Clark model in the general situation
Clark model in the general situation
复制标题
一般情况下的克拉克模型
DOI：
10.1007/s11854-016-0038-4
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
S. Treil
中科院分区：
文献类型：
--
作者：
C. Liaw;S. Treil
For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ǀγǀ = 1. Namely, all such unitary perturbations are operators Uγ:= U + (γ − 1)( ·, b1)Hb, where b ∈ H, ǁbǁ = 1, b1 = U−1b, ǀγǀ = 1. For ǀγǀ < 1, the operators Uγ are contractions with one-dimensional defects.Restricting our attention to the non-trivial part of perturbation, we assume that b is a cyclic vector for U, i.e., $${\rm H} = \overline {span} \left\{ {{U^n}b:n \in \mathbb{Z}} \right\}$$Η=span¯{Unb:n∈ℤ}. In this case, the operator Uγ, ǀγǀ < 1 is a completely non-unitary contraction and thus unitarily equivalent to its functional model Mγ, which is the compression of the multiplication by the independent variable z onto the model space $${K_{{\theta _\gamma }}}$$Kθγ; here, θγ is the characteristic function of the contraction Uγ.The Clark operator Φγ is a unitary operator intertwining the operator Uγ, ǀγǀ < 1 (in the spectral representation of the operator U) and its model Mγ, MγΦγ = ΦγUγ. In the case when the spectral measure of U is purely singular (equivalently, the characteristic function θγ is inner), the operator Φγ was described from a slightly different point of view by D. Clark [3]. The case where θγ is an extreme point of the unit ball in H∞ was treated by D. Sarason [18], using the sub-Hardy spaces H(θ) introduced by L. de Branges.In this paper, we treat the general case and give a systematic presentation of the subject. We first find a formula for the adjoint operator Φγ*, which is represented by a singular integral operator, generalizing in a sense the normalized Cauchy transform studied by A. Poltoratskii [16]. We begin by presenting a “universal” representation that works for any transcription of the functional model. We then give the formulas adapted for specific transcriptions of the model, such as Sz.-Nagy–Foiaş and the de Branges–Rovnyak transcriptions. Finally, we obtain the representation of Φγ.
For a unitary operator U in a Hilbert space H the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter γ, ǀγǀ = 1. Namely, all such unitary perturbations are operators Uγ:= U + (γ − 1)( ·, b1)Hb, where b ∈ H, ǁbǁ = 1, b1 = U−1b, ǀγǀ = 1. For ǀγǀ < 1, the operators Uγ are contractions with one-dimensional defects.Restricting our attention to the non-trivial part of perturbation, we assume that b is a cyclic vector for U, i.e., $${\rm H} = \overline {span} \left\{ {{U^n}b:n \in \mathbb{Z}} \right\}$$Η=span¯{Unb:n∈ℤ}. In this case, the operator Uγ, ǀγǀ < 1 is a completely non-unitary contraction and thus unitarily equivalent to its functional model Mγ, which is the compression of the multiplication by the independent variable z onto the model space $${K_{{\theta _\gamma }}}$$Kθγ; here, θγ is the characteristic function of the contraction Uγ.The Clark operator Φγ is a unitary operator intertwining the operator Uγ, ǀγǀ < 1 (in the spectral representation of the operator U) and its model Mγ, MγΦγ = ΦγUγ. In the case when the spectral measure of U is purely singular (equivalently, the characteristic function θγ is inner), the operator Φγ was described from a slightly different point of view by D. Clark [3]. The case where θγ is an extreme point of the unit ball in H∞ was treated by D. Sarason [18], using the sub-Hardy spaces H(θ) introduced by L. de Branges.In this paper, we treat the general case and give a systematic presentation of the subject. We first find a formula for the adjoint operator Φγ*, which is represented by a singular integral operator, generalizing in a sense the normalized Cauchy transform studied by A. Poltoratskii [16]. We begin by presenting a “universal” representation that works for any transcription of the functional model. We then give the formulas adapted for specific transcriptions of the model, such as Sz.-Nagy–Foiaş and the de Branges–Rovnyak transcriptions. Finally, we obtain the representation of Φγ.