On the Spectra of the Operator of the First Difference on the Spaces W_tau and W_tau^0 and Application to Matrix Transformations

Given any sequence tau =(tau) _n≥1of positive real numbers and any set E of complex sequences, we write E_tau for the set of all sequences x=(x_n)_n≥1 such that x/a=(x_n/a)_n≥1in E. We define the sets W_tau=w_infty\right) _tau and W_tau^0=lw_0\right) _tau , where w_infty is the set of all sequences such that sup_n\left( n^-1sum_m=1^nleft\vert x_mright\vert \right) infty , and w_0 is the set of all sequences such that lim_n\rightarrow \infty \left( n^-1\sum_m=1^n\left\vert x_m\right\vert \right) =0. Then we explicitly calculate the spectra sigma \left( \Delta ,W_tau \right) and sigma \left( \Delta ,W_tau ^0\right) of the operator of the first difference on each of the sets W_tau and W_tau ^0. We then determine the sets left( E,F\right) of all matrix transformations mapping E to F, with E=W_tau left( left Delta -lambda I\right) ^h\right) , or W_tau ^0left \left( \Delta -\lambda I\right) ^h\right) and F=s_xi, or s_\xi ^0 for complex numbers lambda and h and obtain simplifications of these sets for some values of lambda .