The analytic model of a hyponormal operator with rank one self-commutator

A Model for Some Class of Operators

“…We only have to study the case that A is hyponormal satisfying (5.10). as in [3]. Actually this S(·, ·) in (5.12) is the same as in the section 1 of this paper.…”

“…If H 1 f ∈ H g for f ∈ H g (in the statement of Theorem 1 of [3], also see [4], this condition did not appear, but if we check the proof of this theorem, this condition has to be counted), then there is a unitary operator U from H onto H g which satisfies (U (λ − H) −1 x)(z) = (λ − z) −1 x, for x ∈ [H * , H]H and λ ∈ ρ(H) such that H = U HU −1 and H 1 = U H * U −1 . Then these H and H 1 are the analytic model of H and H * .…”

“…In [3] and [4], J.D. Pincus, J. Xia and the author studied the analytic model of a pure hyponormal operator H with rank one self -commutators [H * , H].…”

Seminormal systems of operators in Clifford environments

“…Such models generalize to higher dimension the Hilbert transform models of hyponormal operators with a self-commutator of rank 1 discovered by Xia [42] and Pincus [33], analyzed in more detail by Pincus and Xia [34] and Picus and Xia and Xia [35], and afterwards set up in full generality for pure hyponormal operators by Kato [17] and Muhly [32]. We should point out that the natural framework for developing Riesz transforms models is provided by n-tuples of decomposable linear operators on a direct integral Hilbert space…”

“…For more details in this regard we refer to Martin [23,34]. However, the basic features of the models can be fully illustrated by assuming that each of the spaces H(x), x ∈ Ω, is one-dimensional, i.e., by taking the Lebesgue space H = L 2 (Ω), and that is exactly what we will be doing in this section.…”