On the Banach algebra ℬ(l_p(α))

Abstract

We give some properties of the Banach algebra of bounded operatorsℬ(lp(α))for1≤p≤∞, wherelp(α)=(1/α)−1∗lp. Then we deal with the continued fractions and give some properties of the operatorΔhforh>0or integer greater than or equal to one mappinglp(α)into itself forp≥1real. These results extend, among other things, those concerning the Banach algebraSαand some results on the continued fractions.

On the set of $ \alpha $, $ p $-bounded variation of order $ h $

“…So A ∈ (bv h (α), bv k u (β)) if and only if ∆ k A ∈ (l 1 (α), l u (β)). From the expression of D 1/β ∆ k AD α in the proof of Theorem 10(i)(a), we conclude that D 1/β ∆ k AD α ∈ (l 1 , l u ) if and only if (10) holds. (ii) The condition A ∈ (bv h (α), l u (β)) means that the series defined by A n (∆ −h X) are convergent for all X ∈ l 1 (α) and for all n and…”

“…which is a Banach space with the norm [5][6][7][8][9][10][11][12]. If s α = s β we get the Banach algebra with identity S α,α = S α , see [5,8,11].…”

“…We obtain the following Lemma 6. ( [10]) Let α ∈ U + . (i) For any given real h > 0, the condition bv h (α) = l 1 (α) is equivalent to…”

Matrix Transformations and Quasi-Newton Methods

“…These matrices are used in many applications, let us cite for instance the case of continued fractions (cf. [3]), or the finite differences method, (cf. [8]).…”