SEQUENCES OF LINEAR OPERATORS RELATED TO CES`ARO - CONVERGENT SEQUENCES

. Given a Ces`aro-convergent sequence of real numbers ( a n ) n ∈ N , a sequence ( ϕ n ) n ∈ N of operators is deﬁned on the Banach space R ( I, F ) of regular functions deﬁned on I = [0 , 1] and having values in a Banach space F ,


INTRODUCTION
Let (a n ) n∈N be a sequence of real numbers.It will be called Cesàro-convergent if the sequence of its Cesàro (arithmetic) means is convergent, i.e. lim n→∞ a 1 + . . .+ a n n ∈ R.
For x ∈ R, x will denote the greatest integer number n ≤ x (the integer part of x).
Given the interval I = [0, 1] and a Banach space F = {0} , we denote by B(I, F ) the Banach space of bounded functions f : I → F endowed with the sup norm.The space B(I, F ) contains as a subspace the set of "step-functions" E(I, F ) = {f : I → F : ∃ t 0 , . . ., t n ∈ I, t 0 = 0 < t 1 < . . .< t n = 1, ∃ u k ∈ F so that f | (t k−1 ,t k ) = u k , k = 1, . . ., n}.In fact each f ∈ E(I, F ) is a finite sum of functions having the form χ [α,β] • u, where 0 ≤ α ≤ β ≤ 1, u ∈ F and χ [α,β] is the characteristic function of the interval [α, β].We denote by R(I, F ) the Banach space of regular functions (which admit side limits at each t ∈ I), endowed with the uniform norm f = sup t∈[0,1] f (t) .We mention that R(I, F ) is the closure in B(I, F ) of the subspace E(I, F ), and it contains the Banach space of continuous functions C(I, F ).More details on these spaces of functions are to be found in [3, p. 137].
We define a sequence of operators associated to (a n ) n∈N , namely ϕ n : R(I, Proposition 1.The operator ϕ n is linear and continuous, and its norm is given by Proof.The linearity is straightforward.Because hence ϕ n is also continuous.To obtain the norm of ϕ n , we use the inequality (3) and the function where u ∈ F and u = 1.We have f 0 ∈ E(I, F ) ⊆ R(I, F ), f 0 = 1 and |a k | • u, hence the equality (2) follows.

MAIN RESULTS
We are interested in finding conditions on the sequence (a n ) n∈N in order to obtain the convergence of the sequence of linear operators (1).The theorem below guarantees the convergence of (ϕ n (f )) n∈N for each regular function f ∈ F(I, F ). Beside the condition of Cesàro-convergence for (a n ) n∈N , the boundedness of a certain sequence related to this is imposed.Theorem 2. Let there be given a regular function f ∈ R(I, F ) and a sequence (a n ) n∈N of real numbers satisfying the conditions: Then the sequence (ϕ n (f )) n∈N is convergent and Proof.At first we shall prove (4) for functions f of the form We have If α = 0 the conclusion follows obviously.
For α > 0 we denote a * n = a 1 +...+an n and we write the two sums in the above formula as where We have lim We consider now the general case f ∈ R(I, F ).The sequence |a 1 |+...+|an| n n∈N being bounded, let us choose M such that |a 1 |+...+|an| n ≤ M for each n ∈ N; let also ε > 0 be an arbitrary constant.From the definition of the space R(I, F ) it follows the existence of the functions f i , i = 1, . . ., p of the type described in (5), with f − p i=1 f i < ε.We have The norm of ϕ n , as given by ( 2), is Taking into account the first part of the proof, for each i = 1, . . ., p there exists and the inequalities (6), ( 7) and (8) imply that It follows that the conclusion holds also for the general case f ∈ F(I, F ).
Remark 1.The Cesàro-convergence of (a n ) n∈N in Theorem 2 does not necessarily imply the boundedness of |a 1 |+...+|an| n n∈N .For example, let the sequence be given by Then The norm of the operators ϕ n in the space C(I, F ) is the same as in (1).Indeed, in the proof of Proposition 1, the function f 0 can be modified to a continuous and piecewise affine one which takes also the values (sign a k )u on the points k n , k = 1, . . ., n. From the principle of uniform boundedness [4, p. 66] the second conclusion follows.
Remark 2. Using a principle of condensation of singularities [2], one can prove that the convergence in (4) does not hold for "typical" continuous functions.Even stronger principles of condensation of singularities [1] may be applied.
In what follows we shall prove that for the class of continuous functions having also a continuous derivative, the condition of boundedness of the sequence |a 1 |+...+|an| n n∈N is no longer necessary.In this setting, the principle of uniform boundedness does not work, because C 1 (I, F ) endowed with the uniform norm is not a Banach space.The norm of ϕ n is still the same.In this case we have Theorem 4. Let there be given a function f ∈ C 1 (I, F ) and a sequence (a n ) n∈N of real numbers which is Cesàro-convergent to a ( lim Proof.We write ϕ n (f ) successively as We bring now into the scene the continuous function g given by g(t) = tf (t) and express ϕ n (f ) in the form Applying Theorem 2 for the function g and for the sequence (a * n ) n∈N convergent to a, for which obviously lim (the last equality is a consequence of an integration by parts).The function f being uniformly continuous on I, given ε > 0 and n sufficiently large, we obtain as a consequence of a mean theorem [3, p. 154] where M is a upper bound for the convergent sequence (|a We take the limit in (10) and get the conclusion.
As an application of Theorem 2 we obtain a somehow surprising result, proved directly for differentiable functions with bounded derivative in [5]: For each a ∈ [0, 1] , there exist ε n ∈ {0, 1} such that lim n→∞ To prove this equality, we choose ε n = a n = (n + 1)a − na , n ∈ N which satisfy ε n ∈ {0, 1} and lim n→∞ a 1 +...+an n = a.Open question.It would be interesting to find out if the conclusion of Theorem 2 also holds for a class of functions more general than the regular ones as, for example, the Riemann integrable real-valued functions.For the class of Lebesgue integrable functions the result does not hold, as the function of Dirichlet type f : I → F = R, f (t) = arbitrary, t ∈ [0, 1] ∩ Q 0, otherwise shows.

Theorem 3 .= a); 2 .
The condition of Cesàro-convergence imposed to the sequence (a n ) n∈N in Theorem 2 is a natural one and cannot be relaxed, neither the boundedness of the sequence |a 1 |+...+|an| n n∈N .In fact, Theorem 2 does admit the following converse: Let (ϕ n ) n∈N be the sequence (1) of linear operators associated to the sequence of real numbers(a n ) n∈N .If lim n→∞ ϕ n (f ) exists for every f ∈ C(I, F ) ⊆ R(I, F ), then: 1. (a n ) n∈N is Cesàro-convergent toa ( lim n→∞ a 1 +...+an n the sequence |a 1 |+...+|an| n n∈N is bounded.Proof.The first conclusion follows by taking f