Functions of subnormal operators

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Virginia Polytechnic Institute and State University

If f is analytic in a neighborhood of ∂D = {z| |z|= 1} and if K = f(∂D), then C-K has only finitely many components; moreover, if U is a bounded simply connected region of the plane, then

∂U = Uj=0n</sup rj

where each rj is a rectifiable Jordan curve and ri ∩ rj is a finite set whenever i ≠ j.

Let μ be a positive regular Borel measure supported on ∂D and let m denote normalized Lebesgue measure on ∂D. If L is a compact set such that ∂L ⊂ K and R(L) is a Dirichlet algebra and if ν = μof⁻¹, then the Lebesgue decomposition of ν|∂V with respect to harmonic measure for L is

ν|∂V = μaof⁻¹|∂V + μsof⁻¹|∂V

where V = intL and μ = μa + μs is the Lebesgue decomposition of μ with respect to m.

Applying Sarason’s process, we obtain P(ν) ≠ L(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|Γ << μaof⁻¹|Γ. If U is a unitary operator with scalar-valued spectral measure μ then f(U) is non-reductive if and only if there is a Jordan curve r ⊂ K such that mof⁻¹|Γ << μaof⁻¹|Γ.

Let G be a bounded region of the plane and B(H) the algebra of bounded operators in the separable Hilbert space H. If π: H(G)→B(H) is a norm-continuous homomorphism such that π(1) = 1 and π(z) is pure subnormal then π is weak-star, weak-star continuous. Moreover, if S is a pure subnormal contraction, the S*n→0 sot.