On the simplicity of operator knots (Q2266213)

The paper gives a general criterion of the simplicity (or the complete nonunitariness) of the product of n knots \(\alpha =\alpha_ n...\alpha_ 2\alpha_ 1\) (or the main operator of the knot \(\alpha)\): The knot \(\alpha\) is simple iff 1. The factorization \(\theta (\zeta)=[\theta_ n(\zeta)...\theta_ k(\zeta)][\theta_{k-1}(\zeta)...\theta_ 1(\zeta)]\) of the characteristic function of the knot \(\alpha\) is regular in the sense of Sz.-Nagy-Foiaş for all \(k=1,2,...,n\), 2. The knots \(\alpha_ 1,\alpha_ 2,...,\alpha_ n\) are simple. This result is applied to the study of the complete nonunitariness of contractions of the form \[ (Tf)(x)=e^{i\phi (x)}f(x)-2e^{i\phi (x)}p(x)\Pi^{-1}(x)\int^{t_ 2}_{x}\Pi (t)P(t)f(t)dt \] in \(L^ 2_ E(t_ 1,t_ 2)\).