Individual Theorems for the Operator S*

  • Nikolaĭ K. Nikol’skiĭ
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 273)


Only a small portion of our knowledge on S*-invariant subspaces can be obtained directly from the canonical representation K = H2 ⊝ ΘH 2 , 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.


Orthogonal Projection Invariant Subspace Toeplitz Operator Blaschke Product Canonical Representation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Nikolaĭ K. Nikol’skiĭ
    • 1
  1. 1.Leningrad Branch of the Steklov Mathematical InstituteLeningardUSSR

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