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Laurent and Toeplitz Operators

In: Basic Classes of Linear Operators

Author

Listed:
  • Israel Gohberg

    (Tel Aviv University, School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences)

  • Seymour Goldberg

    (University of Maryland, Department of Mathematics)

  • Marinus A. Kaashoek

    (Vrije Universiteit Amsterdam, Department of Mathematics and Computer Science)

Abstract

This chapter deals with operators on ℓ2(ℤ) and ℓ2 with the property that the matrix relative to the standard basis in these spaces has a special structure, namely the elements on diagonals parallel to the main diagonal are the same, i.e., the matrix entries a jk depend on the difference j — k only. On ℓ2(ℤ) these operators are called Laurent operators (and in that case the matrix is doubly infinite); on ℓ2 they are called Toeplitz operators. These operators form important classes of operators and they appear in many applications. They also have remarkable properties. For instance, there are different methods to invert explicitly these operators, and to compute their spectra. This chapter reviews these results starting from the simplest class.

Suggested Citation

  • Israel Gohberg & Seymour Goldberg & Marinus A. Kaashoek, 2003. "Laurent and Toeplitz Operators," Springer Books, in: Basic Classes of Linear Operators, chapter 0, pages 135-170, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7980-4_3
    DOI: 10.1007/978-3-0348-7980-4_3
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