Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A class of weighted composition operators whose Range and Null spaces are complemented

Anurag Kumar Patel

Abstract

In this paper, we prove that the null space of a weighted composition operator on $\ell_p~ (1 \leq p < \infty)$ is a complemented subspace. We also give a necessary and sufficient condition for a weighted composition operator on $\ell_p$ whose range space is of finite co-dimension. Thereafter, we characterize a class of weighted composition operators whose range space is a complemented subspace. Lastly, we characterize weighted composition operators on $\ell_p$ which are Fredholm operators.

A class of weighted composition operators whose Range and Null spaces are complemented

Abstract

In this paper, we prove that the null space of a weighted composition operator on is a complemented subspace. We also give a necessary and sufficient condition for a weighted composition operator on whose range space is of finite co-dimension. Thereafter, we characterize a class of weighted composition operators whose range space is a complemented subspace. Lastly, we characterize weighted composition operators on which are Fredholm operators.
Paper Structure (4 sections, 21 theorems, 94 equations)

This paper contains 4 sections, 21 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Weighted composition operators on $\ell_p~(1\leq p < \infty)$ spaces with finite dimensional null space
  3. Weighted composition operator on $\ell_p~(1\leq p<\infty)$ spaces whose range space is of finite co-dimension
  4. Weighted composition operator as Fredholm operator on $\ell_p$ spaces $(1\leq p<\infty)$

Key Result

Theorem 1.2

Necessary and sufficient condition for a complex valued measurable function $u$ and a measurable transformation $\phi$ to induce a bounded operator on $L^p(\mu)(1\le p < \infty)$ defined by $uC_\phi(f)=u \cdot f\circ\phi$ are $\mu\phi^{-1}<<\mu$ and $\frac{d\mu_{u^p}, \phi}{d\mu}$ is essentially bou and $\frac{d\mu_{u^p}, \phi}{d\mu}$ represents the Radon-Nikodym derivative.

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • proof
  • Remark 2.2
  • ...and 38 more