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Three families of matrices

Alexander Pushnitski

Abstract

This paper has an expository nature. We compare the spectral properties (such as boundedness and compactness) of three families of semi-infinite matrices and point out similarities between them. The common feature of these families is that they can be understood as matrices of some linear operations on appropriate Hardy spaces.

Three families of matrices

Abstract

This paper has an expository nature. We compare the spectral properties (such as boundedness and compactness) of three families of semi-infinite matrices and point out similarities between them. The common feature of these families is that they can be understood as matrices of some linear operations on appropriate Hardy spaces.
Paper Structure (17 sections, 40 equations, 1 figure)

This paper contains 17 sections, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. The origins of $A$, $B$ and $C$
  4. The structure of the paper
  5. The matrix $A(\tau,\rho)$
  6. The quadratic form of $A(\tau,\rho)$
  7. The spectral properties of $A(\tau,\rho)$
  8. Justification and discussion
  9. The matrix $B(\tau,\rho)$
  10. The quadratic form of $B(\tau,\rho)$
  11. The spectral properties of $B(\tau,\rho)$
  12. Justification and discussion
  13. The matrix $C(\tau,\rho)$
  14. The quadratic form of $C(\tau,\rho)$
  15. The spectral properties of $C(\tau,\rho)$
  16. ...and 2 more sections

Figures (1)

  • Figure 1: In the (open) shaded region the corresponding matrix is bounded and compact. In the (open) complementary region, the corresponding matrix is unbounded.

Theorems & Definitions (3)

  • proof
  • proof
  • proof