Individual Theorems for the Operator S*

Abstract

Only a small portion of our knowledge on S*-invariant subspaces can be obtained directly from the canonical representation K = H² ⊝ ΘH², Θ an inner function. For instance, one can establish that the hull ∨(K _i : i = 1, 2,) of two such subspaces K₁ and K₂ is again a non-trivial S*-invariant subspace in its own right, because ∨(K _i : i = 1, 2)^⊥ = Θ₁ H² ∩ Θ₂ H² = ΘH² (≠{O} where Θ is the LCM of Θ₁ and Θ₂. However many other theorems require a deeper analysis. This Lecture conventionally falls into two parts: the first one is devoted to a detailed analysis of S*-cyclic vectors (that is, elements f in H² such that $E_f^{ * \underline{\underline {def}} } \vee \left( {{S^{ * n}}f:n \geqslant \left. 0 \right) = {H^2}} \right)$ and the second one to an approximation theoretic characterization of S*-invariant subspaces in terms of the root vectors of that operator.