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Orthogonality of bounded linear operators on complex Banach spaces

Kallol Paul, Debmalya Sain, Arpita Mal, Kalidas Mandal

Abstract

We study Birkhoff-James orthogonality of bounded linear operators on complex Banach spaces and obtain a complete characterization of the same. By means of introducing new definitions, we illustrate that it is possible in the complex case, to develop a study of orthogonality of bounded (compact) linear operators, analogous to the real case. Furthermore, earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case, can be obtained as simple corollaries to our present study. In fact, we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case, in order to distinguish the complex case from the real case. We also study the left symmetric linear operators on complex two-dimensional $l_p$ spaces. We prove that $ T $ is a left symmetric linear operator on $ \ell_p^2{(\mathbb{C})}$ if and only if $ T $ is the zero operator.

Orthogonality of bounded linear operators on complex Banach spaces

Abstract

We study Birkhoff-James orthogonality of bounded linear operators on complex Banach spaces and obtain a complete characterization of the same. By means of introducing new definitions, we illustrate that it is possible in the complex case, to develop a study of orthogonality of bounded (compact) linear operators, analogous to the real case. Furthermore, earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case, can be obtained as simple corollaries to our present study. In fact, we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case, in order to distinguish the complex case from the real case. We also study the left symmetric linear operators on complex two-dimensional spaces. We prove that is a left symmetric linear operator on if and only if is the zero operator.

Paper Structure

This paper contains 2 sections, 21 theorems, 18 equations.

Table of Contents

  1. Introduction.
  2. Main Results

Key Result

Theorem 1.1

Let $\mathbb{X}$ be a finite-dimensional real Banach space. Let $T, A \in \mathbb{L}(\mathbb{X}).$ Then $T \perp_B A$ if and only if there exists $x, y \in M_T$ such that $Ax \in Tx^{+}$ and $Ay \in Ty^{-}$.

Theorems & Definitions (40)

  • Theorem 1.1: Theorem 2.2 of S
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.3.1
  • proof
  • Theorem 2.4
  • ...and 30 more