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On complex symmetric weighted shifts

Chafiq Benhida, Piotr Budzyński

Abstract

Unbounded complex symmetric weighted shifts are studied. Complex symmetric unilateral weighted shifts whose $C^\infty$ vectors contain the image of the canonical orthonormal basis under the conjugation are shown to be decomposable into an orthogonal sum of infinitely many complex selfadjoint truncated weighted shifts, which generalizes a result of S. Zhu and C. G. Li. The bilateral case is discussed as well. Additional results, examples, and open problems are supplied.

On complex symmetric weighted shifts

Abstract

Unbounded complex symmetric weighted shifts are studied. Complex symmetric unilateral weighted shifts whose vectors contain the image of the canonical orthonormal basis under the conjugation are shown to be decomposable into an orthogonal sum of infinitely many complex selfadjoint truncated weighted shifts, which generalizes a result of S. Zhu and C. G. Li. The bilateral case is discussed as well. Additional results, examples, and open problems are supplied.

Paper Structure

This paper contains 6 sections, 15 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical unilateral weighted shifts
  4. Complex symmetry
  5. Complex symmetry of weighted shifts
  6. The bilateral case

Key Result

Lemma 3

Let $T$ be a complex symmetric operator such that for a given $k\in\mathbb{N}$, $T^k$ is densely defined. Then the following conditions are satisfied:

Theorems & Definitions (37)

  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Remark 4
  • Theorem 8
  • proof
  • Theorem 9: g-p-tams-2007
  • Lemma 10
  • proof
  • ...and 27 more