) a subspace of L 1 (μ). Then the map ϕ → E(ϕ|A) (the conditional expectation of ϕ with respect to A) is a continuous linear operator from L 1 (μ) to L 1 (μ|A). Therefore, for every operator F ∈ P(X ∗ , L 1 (μ)) one may define E(F|A) in P(X ∗ , L 1 (μ|A)), the conditional expectation of F with respect to A, by the formula E(F|A)(x ∗ ) = E(F(x ∗ )|A) (x ∗ ∈ X ∗ ). Then for all A ∈ A and x ∗ ∈ X ∗ one has     x ∗ , (P) E(F|A)dμ = E(F(x ∗ )|A)dμ A A  F(x ∗ )dμ = A    = x ∗ , (P) Fdμ . A

Remark 2.11 One readily checks that if A ⊂ B are sub-σ -algebras in , then E(F|A) =  E E(F|B)|A .

3 Pettis operators versus vector-valued measures Let us denote by caμ (, X ) the Banach space of all μ-continuous countably additive vector measures m :  → X , endowed with the norm given by the semivariation m = supx ∗ 1 |x ∗ m|(S). Let us include here the known identification between this space and the space of Pettis operators. Theorem 3.1 The spaces P(X ∗ , L 1 (μ)) and caμ (, X ) are isometrically isomorphic.

123

Extension of Pettis integration

Proof Let F ∈ P(X ∗ , L 1 (μ)) and consider the indefinite Pettis μ-integral of F given by  m F (A) = (P) Fdμ, A ∈ . A

In this way we get a finitely additive vector measure m F :  → X . Since  F(x ∗ )dμ for all A ∈  and x ∗ ∈ X ∗ , x ∗ , m F (A) = A

the measure m F is countably additive (by the Orlicz-Pettis theorem), and since m F (A) = 0 whenever μ(A) = 0, it is μ-continuous. It is elementary to see that the norm of F corresponds to the semivariation of m F . Conversely, let m ∈ caμ (, X ), and let F = dm/dμ, the Radon-Nikodym derivative of m with respect to μ, be the operator from X ∗ into L 1 (μ) given by the formula F(x ∗ ) =

dm ∗ d(x ∗ m) (x ) = . dμ dμ

Then for every ψ ∈ L ∞ (μ) and x ∗ ∈ X ∗ one has      x ∗ , ψdm = ψd(x ∗ m) = ψ F(x ∗ )dμ = x ∗ , F ∗ (ψ). S

P(X ∗ , L

Hence F ∈ course m F = m.

S 1 (μ)),

F∗

S

coincides with the operator ψ →

 S

ψdm on L ∞ (μ), and of 

We present now a useful convergence result. Theorem 3.2 Let (Fn ) be a sequence in P(X ∗ , L 1 (μ)) such that the sequence of measures m n = m Fn (n ∈ N) is pointwise weakly convergent; that is, for each A ∈  the sequence (m n (A)) has a weak limit, call it m(A), in X . Then m ∈ caμ (, X ), and the sequence (Fn ) is pointwise weakly convergent to an operator F ∈ P(X ∗ , L 1 (μ)) with m F = m. Proof Since, for every x ∗ ∈ X ∗ , x ∗ m n (A) → x ∗ m(A) as n → ∞ for all A ∈ , x ∗ m is a μ-continuous scalar measure. It follows that m is a μ-continuous vector measure (by the Vitali-Hahn-Sacks theorem). This proves the first assertion. Next note that since for every x ∗ ∈ X ∗ and A ∈  the sequence (x ∗ m n (A)) is bounded, the measures x ∗ m n (n ∈ N) are uniformly bounded or, which is the same, have uniformly bounded total variations (by the Nikodym boundedness theorem). In consequence, also the measures m n (n ∈ N) are uniformly bounded or, equivalently, have uniformly bounded semivariations.   Obviously, for each x ∗ , S ψd x ∗ m n → S ψd x ∗ m for all simple functions ψ ∈ L ∞ (μ). Since such functions are dense in L ∞ (μ), and the integral operators ψ → S ψd x ∗ m n (n ∈ N) on L ∞ (μ) are uniformly bounded (their  norms are equal to the total variations |x ∗ m n |(S) of x ∗ m n ), it follows that S ψdm n → S ψdm weakly in X for every ψ ∈ L ∞ (μ).  Now, note that, according to Theorem 3.1, all the operators M : ψ → ψdm n n (n ∈ N) S  and M : ψ → S ψdm from L ∞ (μ) to X are all in L w∗ ,w (L ∞ (μ), X ), and we have just seen that Mn → M pointwise weakly. Therefore (Fn ) = (Mn∗ ) is pointwise weak∗ convergent to the operator M ∗ ∈ L(X ∗ , (L 1 (μ))∗∗ ). However, since L 1 (μ) is weakly sequentially complete, one actually gets that M ∗ = F ∈ P(X ∗ , L 1 (μ)) and Fn = Mn∗ → F := M ∗ pointwise weakly in P(X ∗ , L 1 (μ)). 

123

O. Blasco, L. Drewnowski

It was shown in [20] (see also [17]) that the indefinite Pettis integral of a Pettis integrable function is always a vector measure of σ -finite variation. The situation in this respect, for the setting of the present work, is clarified by our next, fairly routine result. Theorem 3.3 Let F : X ∗ → L 1 (μ) be a Pettis operator. Then its indefinite integral m F :  → X is a vector measure of finite or σ -finite variation if and only if the set F(B(X ∗ )) is order bounded in L 1 (μ) or L 0 (μ), respectively. Proof ⇒) Denote by v-m F the variation measure of m F . Setting ϕ = d(v-m F )/dμ one gets a required order bound for the F(x ∗ )’s, where x ∗ ∈ B(X ∗ ). ⇐) Let ϕ be a nonnegative real-valued measurable function on S such that |F(x ∗ )|  ϕ for all x ∗ ∈ B(X ∗ ). First, suppose  ϕ ∈ L 1 (μ), and  take any A ∈ . Then for each x ∗ ∈ B(X ∗ ) one has |x ∗ m F (A)|  A |F(x ∗ )|dμ  A ϕdμ = μϕ (A). In consequence, m F (A)  μϕ (A) so that, finally, v-m F  μϕ , where μϕ is a finite measure on . In the other case, i.e.. when merely ϕ ∈ L 0 (μ), choose any  partition (Sn ) of S such that, for each n, μ(Sn ) < ∞ and ϕ|Sn is bounded. Proceeding as above one readily arrives at the inequality v-m F  μϕ . Now, obviously, μϕ , is a σ -finite measure on , hence so is v-m F .  Let us denote P0 (X ∗ , L 1 (μ)) the subspace of P(X ∗ , L 1 (μ)) consisting of Pettis operators with indefinite integrals of σ -finite variation. Theorem 3.4 If F ∈ P0 (X ∗ , L 1 (μ)) and X satisfies (w-RNP) then there exists a Pettis integrable function f : S → X such that F = P f . Proof Recall that a Banach space X is said to have the weak-Radon-Nikodym property (see [17]), to be denoted by (w-RNP), if and only if for any vector measure m defined in S of σ finite variation, that is absolutely continuous with respect to μ, there exists a Pettis integrable function f : S → X with F ∈ P0 (X ∗ , L 1 (μ)) and X  m∗ f = m. Hence,∗ assuming that ∗ satisfy (w-RNP) one has A x f dμ = A F(x )dμ for all x ∈ X ∗ and A ∈ , Therefore, F(x ∗ ) = x ∗ f (μ-a.e.).  Evidently, any two functions satisfying the assertion of Theorem 3.4, say f and g, are weakly μ- equivalent, that is, x ∗ f = x ∗ g μ-a.e. for each x ∗ ∈ X ∗ . Therefore, when the Banach space X has (w-RNP), one may consider the map M : P0 (X ∗ , L 1 (μ)) → P1 (μ, X ) given by M(F) = f where F = P f . We shall call it the Musiał operator. From the weak μ—uniqueness it follows readily that M is a linear operator. Moreover, it clearly preserves norms and thus is a linear isometry between P0 (X ∗ , L 1 (μ)) and P1 (μ, X ). Unfortunately, the definition of M is highly nonconstructive.

4 Martingales of Pettis operators Throughout this section μ is assumed to be a finite measure. In addition, J = (J, ) denotes a (nonempty) directed set, and (A j : j ∈ J ) stands for a monotone increasing net of sub-σ  algebras in . We shall write A∞ for the algebra of sets j∈J A j , and B for the σ -algebra generated by A∞ . As is to be expected, by a martingale of Pettis operators from X ∗ to L 1 (μ) subordinated to the net (A j ) of sub-σ -algebras, we shall mean any net (F j : j ∈ J ) of such operators for which Fi = E(F j |Ai ) whenever i, j ∈ J and i  j. Obviously each F j (x ∗ ) is   A j -measurable. Note that then the net (P) A F j : j ∈ J in X is eventually constant for

123

Extension of Pettis integration

each A ∈ A∞ , and hence convergent in norm. A standard example of a martingale as above is obtained by taking any Pettis operator F : X ∗ → L 1 (μ), and setting F j = E(F|A j ) for each j ∈ J . The convergence result below is nothing but a routine adaptation of what is known in the scalar case to the present setting. Being merely an illustration of what can be done in this direction, it should suffice at the moment. Theorem 4.1 Let (F j ) be a martingale of Pettis operators as described above. Then the following are equivalent (a) The martingale (F j ) converges pointwise on X ∗ to some Pettis operator F : X ∗ → L 1 (μ), that is, lim j F j (x ∗ ) = F(x ∗ ) in L 1 (μ) for all x ∗ ∈ X ∗ . (b) There exists a Pettis operator G : X ∗ → L 1 (μ) such that   lim (P) F j dμ = (P) Gdμ weakly in X for each A ∈ A∞ . j

A

A

Moreover F = E(G|B) if (a) or (b) hold. Proof Assume (a) and take any A ∈  and x ∗ ∈ X ∗ . Then

  x ∗ , (P) Fdμ = F(x ∗ )dμ A A  F j (x ∗ )dμ = lim j A

 ∗ = lim x , (P) F j dμ . j

A

Therefore, (b) holds with G = F.  Now, assume (b) and start by noting that for every x ∗ ∈ X ∗ the net F j (x ∗ ) : j ∈ J is a martingale in L 1 (μ) subordinated to the net (A j : j ∈ J ). In addition, as follows from (b), for all x ∗ ∈ X ∗ and A ∈ A∞ ,   F j (x ∗ )dμ = G(x ∗ )dμ. lim j

A

A

Therefore, by [7, Ch. 5, Sec. 2, Theorem 1] (for the case X =scalars), and its proof, putting F = E(G|B) one has E(G|B)(x ∗ ) = E(G(x ∗ )|B) = lim F j (x ∗ ) = F(x ∗ ) j



and (a) follows.

5 Fubini theorem for Pettis operators Let the measure space (S, , μ) be the product (S1 , 1 , μ1 ) ⊗ (S2 , 2 , μ2 ) of two σ -finite measure spaces; thus S = S1 × S2 ,  = 1 ⊗ 2 , and μ = μ1 ⊗ μ2 . Also, let F ∈ P(X ∗ , L 1 (μ)). By the standard Fubini theorem, for every ϕ ∈ L 1 (μ),    ϕdμ = ϕdμ2 dμ1 S S1 S2   = ϕdμ1 dμ2 . S2

S1

123

O. Blasco, L. Drewnowski

Here we have only stated the Fubini’s formula, omitting the descriptive part of the theorem. Also, we suppressed the variables for functions and integrals. These, as we expect, should be clear from the context. As a consequence of the above, let us also state the following: Whenever ψ1 ∈ L ∞ (μ1 ) and ψ2 ∈ L ∞ (μ2 ), denoting by (ψ1 ⊗ ψ2 )(s1 , s2 ) = ψ1 (s1 )ψ2 (s2 ) for (s1 , s2 ) ∈ S1 × S2 , then    ϕdμ = ϕdμ2 dμ1 ψ1 ⊗ψ2

ψ1

  =

ψ2

ψ2



ϕdμ1 dμ2 .

ψ1

 The above gives rise to two continuous linear operators: E 1 : ϕ → ψ2 ϕdμ2 from L 1 (μ)  to L 1 (μ1 ), and E 2 : ϕ → ψ1 ϕdμ1 from L 1 (μ) to L 1 (μ2 ). In consequence, for each Pettis operator F : X ∗ → L 1 (μ), also the operators   ∗ F1 := ψ2 Fdμ2 := E 1 F : x → ψ2 F(x ∗ )dμ2 from X ∗ to L 1 (μ1 ), as well as   F2 := ψ1 Fdμ1 := E 2 F : x ∗ → S1 F(x ∗ )dμ1 from X ∗ to L 1 (μ2 ) are both Pettis operators. Now, take any x ∗ ∈ X ∗ . Then

 x ∗ , (P)

ψ1 ⊗ψ2

 Fdμ =

ψ1 ⊗ψ2

F(x ∗ )dμ

  =

ψ1

ψ2

ψ1

ψ2

F(x ∗ )dμ2 dμ1

  =

Fdμ2 (x ∗ )dμ1

  = x ∗ , (P)

ψ1

ψ2



Fdμ2 dμ1 .

Therefore,  (P)

ψ1 ⊗ψ2

  Fdμ = (P) ψ1

ψ2

Fdμ2 dμ1 .

This is our Fubini formula for the Pettis operators. Hence, in particular, taking ψ1 = χ A1 and ψ2 = χ A2 , where A1 ∈ 1 , A2 ∈ 2 , one has  (P)

A1 ×A2

  Fdμ = (P) A1

Fdμ2 dμ1 . A2

Remark 5.1 For any ψ1 ∈ L ∞ (μ1 ) and ψ2 ∈ L ∞ (μ2 ) define 1 and 2 in L ∞ (μ) by 1 (s1 , s2 ) = ψ1 (s1 )χ S2 (s2 ) and 2 (s1 , s2 ) = ψ2 (s2 )χ S1 (s1 ) for all (s1 , s2 ) ∈ S. Then, as easily seen, 1 = E 1∗ (ψ1 ), 2 = E 2∗ (ψ2 ) and,  (P)

ψi

123

 Fi dμi = (P)

i

Fdμ, i = 1, 2.

Extension of Pettis integration

6 Pettis operators versus vector-valued harmonic functions In this section, X is a real Banach space, and D and T are the open unit disk and the unit circle in the complex plain C, respectively. We consider T with its normalized Haar measure μT , and write L 1 (T) = L 1 (μT ) and L ∞ (T) = L ∞ (μT ). For z ∈ D, we denote by P : D → R+ the harmonic function P(z) =  1+z 1−z and by Pz the Poisson kernel on T as Pz (η) = P(z η) ¯ =

1 − |z|2 (η ∈ T). |1 − z η| ¯2

and, as usual, for f : D → X we write fr (ξ ) = f (r ξ ) for ξ ∈ T and 0 < r < 1. In particular Pr (ξ η) ¯ = Pr ξ (η) = Pr ξ (η). Of course for φ ∈ L 1 (T), 0 < r < 1, ξ ∈ T and z = r ξ ,  Pr ∗ φ(ξ ) = Pr (ξ η)φ(η)dμ ¯ T (η) T φ(η)Pr ξ (η)dμT (η) = T

= φ, Pz  = P(φ)(z) where P(φ) is the Poisson integral of φ. 1 (D) stand for the spaces of harmonic functions φ in the unit disc Let h 1 (D) and Hmax such that φh 1 = sup0
(1)

To see that P(T ) is a vector-valued harmonic function it suffices to observe that x ∗ P(T ) is harmonic for any x ∗ ∈ X ∗ . This easily follows by noticing that for each x ∗ ∈ X ∗ and T ∈ L(C(T), X ) one has that x ∗ T ∈ (C(T))∗ , and hence it can be identified with a measure νx ∗ ∈ M(T), and clearly P(νx ∗ )(z) = x ∗ , P(T )(z) is harmonic. Proposition 6.1 Let F ∈ P(X ∗ , L 1 (T)). Then (a) f = P(F ∗ ) ∈ wh 1 (D, X ). (b) limr →1 P fr = F in the strong operator topology, i.e. limr →1 P fr (x ∗ ) = F(x ∗ ) for all x ∗ ∈ X ∗.

123

O. Blasco, L. Drewnowski

(c) If F is compact, then limr →1 P fr = F in the uniform norm of operators. Proof (a) Notice that F ∗ : L ∞ (T) → X is bounded and weak∗ –weakly continuous. In particular, F ∗ ∈ L(C(T), X ). In [1, Theorem 9] it was shown that actually wh 1 (D, X ) is isometrically isometric to L(C(T), X ) via the Poisson integral defined by (1). In particular, since Pz ∈ C(T) for each z ∈ D one gets that f = P(F ∗ |C(T) ) ∈ wh 1 (D, X ). (b) Since f = P(F ∗ ) : D → X is harmonic then fr ∈ C(T, X ), and in particular, Bochner (and hence Pettis) integrable. Let us see that P fr = Pr ∗ F. For each 0 < r < 1, ψ ∈ L ∞ (T) and x ∗ ∈ X ∗ , since Pr = P˜r one has (Pr ∗ F)(x ∗ ), ψ = F(x ∗ ), Pr ∗ ψ = x ∗ , F ∗ (Pr ∗ ψ). On the other hand, for each 0 < r < 1, the function ξ → Pr ξ belongs to C(T, L ∞ (T)). Hence, for each ψ ∈ L ∞ (T),  Pr ∗ ψ = ψ(ξ )Pr ξ dμT (ξ ) ∈ L ∞ (T) T

and therefore



∗

F (Pr ∗ ψ) =

T

F

∗



Pr ξ ψ(ξ )dμT (ξ ) =

Now for each x ∗ ∈ X ∗ we have ∗





T

f (r ξ )ψ(ξ )dμT (ξ ).



x ∗ , f (r ξ )ψ(ξ )dμT (ξ )  f (r ξ )ψ(ξ )dμT (ξ ). = x ∗ ,

(P fr )(x ), ψ =

T

T

Combining the previous formulas one has (Pr ∗ F)(x ∗ ), ψ = x ∗ , F ∗ (Pr ∗ ψ)  ∗ = x , f (r ξ )ψ(ξ )dμT (ξ ) T

= (P fr )(x ∗ ), ψ. From this one gets lim P fr (x ∗ ) = lim F(x ∗ ) ∗ Pr = F(x ∗ )

r →1

x∗

r →1

X ∗.

for any ∈ (c) It was shown in [12, Proposition 5] that Pr ∗ F ∗ → F ∗ whenever F (or equivalently F ∗ ) is assumed to be compact.  1 (D, X ) then there exists F ∈ P(X ∗ , L (T)) such that f = Theorem 6.2 If f ∈ w Hmax 1 ∗ P(F ).

Proof Let us define F : X ∗ → L 1 (T) by F(x ∗ ) = (x ∗ f )◦ where we have used the inclusion 1 (D) ⊆ L (T) given φ → φ ◦ . Hence for 0 < r < 1 and ξ ∈ T, Hmax 1 x ∗ , f (r ξ ) = Pr ∗ (x ∗ f )◦ (ξ ). Note that (x ∗ f )(z) = P((x ∗ f )◦ )(z) = Pz , F(x ∗ ) = F ∗ (Pz ), x ∗ .

123

Extension of Pettis integration

Hence for x ∗ ∈ X ∗ and z ∈ D one has F ∗ (Pz ) ∈ X and F ∗ (Pz ) = f (z). We shall show that F ∈ P(X ∗ , L 1 (T)). Let (rn ) be a sequence converging to 1 and consider Fn = Prn ∗ F and m n = m Fn the corresponding measures. Notice that Fn∗ = F ∗ ∗ Prn : L ∞ (T) → X ∗∗ satisfies that Fn∗ (ψ), x ∗  = ψ, Prn ∗ F(x ∗ )  = ψ(ξ )(x ∗ f )(rn ξ )dμT (ξ )  = x ∗ , ψ(ξ ) f (rn ξ )dμT (ξ ).  Hence Fn∗ (ψ) = ψ(ξ ) f (rn ξ )dμT (ξ ) ∈ X. This gives that Fn are Pettis operators. On the 1 (D, X ) for each x ∗ ∈ X ∗ , other hand, since f ∈ Hmax limx ∗ , f (rn ξ ) = (x ∗ f )◦ (ξ ), for μT - a.e. ξ ∈ T n

and for each ψ ∈ L ∞ (T) one has sup |ψ x ∗ ◦ frn | ∈ L 1 (T). n

This allows us to conclude that Fn∗ (ψ) is weakly convergent to F ∗ (ψ) for any ψ ∈ L ∞ (T). In particular, m n converges pointwise weakly to m. Finally, invoking Theorem 3.2, we obtain that F is a Pettis operator. 

7 Application to vector-valued Hardy spaces: Radial limits Pettis operators In this section, X is a complex Banach space and we shall denote by H (D) and H (D, X ) the spaces of all analytic complex—and X -valued functions on D, respectively. In the previous section functions were considered to be real-valued and spaces over the real field, but similar results hold for complex valued harmonic functions and complex spaces. The reader should simply notice that the duality between L 1 (T)) and L ∞ (T) should be replaced by φ, ψ =  φ(ξ )ψ(ξ )dμ(ξ ) for φ ∈ L 1 (T) and ψ ∈ L ∞ (T). Furthermore, we let H 1 (D) = h 1 (D) ∩ T H (D). In what follows, we tacitly use some standard facts concerning functions in H 1 (D) 1 (D) (see [11, Theorem 1.9]). that can be found in [11,14]. It is known that H 1 (D) ⊂ Hmax 1 In particular we know that if φ ∈ H (D) then there exists limr →1 φ(r ξ ) = φ ◦ (ξ ) a.e, φ ◦ ∈ L 1 (T), φ ◦ 1 = φ H 1 and φr − φ ◦ 1 → 0 as r → 1. We denote by w H 1 (D, X ) the weak vector-valued Hardy space on D, consisting of all analytic functions f : D → X such that x ∗ f belongs to H1 (D) for all x ∗ ∈ X ∗ ; see [1,15] and the references therein. It is a Banach space under the norm  f w,1 := sup{x ∗ f  H1 (D) : x ∗   1}. For z ∈ D, we define the Cauchy kernel C z as C z (ξ ) =

1 , z ∈ D, ξ ∈ T 1 − z ξ¯

and it is known (see [11, Theorem 3.6]) that ψ(z) = ψ, C z  for any ψ ∈ H 1 (D). The aim of this section is to relate the space w H 1 (D, X ) with the subspace of P(X ∗ , L 1 (T)) formed by analytic operators. For each integer n, we let en (ξ ) = ξ −n for ξ ∈ T and for any bounded linear operator T : L ∞ (T) → X we consider the nth Fourier coefficient of T , to be denoted Tˆ (n) = T (en ) .

123

O. Blasco, L. Drewnowski

Definition 7.1 Let F ∈ P(X ∗ , L 1 (T)). We shall say that F is an analytic Pettis operator ∗ (n) = F ∗ (en ) = 0 for n < 0. We shall write Pa (X ∗ , L 1 (T)) for the subspace whenever F of analytic Pettis operators. 1 (D, X ) we can apply Theorem 6.2 to get that Using the fact that w H 1 (D, X ) ⊂ w Hmax ◦ ∗ ∗ ◦ ∗ the equality f (x ) = (x f ) for every x ∈ X ∗ defines a Pettis operator in P(X ∗ , L 1 (T)).

Definition 7.2 Let f ∈ w H 1 (D, X ). We call f ◦ the radial boundary operator for the function f . Theorem 7.3 The spaces w H 1 (D, X ) and Pa (X ∗ , L 1 (T)) are isometrically isomorphic. Proof Let f ∈ w H 1 (D, X ). We already know that f ◦ ∈ P(X ∗ , L 1 (T)). Let us show that f ◦ is analytic and  f w.1 =  f ◦ . Obviously, for each 0 < r < 1, fr ∈ w H1 (D, X ) and (x ∗ fr )◦ (ξ ) = (x ∗ f )(r ξ ) for all ξ ∈ T and x ∗ ∈ X ∗ . Thus fr◦ is the Pettis operator from X ∗ to L 1 (T) associated with the continuous function fr . For each x ∗ ∈ X ∗ and n < 0, since x ∗ f ∈ H 1 (D), we obtain  ∗ f (n) = 0. x ∗ , ( fr◦ )∗ (en ) = x ∗ , fr (ξ )en (ξ )dμT (ξ ) = x r T

( fr◦ )∗ (en )

Due to Proposition 6.1 → ( f ◦ )∗ (en ) weakly for any n ∈ Z and we conclude that ◦ f is indeed analytic. Let us show now that  f w.1 =  f ◦ . Of course fr◦ = Pr ∗ f ◦ and therefore  fr◦  ≤  f ◦  for any 0 < r < 1. Moreover, for x ∗   1,  f ◦ (x ∗ )1 = lim  fr◦ (x ∗ )1  sup  fr◦ (x ∗ )   f w,1 . r →1

0≤r <1

 f ◦ .

This shows that  f w.1 = Conversely, let F ∈ Pa (X ∗ , L 1 (T)). In particular, due to Proposition 6.1 f (z) = ∗ F (Pz ) ∈ wh 1 (D, X ). To obtain that f ∈ w H 1 (D, X ) it suffices to show that f ∈ H (D, X ). ∗ )(n) = 0 for n < 0. This shows that  Note that F(x   ∗ F(x )(ξ )Pz (ξ )dm(ξ ) = F(x ∗ )(ξ )C z (ξ )dm(ξ ) ∈ H 1 (D). T

T

Hence f (z) = F ∗ (C z ) for z ∈ D is an analytic function.



Remark 7.4 If instead of w H 1 (D, X ) one considers the usual Hardy space H 1 (D, X ), and X has the analytic Radon-Nikodym property (a-RNP) (see [5,8]), then for each f ∈ H 1 (D, X ) the norm radial limit function f • of f exists m-a.e. on T and is Bochner integrable. In this case, of course, one has f ◦ (x ∗ ) = x ∗ f • m-a.e. for all x ∗ ∈ X ∗ , and the Pettis integrals involving f ◦ may well be understood as Bochner integrals involving f • . Acknowledgements The first author has been supported by grant MTM2014-53009-P (MINECO, Spain). Both the authors are grateful to A. Michalak, K. Musiał, M. Nawrocki and W. Ruess for their interest in this work and numerous helpful comments. We also thank the referee for his/her interesting comments which helped us to improve the paper.

References 1. Blasco, O.: Boundary values of vector-valued harmonic functions considered as operators. Studia Math. 86, 19–33 (1987)

123

Extension of Pettis integration 2. Blasco, O.: Convolution of operators and applications. Math. Z. 199, 109–114 (1988) 3. Blasco, O.: Hardy spaces of vector-valued functions: duality. Trans. Am. Math. Soc. 308, 495–507 (1988) 4. Blasco, O.: Boundary values of functions in vector-valued Hardy spaces and geometry of Banach spaces. J. Funct. Anal. 78, 346–364 (1988) 5. Bukhvalov, A.V., Danilevich, A.A.: Boundary properties of analytic and harmonic functions with values in Banach spaces. Mat. Zametki 31, 203–214 (1982). English translation: Math. Notes 31, 104–110 (1982) 6. Collins, H.S., Ruess, W.: Weak compactness in spaces of compact operators and of vector valued functions. Pac. J. Math. 106, 45–71 (1983) 7. Diestel, J., Uhl, J.J.: Vector Measures, Mathematical Surveys, vol. 15. Am. Math. Soc, Providence (1977) 8. Dowling, P.N., Edgar, G.A.: Some characterizations of the analytic Radon–Nikodqm property in Banach spaces. J. Funct. Anal. 80, 349–357 (1988) 9. Drewnowski, L.: On the Dunford and Pettis integrals. In: Proceedings of Conference “Probability and Banach Spaces”, June 17–21, 1985, Zaragoza (Spain). Lecture Notes in Mathematics, vol. 1221, pp. 1–15 (1986) 10. Drewnowski, L., Lipecki, Z.: On vector measures which have everywhere infinite variation or noncompact range. Diss. Math. 339, 3–39 (1995) 11. Duren, P.L.: Theory of H p Spaces. Academic Press, New York (1970) 12. Freniche, F., García-Vázquez, J.C., Rodrguez-Piazza, L.: The failure of Fatou’s theorem on Poisson integrals of Pettis integrable functions. J. Funct. Anal. 160(1), 28–41 (1998) 13. Graves, W.H., Ruess, W.: Compactness and weak compactness in spaces of compact range vector measures. Can. J. Math. 36, 1000–1020 (1984) 14. Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981) 15. Laititila, J., Tylli, H.-O., Wang, M.: Composition operators from weak to strong spaces of vector-valued analytic functions. J. Oper. Theory 62, 281–295 (2009) 16. Michalak, A.: On the Fubini theorem for the Pettis integral of bounded functions. Bull. Pol. Acad. Sci. Math. 49, 1–14 (2001) 17. Musiał, K.: The weak Radon–Nikodým property in Banach spaces. Studia Math. 64, 151–174 (1979) 18. Musiał, K.: Topics in the theory of Pettis integration. Rend. Istit. Mat. Univ. Trieste 23, 177–262 (1991) 19. Musiał, K.: Pettis integral. In: Pap, E. (ed.) Handbook of Measure Theory, Chap. 12, pp 531–586. North Holland (2002) 20. Rybakov, V.I.: Conditional mathematical expectation for functions that are integrable in the sense of Pettis. Mat. Zametki 10, 58–64 (1971). (in Russian) 21. Talagrand, M.: Pettis integral and measure theory. Mem. Am. Math. Soc. 307, 224 (1984) 22. Thomas, E.: The Lebesgue–Nikodým theorem for vector valued radon measures. Mem. Am. Math. Soc. 139, 101 (1974)

123