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Arithmetic Bohr radius of bounded linear operators

Vasudevarao Allu, Subhadip Pal

Abstract

In this paper, we investigate the arithmetic Bohr radius of bounded linear operators between arbitrary complex Banach spaces. We establish the close connection between the classical Bohr radius and the arithmetic Bohr radius of bounded linear operators. Further, we study the asymptotic estimates of arithmetic Bohr radius for identity operator on infinite dimensional complex Banach spaces. Finally, we obtain the correct asymptotic behavior of Bohr radii of operators between sequence spaces.

Arithmetic Bohr radius of bounded linear operators

Abstract

In this paper, we investigate the arithmetic Bohr radius of bounded linear operators between arbitrary complex Banach spaces. We establish the close connection between the classical Bohr radius and the arithmetic Bohr radius of bounded linear operators. Further, we study the asymptotic estimates of arithmetic Bohr radius for identity operator on infinite dimensional complex Banach spaces. Finally, we obtain the correct asymptotic behavior of Bohr radii of operators between sequence spaces.

Paper Structure

This paper contains 4 sections, 9 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Proof of the Main results

Key Result

Proposition 3.1

Let $1\leq p<\infty$ and $T:X\rightarrow Y$ be a bounded operator between complex Banach spaces $X$ and $Y$ with $\left\lVert T\right\rVert<\lambda^{1/p}$. Then we have

Theorems & Definitions (11)

  • Definition 1.1
  • Definition 1.2
  • Proposition 3.1
  • Lemma 3.2
  • Theorem A
  • Theorem 3.1
  • Theorem B
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • ...and 1 more