Similarity of certain operators in $l^{p}$

Abstract:

Let M be the multiplication operator in ${l^p},1 \leqslant p \leqslant \infty$, i.e., $M:x = \{ {x_k}\} \to \{ k{x_k}\}$. Let $w = \{ {w_j}\} _{j = 0}^\infty$ be a weight, i.e., a positive sequence such that ${w_1} < {w_0} = 1$ and ${w_{n + m}} \leqslant {w_n}{w_m}$. For $\zeta \in C$, define $N_w^\zeta$ on ${l^p}$ by \[ {(N_w^\zeta x)_k} = \sum \limits _{j = 1}^k {\left ( {\frac {{{w_k}}}{{{w_j}}}} \right )} \left ( {\begin {array}{*{20}{c}} {\zeta - 1 + k - j} \\ {k - j} \\ \end {array} } \right ){x_j}\quad (k = 1,2, \ldots ).\] Then $\{ N_w^\zeta ;\zeta \in C\}$ is a holomorphic group of operators, and for any function g holomorphic on the spectrum of $N_w^\zeta ,M + g(N_w^\zeta )$ is similar to $M + g(1)I$.