Banach space valued Mean Periodic Functions.

For a Banach space B over the complex field C, let C(R, B) denote the set of all continuous functions defined on the real line R taking values in B with the compact convergence topology. When B = Cwe write C(R) for C(R, C). For a function [empty set] in C(R, B) let T ([empty set]) denote the closure in C(R, B) of the span of all translates of [empty set].

Definitiom 1. A function [empty set] in C(R, B) is said to be mean periodic if T ([empty set]) [not equal to] C(R, B).

We prove the following theorem.

Theorem A Banach space B is separable if and only if not all functions in C(R, B) are mean periodic.

Preliminaries Let B* be the dual of B and let [B*.sub.s] denote the space B* along with the weak * topology on it. Let [mu] be a countably additive, regular, vector valued, Borel measure on R taking values in [B *.sub.s]. For a Borel set b' [??]R, set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This is the norm in the Banach space B*. For a Borel set b in R let P(b) be the set of all finite Borel partitions of b. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where the sum is taken over {b'} which form a finite Borel partition of b, for all Borel sets bin B; then [mu] is said to be of bounded variation.

Let M(R,[B*.sub.s]) be the space of all vector valued Borel measures [mu] on R taking values in [B*.sub.s] with the following properties:

1. [mu] is countably additive and regular,

2. [mu] has compact support in R,

3. [mu] is of bounded variation,

4. there exists a constant C > 0 such that for any Borel set b of R, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the norm in the Banach space B*.

Let [mu] [member of] M(R,[B*.sub.s]) with support K. Let [b.sub.1], [b.sub.2], ... [b.sub.n] be a Borel partition of K and [[upsilon].sub.1], [[upsilon].sub.2], ... [[upsilon].sub.n] be arbitrary elements of B. Then [empty set] = [[SIGMA].sub.i=1.sup.n]] [[upsilon].sub.i][chi][b.sub.i] is a simple function from R to B. Then [integral] [empty set]d[mu] is defined as

[integral] [empty set]d[mu] = [[SIGMA].sub.i=1.sup.n]<[v.sub.i], m([b.sub.i])>

Now let [empty set]: R [right arrow] B be any continuous function and let [mu] and K be as before. For any [epsilon] > 0 we can get a finite Borel partition {[b.sub.i]} of K such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all x, y in [b.sub.i] for all i. Choose any point [x.sub.i] [member of] [b.sub.i]. Define the simple function [[empty set].sub.[epsilon] = [SIGMA][empty set]([x.sub.i])[chi][b.sub.i]. Then [integral] [empty set]d[mu] = [lim.sub.[epsilon][right arrow]0] [integral] [[empty set].sub.[epsilon]]d[mu]. This limit exists and is independent of the choice of the Borel partition {[b.sub.i]} and also of the choice of the points {[x.sub.i]} (see (2)).

For a function f [member of] C(R) and a measure [mu] [member of] M(R,[B*.sub.s]) we define the integral [integral] fd[mu] in a similar way: for a simple function f = [[SIGMA].sub.i] [y.sub.i][chi][b.sub.i] where [y.sub.i] [member of] C and {[b.sub.i]} is a Borel partition of K, the support of [mu], define [integral] fd[mu] = [[SIGMA].sub.i][y.sub.i][mu]([b.sub.i]) [member of] [B*.sub.s]. For f [member of] C(R), [integral] fd[mu] = [lim.sub.[epsilon][right arrow]0] [integral] [f.sub.[epsilon]]d[mu], where [f.sub.[epsilon] are simple functions defined as before. Therefore [integral] fd[mu] is an element of [B*.sub.s].

Singer's Theorem (see (2)): The dual of C(R;B) is the space M(R,[B*.sub.s]).

Using Hahn Banach Theorem it is easy to see that a function [empty set]is mean periodic if and only if there exists a nonzero measure [mu] in M(R,[B*.sub.s]) such that [empty set] * [mu] = 0.

Proof of the theorem Suppose B is not separable. Let [empty set] be any function in C(R, B). Since R is separable, [empty set](R)is separable. Let [B.sub.1] be the closure of the space generated by [empty set](R). It is easy to see that [B.sub.1] is also separable. More-over, range of any function in T([empty set]) is contained in [B.sub.1]: Since [B.sub.1] is a proper subspace of B we can find a function in C(R,B) whose range is not contained in [B.sub.1]. Hence T([empty set]) [not equal to] C(R,B).

Conversely, if B is separable we will show that there exists a function in C(R,B) which is not mean periodic. First we will take B to be [l.sup.1], the space of absolutely summable sequences in C. For each n [empty set] N, let [e.sub.n] = (0, ... 1, 0, ... ) be the sequence in [l.sup.1] where 1 occurs only at the n th entry. For each n [member of] N we define an element [{[a.sub.j](n)}.sub.j=1.sup.[infinity]] in [l.sup.1] and a sequence of real numbers {[[lambda].sub.j](n)}.sub.j=1.sup.[infinity]] such that:

1. [a.sub.j](n) is nonzero for all j and n.

2. If we denote [[SIGMA].sub.j=1.sup.[infinity]] [absolute value of [a.sub.j](n)] = a(n), then [[SIGMA].sub.n=1.sup.[infinity]] a(n) < [infinity] (that is, {[a(n)}.sub.n=1].sup.[infinity]] is also in [l.sup.1].

3. [[lambda].sub.j](n) are all different. that is, if [[lambda].sub.j](n) = [[lambda].sub.i](m) then j = i and n = m:

4. The sequence [{[[lambda].sub.j][(n)}.sub.j=1.sup.[infinity]]] converges to a real number, say [lambda](n).

For a real number s [member of] R define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [f.sub.n]: R [right arrow] C is continuous. Put [empty set](s) = [[summation].sub.n=1.sub.[infinity]] [f.sub.n](s)[e.sub.n]. It is easy to see that [empty set]: R [right arrow] [l.sup.1] is continuous. The series [[summation].sub.n=1.sup.[infinity]] [f.sub.n](s)[e.sub.n] converges to the function [empty set] uniformly on R. We will show that for this function [empty set] [member of] C(R,[l.sup.1]), T([empty set]) = C(R,[l.sub.1]).

Suppose not. Then by Singer's theorem there exists a nonzero measure [mu] [member of] M(R,[l.sub.s.sup.[infinity]]) such that [empty set] * [micro] = 0. Let K be the compact support of [mu].

0 = [phi] * [mu] = [[infinity][summation over(n=1)]]([f.sub.n][e.sub.n]*[mu]]) = [[infinity].summation over(n=1)] < [e.sub.n],[f.sub.n]*[mu]>.

For the last equality refer to theorem III.2.6 in (2). For n [member of] Nand a Borel set b in R, put [[mu].sub.n](b) =< [e.sub.n], [mu](b) >. Then [[mu].sub.n] is a countably additive, regular complex measure with compact support contained in K(see the proof of Singer's theorem in (2)). We can write [mu](b) = ([[mu].sub.1](b), [[mu].sub.2](b), ...) [member of] [l.sub.s.sup.[infinity]]. For any f [member of] C(R) we have < [e.sub.n], [integral] fd[mu] >=[integral] fd[[mu.sub.n]: this is true for characteristic functions so also for simple functions. By the limiting process one can prove it for any continuous function on R. So by (1) we get [SIGMA] [f.sub.n] *[[mu].sub.n] = 0. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCABLE IN ASCII]

Since [^.[mu].sub.n]([[lambda].sub.j](n)) are uniformly bounded by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the left hand side is an almost periodic function. Because [a.sub.j](n) [not equal to] 0 for all j, n, this implies [^.[[mu]].sub.nj](n)) = 0. The zero's of the holomorphic function [^.[[mu]].sub.n] have a limit point [lambda](n): Therefore [[mu].sub.n] = 0 for all n and hence [mu] = 0, a contradiction.

Now let B be an arbitrary separable Banach space. Since B is separable we can find a countable set {[h.sub.n]} in B such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all nand the subspace H generated by {[h.sub.n]} is dense in B. Define a function [psi]: R [right arrow] Bby [psi] (s) = [[SIGMA].sub.n=1.sup.[infinity]] [f.sub.n](s)[h.sub.n] where [f.sub.n] 's are defined as before .Then this series converges in Band the function [psi] is continuous. Let g [member of]C(R) be any function. Then we claim that [gh.sub.1] [member of] T([psi]):

Since t([empty set]) = C(R; [l.sup.1]) we know that [ge.sub.1] is a limit (in C(R,[l.sub.1])) of a sequence of linear combinations of translates of [empty set]. That is, [ge.sub.1] = [lim.sub.m[right arrow][infinity]] [[PHI].sub.m] where for each m [member of] N, [[PHI.sub.m] is a finite sum [[PHI.sub.m] = [SIGMA][c.sub.i][empty set][y.sub.i]. Define [[PSI].sub.m] = [SIGMA][c.sub.i][[psi].sub.yi] for each m: Then [gh.sub.1] = [lim.sub.m[right arrow][infinity]] [[PSI].sub.m]: Let [epsilon] > 0 and C be a compact subset of R. For any s [member of] C

[MATHEMATICAL EXPRESSION NOT REPRODUCABLE IN ASCII]

for all m suffciently large.

Similarly we can prove that any finite sum [SIGMA][g.sub.i][h.sub.i] in C(R) [cross product] H is in T([psi]). Since T([psi]) is closed and C(R) [cross product] His dense in C(R) [cross product] B, we get C(R) [cross product] B [??] T([psi]): But C(R) [cross product] Bis dense in C(R,B) (see [2]). Hence T([psi]) = C(R,B).

Received December 22, 2005, Accepted April 11, 2006.

References

(1) J. P. Kahane, Lectures on Mean Periodic Functions, Tata Institute of Fundamental Research, Mumbai, 1959.

(2) J. Schmets, Spaces of Vector Valued Continuous Functions, Lecture Notes in Mathematics 1003, Springer Verlag, Berlin, 1983.

(3) F. Treves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967.

K. GowriNavada

Institute of Applied Mathematics, Faculty of Science Anshan University of Science and Technology Anshan 114044, Liaoning, China

Received December 22, 2005, Accepted April 11, 2006.