Cyclic Vectors

Abstract

For the simple shift, consider a vector 〈ξ₀, ξ₁, ξ₂,…〉 such that

$$ \mathop {\lim }\limits_k \,\frac{1}{{{{\left| {{\xi _k}} \right|}^2}}}\,{\sum\limits_{n = 1}^\infty {\left| {{\xi _{n\, + k}}} \right|} ^2}\, = \,0.$$

For shifts of higher multiplicity, form vectors whose components are subsequences of this sequence {ξ_n}.