On commutators of idempotents

Drnovšek, Roman (author)

Let $T$ be an operator on a Banach space $X$ that is similar to $- T$ via an involution $U$. Then $U$ decomposes the Banach space $X$ as $X = X_1 \oplus X_2$ with respect to which decomposition we have $U = \left(\begin{matrix} I_1 & 0 \\ 0 & -I_2 \end{matrix} \right)$, where $I_i$ is the identity operator on the closed subspace $X_i$ ($i=1, 2$). Furthermore, $T$ has necessarily the form $T = \left(\begin{matrix} 0 & * \\ * & 0 \end{matrix} \right)$ with respect to the same decomposition. In this note we consider the question when $T$ is a commutator of the idempotent $P = \left(\begin{matrix} I_1 & 0 \\ 0 & 0 \end{matrix} \right)$ and some idempotent $Q$ on $X$. We also determine which scalar multiples of unilateral shifts on $l^p$ spaces ($1 \le p \le \infty$) are commutators of idempotent operators.

• Language: English
• Keywords: Banach spaces, operators, idempotents, commutators
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: Str. 649-654
• Numbering: Vol. 73, no. 4
• PID: 20.500.12556/RUL-168615
• UDC: 517.9
• ISSN on article: 0308-1087
• DOI: 10.1080/03081087.2024.2368734
• COBISS.SI-ID: 233159171
• Publication date in RUL: 18.04.2025

Record is a part of a journal

• Title: Linear and multilinear algebra
• Shortened title: Linear multilinear algebra
• Publisher: Taylor & Francis
• ISSN: 0308-1087
• COBISS.SI-ID: 25872128

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0222
• Name: Algebra, teorija operatorjev in finančna matematika