{
  "id": "2311.16724",
  "version": "v1",
  "published": "2023-11-28T12:08:02.000Z",
  "updated": "2023-11-28T12:08:02.000Z",
  "title": "On commutators of idempotents",
  "authors": [
    "Roman Drnovšek"
  ],
  "comment": "7 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $T$ be an operator on 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-28T12:08:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B47",
      "39B42"
    ],
    "keywords": [
      "commutator",
      "banach space",
      "scalar multiples",
      "idempotent operators",
      "decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}