Abstract. In this paper we recall recent results that are direct consequences of the fact that $(w_{\infty }(\lambda),w_{\infty}(\lambda)) $ is a Banach algebra. Then we define the set $W_{\tau }=D_{\tau }w_{\infty }$ and characterize the sets $W_{\tau}(A) $ where $A$ is either of the operators $\Delta $, $\Sigma $, $\Delta(\lambda) $, or $C(\lambda)$. Afterwards we consider the sets $[A_{1},A_{2}] _{W_{\tau }}$ and give conditions for these sets to be in the form $W_{\tau }$. Finally we apply the previous results to obtain characterizations of matrix transformations in sets that generalize the sets of lacunary sequences such as $(N_{\theta }^{\infty }(\Delta ),N_{\xi }^{\infty }(\Delta ^{+h}))$, $(N_{\theta }^{\infty }(D_{1/\tau }C^+(\lambda )),N_{\xi }^{\infty })$ and $(N_{\theta }^{\infty }(\Delta(\mu )),N_{\xi }^{\infty })$.

AMS Subject Classification (1991): 40C05, 40J05, 46A15

Received April 21, 2007, and in revised form May 9, 2007. (Registered under 6008/2009.)