Subnormal operators, hyponormal operators, and mean polynomial approximation

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Virginia Tech

We prove quasisimilar subdecomposable operators without eigenvalues have equal essential spectra. Therefore, quasisimilar hyponormal operators have equal essential spectra. We obtain some results on the spectral pictures of cyclic hyponormal operators. An algebra homomorphism π from H(G) to L(H) is a unital representation for T if π(1) = I and π(x) = T. It is shown that if the boundary of G has zero area measure, then the unital norm continuous representation for a pure hyponormal operator T is unique and is weak star continuous. It follows that every pure hyponormal contraction is in C.0

Let μ represent a positive, compactly supported Borel measure in the plane, C. For each t in [1, ∞ ), the space Pt(μ) consists of the functions in Lt(μ) that belong to the (norm) closure of the (analytic) polynomials. J. Thomson in [T] has shown that the set of bounded point evaluations, bpe μ, for Pt(μ) is a nonempty simply connected region G. We prove that the measure μ restricted to the boundary of G is absolutely continuous with respect to the harmonic measure on G and the space P2(μ)∩C(spt μ) = A(G), where C(spt μ) denotes the continuous functions on spt μ and A(G) denotes those functions continuous on G ¯ that are analytic on G.

We also show that if a function f in P2(μ) is zero a.e. μ in a neighborhood of a point on the boundary, then f has to be the zero function. Using this result, we are able to prove that the essential spectrum of a cyclic, self-dual, subnormal operator is symmetric with respect to the real axis. We obtain a reduction into the structure of a cyclic, irreducible, self-dual, subnormal operator. One may assume, in this inquiry, that the corresponding P2(μ) space has bpe μ = D. Necessary and sufficient conditions for a cyclic, subnormal operator Sμ with bpe μ = D to have a self-dual are obtained under the additional assumption that the measure on the unit circle is log-integrable.