Lacunary Power Sequences and Extremal Vectors

This dissertation has two parts. The first four chapters deal with lacunary power sequences. In 1966, V.I. Gurariy and V.I. Matsaev showed that a sequence {t^λ_k} is a basic sequence in the spaces C[0, 1] and L_p[0, 1], (1 ≤ p < ∞) if and only if {λ_k} is a lacunary sequence. Here, we use various methods to generalize this result to sequences {h^λ_kf} in the spaces C[a, b] and L_p[a, b], where 1 ≤ p < ∞ and 0 ≤ a < b.

The fifth chapter is on extremal vectors. In 1996 P. Enflo introduced backward minimal vectors to study invariant subspaces. If a bounded linear operator T on a Hilbert space H has dense range, then for each non-zero element x₀ of H, each positive number epsilon; with ε ≤ ‖x₀‖ and each natural number n, there exists a unique vector y_ε = y(x₀ , ε , n), called backward minimal vector, such that ‖Tⁿy_ε - x₀‖ ≤ ε and y = inf{‖y‖ : ‖Tⁿy - x₀‖ ≤ ε}. Here, we investigate rectifiability properties of the curve γ : ε → Ty_ε for the multiplication operator T on L₂[0, 1].