Compact endomorphisms of H ∞ (D)
نویسنده
چکیده
Let D be the open unit disc and, as usual, let H(D) be the algebra of bounded analytic functions on D with ‖f‖ = supz∈D |f(z)|. With pointwise addition and multiplication, H(D) is a well known uniform algebra. In this note we characterize the compact endomorphisms of H(D) and determine their spectra. We show that although not every endomorphism T of H(D) has the form T (f)(z) = f(φ(z)) for some analytic φ mapping D into itself, if T is compact, there is an analytic function ψ : D → D associated with T . In the case where T is compact, the derivative of ψ at its fixed point determines the spectrum of T. The structure of the maximal ideal spaceMH∞ is well known. Evaluation at a point z ∈ D gives rise to an element in MH∞ in the natural way. The remainder of MH∞ consists of singleton Gleason parts and Gleason parts which are analytic discs. An analytic disc, P (m), containing a point m ∈ MH∞ , is a subset of MH∞ for which there exists a continuous bijection Lm : D → P (m) such that Lm(0) = m and f̂(Lm(z)) is analytic on D for each f ∈ H(D). Moreover, the map Lm has the form Lm(z) = w ∗ lim z + zα 1 + zαz for some net zα → m in the w*topology, whence f̂(Lm(z)) = lim f( z + zα 1 + zαz )
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