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Counterexamples in Operator Theory

  • Textbook
  • © 2022

Overview

Authors:
  1. Mohammed Hichem Mortad

    1. Department of Mathematics, University of Oran 1, Oran, Algeria

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  • Presents over 500 counterexamples in operator theory at varying levels of difficulty

  • Includes counterexamples on both bounded and unbounded linear operators, many of which are the author’s original work

  • Divided into brief subsections, making it easy for readers to navigate the wide variety of topics included

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Table of contents (32 chapters)

  1. Front Matter

    Pages i-xxxvi
    Download chapter PDF

  2. Bounded Linear Operators

    1. Some Basic Properties

      • Mohammed Hichem Mortad
      Pages 3-20

    2. Basic Classes of Bounded Linear Operators

      • Mohammed Hichem Mortad
      Pages 21-37

    3. Operator Topologies

      • Mohammed Hichem Mortad
      Pages 39-45

    4. Positive Operators

      • Mohammed Hichem Mortad
      Pages 47-58

    5. Matrices of Bounded Operators

      • Mohammed Hichem Mortad
      Pages 59-73

    6. (Square) Roots of Bounded Operators

      • Mohammed Hichem Mortad
      Pages 75-94

    7. Absolute Value, Polar Decomposition

      • Mohammed Hichem Mortad
      Pages 95-119

    8. Spectrum

      • Mohammed Hichem Mortad
      Pages 121-143

    9. Spectral Radius, Numerical Range

      • Mohammed Hichem Mortad
      Pages 145-157

    10. Compact Operators

      • Mohammed Hichem Mortad
      Pages 159-167

    11. Functional Calculi

      • Mohammed Hichem Mortad
      Pages 169-177

    12. Fuglede-Putnam Theorems and Intertwining Relations

      • Mohammed Hichem Mortad
      Pages 179-189

    13. Operator Exponentials

      • Mohammed Hichem Mortad
      Pages 191-205

    14. Nonnormal Operators

      • Mohammed Hichem Mortad
      Pages 207-262

    15. Similarity and Unitary Equivalence

      • Mohammed Hichem Mortad
      Pages 263-279

    16. The Sylvester Equation

      • Mohammed Hichem Mortad
      Pages 281-284

    17. More Questions and Some Open Problems

      • Mohammed Hichem Mortad
      Pages 285-290

  3. Unbounded Linear Operators

    1. Front Matter

      Pages 291-291
      Download chapter PDF

    2. Basic Notions

      • Mohammed Hichem Mortad
      Pages 293-306
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Keywords

About this book

This text is the first of its kind exclusively devoted to counterexamples in operator theory and includes over 500 problems on bounded and unbounded linear operators in Hilbert spaces. This volume is geared towards graduate students studying operator theory, and the author has designated the difficulty level for each counterexample, indicating which ones are also suitable for advanced undergraduate students.


The first half of the book focuses on bounded linear operators, including counterexamples in the areas of operator topologies, matrices of bounded operators, square roots, the spectrum, operator exponentials, and non-normal operators. The second part of the book is devoted to unbounded linear operators in areas such as closedness and closability, self-adjointness, normality, commutativity, and the spectrum, concluding with a chapter that features some open problems. Chapters begin with a brief “Basics” section for the readers’ reference, and many of the counterexamples included are the author’s original work. 


Counterexamples in Operator Theory can be used by students in graduate courses on operator theory and advanced matrix theory. Previous coursework in advanced linear algebra, operator theory, and functional analysis is assumed. Researchers, quantum physicists, and undergraduate students studying functional analysis and operator theory will also find this book to be a useful reference.




Reviews

“The book … offers a wide range of examples and counterexamples in the theory of linear operators acting on Hilbert spaces. … The books thus covers a broad range of largely classical topics in operator theory … . The book is most likely … a reference work for its intended audience of advanced students and researchers in operator theory, as well as, no doubt, those setting exam questions on the subject.” (David Seifert, zbMATH 1512.47002, 2023)

Authors and Affiliations

  • Department of Mathematics, University of Oran 1, Oran, Algeria

    Mohammed Hichem Mortad

About the author

Mohammed Hichem Mortad's primary research interest is operator theory. He received his Ph.D. in mathematical analysis from the University of Edinburgh in 2003. 

Bibliographic Information

  • Book Title: Counterexamples in Operator Theory

  • Authors: Mohammed Hichem Mortad

  • DOI: https://doi.org/10.1007/978-3-030-97814-3

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022

  • Hardcover ISBN: 978-3-030-97813-6Published: 04 May 2022

  • Softcover ISBN: 978-3-030-97816-7Published: 05 May 2023

  • eBook ISBN: 978-3-030-97814-3Published: 03 May 2022

  • Edition Number: 1

  • Number of Pages: XXXVI, 598

  • Topics: Operator Theory, Functional Analysis

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