A unicellular universal quasinilpotent weighted shift

Abstract

For a suitably chosen sequence of weights { α n } \{ {\alpha _n}\} , the unilateral weighted shift Q Q on l p ( 1 ≤ p > ∞ ) {l^p}(1 \leq p > \infty ) , defined by Q e n = α n e n + 1 ( n ≥ 1 ) Q{e_n} = {\alpha _n}{e_{n + 1}}(n \geq 1) , is a unicellular quasinilpotent operator such that Q k {Q^k} is not compact for any power k ≥ 1 k \geq 1 . As a corollary, several applications to approximation of Hilbert space operators are given.