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Singular Integrals, Rank One Perturbations and Clark Model in General Situation

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Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2)

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 5))

Abstract

We start with considering rank one self-adjoint perturbations A α  = A +α( ⋅ , φ)φ with cyclic vector \(\varphi \in \mathcal{H}\) on a separable Hilbert space \(\mathcal{H}\). The spectral representation of the perturbed operator A α is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators A and A α .

Similar results will be presented for unitary rank one perturbations of unitary operators, leading to singular integral operators on the circle.

This motivates the study of abstract singular integral operators, in particular the regularization of such operator in very general settings.

Further, starting with contractive rank one perturbations we present the Clark theory for arbitrary spectral measures (i.e. for arbitrary, possibly not inner characteristic functions). We present a description of the Clark operator and its adjoint in the general settings. Singular integral operators, in particular the so-called normalized Cauchy transform again plays a prominent role.

Finally, we present a possible way to construct the Clark theory for dissipative rank one perturbations of self-adjoint operators.

These lecture notes give an account of the mini-course delivered by the authors, which was centered around (Liaw and Treil, J Funct Anal 257(6):1947–1975, 2009; Rev Mat Iberoam 29(1):53–74, 2013; J Anal Math). Unpublished results are restricted to the last part of this manuscript.

To the memory of Cora Sadosky

The work of Constanze Liaw is supported by the Simons Foundation Collaboration Grant for Mathematicians #426258. Work of S. Treil is supported by the National Science Foundation under the grant DMS-1301579.

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Notes

  1. 1.

    We did not discuss singular integral operators with vector-valued kernels, but the extension of the theory presented in section “Singular Integral Operators” to the case of kernels with values in \(\mathbb{R}^{d}\) or \(\mathbb{C}^{d}\) is trivial and we omit it.

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Authors and Affiliations

  1. CASPER and Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX, 76798, USA

    Constanze Liaw

  2. Department of Mathematics, Brown University, 151 Thayer Str./Box 1917, Providence, RI, 02912, USA

    Sergei Treil

Authors
  1. Constanze Liaw
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  2. Sergei Treil
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Correspondence to Constanze Liaw .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    María Cristina Pereyra

  2. Department of Mathematics, Venezuelan Institute for Scientific Research, Caracas, Venezuela

    Stefania Marcantognini

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  4. Department of Mathematical and Actuarial Sciences, Roosevelt University Chicago, Chicago, Illinois, USA

    Wilfredo Urbina

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Liaw, C., Treil, S. (2017). Singular Integrals, Rank One Perturbations and Clark Model in General Situation. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds) Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2). Association for Women in Mathematics Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-51593-9_4

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