Lemma 35 ([Analytic extension of a frequency-preserving multiplier]). Suppose multiplication by a bounded field B(t) on L²(,H) preserves P₀ K . Then it is the boundary field of a bounded, operator-norm holomorphic function on the upper …

Write w=(t-i)/(t+i) and give the unit circle normalized measure dθ/(2π) . The transformation (Jf)(t)=(1)/(√π(t+i)) f((t-i)/(t+i)) is unitary from L^2(𝕋,H) onto K : the change of variables has dθ/(2π)=dt/(π(1+t^2)) . For n≥0 , J(w^n) is a linear combination of (t+i)^ -k , 1≤k≤n+1 , whose Fourier transforms are supported in [0,∞) . For n<0 its expression is a linear combination of (t-i)^ -k , whose Fourier transforms are supported in (-∞,0] . These support statements follow directly from (t+i)^ -k =((-i)^k)/((k-1)!) ∫_0^∞u^ k-1 e^ -u e^ itu \,du and its complex conjugate, interpreted under the L^2 Fourier transform. Since the images of all circle modes form an orthonormal basis, J identifies the nonnegative-mode subspace with P_0 K . Conjugating the multiplier by J gives the circle multiplier B(w)=B(t) . Apply its invariant-subspace property to every constant vector. All negative Fourier coefficients of B vanish. If b_n denotes its n th coefficient, defined by matrix coefficients, then F(w)=∑_ n≥0 b_nw^n, |w|<1, converges in operator norm locally uniformly and is operator-norm holomorphic. Its value at w=ρe^ iθ is the circle Poisson integral of B , since all its negative coefficients vanish. The Poisson kernel is positive and has integral one, so F(w)≤B_∞ . Changing variables in the circle Poisson integral gives exactly ; explicitly, harmonic measure at t+ir pushes forward to r\,ds/(π((t-s)^2+r^2)) . For completeness, multiplication of these extensions can be checked without presupposing pointwise boundary limits. The radial Poisson integrals converge in L^2(𝕋,H) on each fixed vector, for both B and B^* , and likewise for S and S^* . Choose a sequence of radii tending to one along which these convergences hold pointwise for a countable dense family of vectors. Uniform boundedness extends the pointwise convergence to every vector. Along this sequence the products converge strongly-* almost everywhere to B(w)xS(w) . Each Fourier coefficient of the bounded analytic product F(w)xG(w) therefore converges, by dominated convergence on matrix coefficients, to the corresponding coefficient of that boundary product. In particular its negative coefficients vanish, and its nonnegative coefficients are precisely the Taylor coefficients of F xG . This proves the asserted product identity. The same argument applies to products with a scalar bounded analytic function.