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A(H+ According to Corollary According to Theorem admits a J-polar analytic continuation angular operator ator Ax,Ax Ax,x biholomorphic map bijection bounded linear operator canonical decomposition compact consequently contained double commutant eigenvalues following statements follows from Corollary follows from Theorem fractional-linear map gonal hence Hilbert space idempotent implies injective inner product x,y J-adjoint J-biexpansive operator J-expansive J-isometry J-negative J-orthogonal J-partial isometry J-polar decomposition J-positive operator J-projection J-unitary operator ker(A ker(U Krein and Smuljan Krein space Let us define maxi maximal negative subspace maximal positive subspace maximal strictly positive maximal uniformly positive mutually equivalent non-negative open unit operator A transforms orthoprojection P+(M positive invertible positive vector proof of Theorem proved analogously ran(A ran(E ran(F ran(I-F ran(U Riesz theorem S(H+ selfadjoint operator spectral function spectral projection spectrum suffices to prove Suppose Theorem 1.3 tion tive transforms H+ uniformly negative subspace uniformly positive subspace unitary weak operator topology