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Realization and linearization
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analytic angular operator angular with respect apply Theorem associate main operator assume Banach space bijective bounded linear operator Brodskii J-node Cauchy contour characteristic operator function closed subspace corresponding decomposition defined denoted divisors eigenvalue eigenvector equal exists fact factorization with respect finite dimensional node follows formula geometric multiplicity given hence Hilbert space implies invertible external operator invertible operator Jordan chains Krein J-node Let 9 McMillan degree minimal factorizations minimal node minimal polynomial minimal realization Mobius transformation monic node monic operator polynomials node 9 node similarity non-trivial Note nxn matrix function Observe operator norm pole polynomial node projection for 9 proof is complete prove rank rational nxn matrix real line Riccati equation Riemann sphere Riesz projection right canonical factorization spectral subspace splits the spectrum stable invariant subspace supporting subspace Theorem l.l tion trace class transfer function unitary operator vector zero