On quasi-similarity of subnormal operators

Abstract

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R 2 (K, ν) denote the closure of Rat(K) in L 2 (ν) and let Sν denote the operator of multiplication by the inde-Suppose Ω is a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H 2 (Ω). Let σ denote a harmonic measure for Ω. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R 2 (Ω, σ). We show that for a given ν supported on Ω, Sν is quasi-similar to Sσ if and only if ν|∂Ω σ and log( dν dσ ) ∈ L 1 (σ). Our result extends a well-known result of Clary on the unit disk.

Beurling’s theorem and invariant subspaces for the shift on Hardy spaces

“…In 1991 J. Thomson first proved Theorem 1 for P 2 (µ) except 4) and 5) in [20]. Some applications of Theorem 1 can be found in [14,17].…”

Carleson measures on planar sets

“…Originally motivated by a research problem in operator theory suggested by Ronald Douglas: Can the similarity and quasi-similarity of the well-known results of W. Clary be extended to multiply connected domains? The readers may consult [4][ p. 370-373] and [5,6,7,8,9] for the topics. The quasi-similarity case was first investigated in [5] and then later in [8], but the similarity case was only to be done recently in [9] and the extension was completed because of the research in this work.…”

“…The readers may consult [4][ p. 370-373] and [5,6,7,8,9] for the topics. The quasi-similarity case was first investigated in [5] and then later in [8], but the similarity case was only to be done recently in [9] and the extension was completed because of the research in this work. We consider the following problem: What are the natural extension of the Carleson measures from circular domains to general domains.…”

The Structure of the Closure of the Rational Functions in Lq(μ)

“…One needs to have them when applying it to operator theory. For example, see [17] and [24]). In [6], Conway and Elias studied the same problem as that in this paper.…”