Operations on Infinite x Infinite Matrices and Their Use in Dynamics and Spectral Theory

By first looking at the orthonormal basis: Γ = {∑_i 4 ⁱb_i ∈{0, 1}, finite sums} and the related orthonormal basis 5Γ = {5∑_i 4ⁱ b_i : b_i ∈ {0, 1}, finite sums} we find several interesting relationships with the unitary matrix Uα,β arising from the operator U: Γ → 5Γ. Further, we investigate the relationships between U and the operators S_o : Γ → 4Γ defined by S_oe_4γ where e_γ = e^2ΠiΓ and S₁: Γ → 4Γ+1 defined by S₁e_γ = e_4γ+1.

Most intriguing, we found that when taking powers of the aforementioned Uα,β matrix that although there are infinitely many 1's occurring in the entries of Uα,β only one such 1 occurs in the subsequent higher powers U^kα,β. This means that there are infinitely many γ ∈ Γ ∩ 5Γ, but only one such γ in the intersection Γ and 5^kΓ, for k ≥ 2.