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Commutative Dilation Theory

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Operator Theory

Abstract

Dilation theory of single Hilbert space contractions is an important and very useful part of operator theory. By the main result of the theory, every Hilbert space contraction has the uniquely determined minimal unitary dilation. In many situations this enables to study instead of a general contraction its unitary dilation, which has much nicer properties.The present paper gives a survey of dilation theory for commuting tuples of Hilbert space operators. The paper is organized as follows:

  1. 1.

    Introduction

  2. 2.

    Dilation theory of single contractions

  3. 3.

    Regular dilations

  4. 4.

    The Ando dilation and von Neumann inequality

  5. 5.

    Spherical dilations

  6. 6.

    Analytic models

  7. 7.

    Further examples

  8. 8.

    Concluding remarks

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Academy of Czech Republic, Prague, Czech Republic

    Caline Ambrozie

  2. Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic

    Vladimír Müller

Authors
  1. Caline Ambrozie
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  2. Vladimír Müller
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Corresponding authors

Correspondence to Caline Ambrozie or Vladimír Müller .

Editor information

Editors and Affiliations

  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

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Ambrozie, C., Müller, V. (2015). Commutative Dilation Theory. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_58

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