ON THE SPECTRA OF THE OPERATOR B((r)over-tilde,(s)over-tilde) MAPPING IN (w(infinity) (lambda))(a) AND (w(0) (lambda))(a) WHERE lambda IS A NONDECREASING EXPONENTIALLY BOUNDED SEQUENCE

Given any sequence a = (a(n))(n >= 1) of positive real numbers and any set E of complex sequences, we write E-a for the set of all sequences x = (x(n))(n >= 1) such that x=a = (x(n)=a(n))(n>1) is an element of E. We denote by W-a(lambda) = (w(infinity) (lambda)) a and W-a(0) (lambda) = (w(0) (lambda))(a) the sets of all sequences x such that sup(n) (lambda(-1)(n) Sigma(n)(k=1) vertical bar x(k)vertical bar /a(k)) < infinity and lim(n ->infinity) (lambda(-1)(n) Sigma(n)(k=1) vertical bar x(k)vertical bar /a(k)) = 0, where lambda is a nondecreasing exponentially bounded sequence. In this paper we recall some properties of the Banach algebras (W-a (lambda);W-a(lambda)), and W-a(0)(lambda), where a is a positive sequence. We then consider the operator Delta(rho), defined by [Delta(rho)x](n) = x(n) rho(n-1)x(n-1) for all n >= 1 with the convention x(0), rho(0) = 0, and we give necessary and sufficient conditions for the operator Delta(rho) : E -> E to be bijective, for E = w(0) (lambda), or w(infinity) (lambda). Then we consider the generalized operator of the first difference B ((r) over tilde,(s) over tilde), where (r) over tilde,(s) over tilde are two convergent sequences, and defined by [B ((r) over tilde,(s) over tilde) x](n) = r(n)x(n) + s(n-1)x(n-1) for all n >= 1 with the convention x(0), s(0) = 0. Then we deal with the operator B ((r) over tilde,(s) over tilde) mapping in either of the sets W-a (lambda), or W-a(0)(lambda). We then apply the previous results to explicitly calculate the spectrum of B ((r) over tilde,(s) over tilde) over each of the spaces E-a, where E = w(0) (lambda), or w(infinity) (lambda). Finally we give a characterization of the identity (W-a (lambda))(B(r,s)) = W-b (lambda).