ON THE SPECTRA OF THE OPERATOR B(r~,s~) MAPPING IN (w∞(λ))a AND (w0(λ))a WHERE λ IS A NONDECREASING EXPONENTIALLY BOUNDED SEQUENCE

Abstract

Given any sequence $a=(a_n{)}_{n\ge 1}$ of positive real numbers and any set $E$ of complex sequences, we write $E_a$ for the set of all sequences $x=(x_n{)}_{n\ge 1}$ such that $x/a=(x_n/a_n{)}_{n\ge 1}\in E$. We denote by $W_a\left(\lambda \right)$ $={\left(w_{\mathrm{\infty }}\left(\lambda \right)\right)}_a$ and $W_a^0\left(\lambda \right)={\left(w_0\left(\lambda \right)\right)}_a$ the sets of all sequences $x$ such that $\underset{n}{sup}\left({\lambda }_n^{-1}\sum _{k=1}^n\left|x_k\right|/a_k\right)<\mathrm{\infty }$ and $\underset{n\to \mathrm{\infty }}{lim}\left({\lambda }_n^{-1}\sum _{k=1}^n\left|x_k\right|/a_k\right)=0$, where $\lambda$ is a nondecreasing exponentially bounded sequence. In this paper we recall some properties of the Banach algebras $(W_a\left(\lambda \right),W_a\left(\lambda \right))$, and $(W_a^0\left(\lambda \right),W_a^0\left(\lambda \right))$, where $a$ is a positive sequence. We then consider the operator ${\mathrm{\Delta }}_{\rho }$, definedby ${\left[{\mathrm{\Delta }}_{\rho }x\right]}_n=x_n-{\rho }_{n-1}{x}_{n-1}$ for all $n\ge 1$ with the convention $x_0$, ${\rho }_0=0$, and we give necessaryand sufficient conditions for the operator ${\mathrm{\Delta }}_{\rho }:E\to E$ to be bijective, for $E=w_0\left(\lambda \right)$, or $w_{\mathrm{\infty }}\left(\lambda \right)$. Then we consider the generalized operator of the first difference $B(\tilde{r},\tilde{s})$, where $\tilde{r},\tilde{s}$ are two convergent sequences, and defined by ${\left[B(\tilde{r},\tilde{s})x\right]}_n=r_n{x}_n+s_{n-1}{x}_{n-1}$ for all $n\ge 1$ with the convention $x_0,s_0=0$. Then we deal with the operator $B(\tilde{r},\tilde{s})$ mapping in either of the sets $W_a\left(\lambda \right)$, or $W_a^0\left(\lambda \right)$. We then apply the previous results to explicitly calculate the spectrum of $B(\tilde{r},\tilde{s})$ over each of the spaces $E_a$, where $E=w_0\left(\lambda \right)$, or $w_{\mathrm{\infty }}\left(\lambda \right)$. Finally we give a characterization of the identity ${\left(W_a\left(\lambda \right)\right)}_{B(r,s)}=W_b\left(\lambda \right)$.