## Functional Calculus and Common Invariant Subspaces |

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A-measures A-topology absolutely continuous assume backward shift bands of measures Borel set Chapter class Kq closed subspace coisometric extension common invariant subspaces commuting contractions commuting operators compact set completes the proof continuous with respect contractive representation convergence Corollary 2.2 decomposes defined denote direct sum dominating doubly commute dual algebra elementary measures essential spectrum following decomposition following THEOREM function algebra functional calculus G M(X Gleason Hence Hilbert space invariant subspace results isometry iV-tuple Joint spectra Lemma measures orthogonal minimal Moreover mutually singular N-tuple of commuting natural representation Neumann inequality non-peak points norm nullset obtain operator topology pair of commuting polydisc polynomials property F Proposition 2.2 R(Xj reducing band reflexivity representing measures resp sequence spectral measure spectral set subset system of elementary Theorem 2.1 Tj G tuple unitary dilation unitary operator vanishing conditions vectors weak-star topology Wold-type extension Wold-von Neumann decomposition