OPERATORS COMMUTING WITH THE SHIFT ON SEQUENCE SPACES

A complete characterization of shift-invariant operators that are isomorphisms is given in certain sequence spaces. Also given is a sufficient condition for an operator commuting with a shift-invariant operator to be shift invariant.


Introduction.
There is a long-standing interest in linear continuous operators commuting with the right shift operator, weighted or not.The study of these operators is closely related to the study of operators commuting with the differentiation operator (weighted left shift).Several authors have treated topics connected with these operators; for instance, for unweighted shifts, a good reference is [10], while a good source for weighted shift operators is the papers [13,14,15] and the book of Halmos [9].
The concrete problem of determining the spectrum of a weighted right shift operator was studied mainly by Gellar [3,4,5].There is a strong relationship between the spectrum of such an operator and the question of whether or not an operator which commutes with it is an isomorphism.In [17] the spectrum of the differentiation operator on certain sequence spaces was computed directly, although it could have been deduced using [4, theorem 10].
In the so-called umbral calculus appears the concept of a delta operator [2], which is invariant by differentiation and so connected with shift-invariant operators.In fact, in [2], the relationship between Sheffer operators, differentiation-invariant operators, and shift-invariant operators, as well as the importance of the spectrum in the characterization of isomorphisms, was shown.Similar questions were studied in the papers [6,7,8]; for differentiation-invariant operators, see [1,2,12,16].
In the present paper, we consider shift-invariant operators on infinite power series spaces.Necessary and sufficient conditions for an operator to be continuous are given for any infinite power series space.Also given is a complete characterization of isomorphisms when the space is nuclear and a projective limit of Banach algebras.In addition, we give a sufficient condition for an operator commuting with a shift-invariant operator also to be shift invariant.

Definitions and notations.
Let Λ ∞ (α) be the infinite power series space with the usual topology, that is, x n e kαn < ∞, k = 1, 2, 3,... , x n e kαn , (2.1) where is a Fréchet space with a canonical basis, noted by (e n ); its topological dual can be identified with the sequence space and the coordinate operators are continuous.These spaces are nuclear if and only if ∀k, ∃N(k) such that e kαn e N(k)αn ∈ l 1 . (2.3) They are projective limits of the Banach spaces l 1 (e kαn ), k ∈ N, which are Banach algebras (with the convolution multiplication) if and only if there exists C > 0 such that α n+m ≤ C +α n +α m , for every m and n.Well-known examples of nuclear infinite power series spaces that are projective limits of Banach algebras are the space H(C) of entire functions on the complex plane (in this case, α n = n) and s, the space of sequences rapidly decreasing to zero (for α n = ln n).
S denotes the shift operator S(e n ) = e n+1 ; it is assumed that S is a continuous operator from Λ ∞ (α) to Λ ∞ (α), so we have the condition sup α n+1 /α n < ∞.As lim inf α s /s = 0, or is equal to a number a > 0, or ∞, the result follows.

Continuous operators commuting with S Theorem 3.1. A linear operator T is continuous on Λ ∞ (α) and commutes with S if and only if
It is obvious that two operators commuting with S commute with each other [4, Corollary 1], but it is not true, in general, that if T 1 commutes with T 2 and T 2 commutes with S, it follows that T 1 commutes with S; take T 2 = S 2 and T 1 given by an infinite two-block matrix a 00 a 01 a 10 a 11 , where a 01 , a 10 are different from zero and a 00 ≠ a 11 .We show in the following proposition that for certain operators T 2 the result is true.Theorem 3.5.Let T 2 be a continuous linear operator from Λ ∞ (α) to Λ ∞ (α) commuting with S, T 2 = ∞ s=0 t s S s , verifying the condition that the sequence t (0) = (t 1 ,t 2, t 3 ,...), t (1) = (0,t 1 ,t 2, t 3 ,...), t (2) = (0, 0,t 1 ,t 2, t 3 ,...),... is a basis of the power series space Λ ∞ (α).
Then any linear continuous operator T 1 commuting with T 2 commutes with S.
Proceeding in an analogous way, the third set of equations can be written as and we get a 21 = a 10 , a 22 = a 11 , a 23 = a 24 = ••• = 0. Thus the operator T 1 is given by a matrix that commutes with the matrix of S.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

First
Round of ReviewsMarch 1, 2009 Condition (3.2) can be simplified for certain infinite power series spaces as it is shown in the propositions below whose proofs are omitted.Assume that Λ ∞ (α) is a nuclear space.Then condition (3.2) is equivalent to the following one: for all k, there exists N(k) and there exists C(k) > 0 such that Assume that Λ ∞ (α) is a nuclear space and a projective limit of Banach algebras.Then conditions (3.2) and(3.3)areequivalent to saying that the sequence (t s ) is an element of Λ ∞ (α).that if T commutes with S, T (x s ) = (t s ) * (x s ), where * represents the convolution product, (or T ( x s z s ) = t s z s • x s z s ) [4, theorem 2], then we have the following proposition.If T is a continuous linear operator on Λ ∞ (α) commuting with S, then the function φ(z) = t s z s is holomorphic on a disk whose radius (finite or not) is greater than or equal to 1.
Proof.Observe that the matrix (t s,k ) ∞ s,k=0 defining T is lower triangular, t s,k = 0, k > s, and t s,k = t s−k , k ≤ s.Therefore compute T e n for all n and the result follows.