{
  "id": "1903.10358",
  "version": "v1",
  "published": "2019-03-25T14:17:50.000Z",
  "updated": "2019-03-25T14:17:50.000Z",
  "title": "On Norm Inequalities and Orthogonality of Commutators of Derivations",
  "authors": [
    "N. B. Okelo",
    "P. O. Mogotu"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $H$ be a complex separable Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper, we give considerable generalizations of the inequalities for norms of commutators of normal operators. Let $S, T \\in B(H)$ be positive normal operators with the cartesian decomposition $S=A+iC$ and $T=B+iD$ such that $a_{1}\\leq A \\leq a_{2},\\, b_{1}\\leq B \\leq b_{2},\\,c_{1}\\leq C \\leq c_{2}$ and $ d_{1}\\leq D \\leq d_{2}$ for some real numbers $a_{1},\\,a_{2},\\,b_{1},\\,b_{2},\\,c_{1},\\,c_{2},\\,d_{1}$ and $d_{2}$ we have shown that $\\|ST-TS\\|\\leq \\frac{1}{2}\\sqrt{(a_{2}-a_{1})^{2}+(c_{2}-c_{1})^{2}}\\sqrt{(b_{2}-b_{1})^{2}+(d_{2}-d_{1})^{2}}.$ Moreover, orthogonality and norm inequalities for commutators of derivation are also established. We have shown that if the pair of operators $(S,T)$ satisfies Fuglede-Putnam's property and $C \\in ker(\\delta _{S,T})$ where $C\\in B(H)$ then $\\|\\delta _{S,T}X+C\\|\\geq \\|C\\|.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-25T14:17:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A12",
      "47A30",
      "47A63"
    ],
    "keywords": [
      "norm inequalities",
      "commutators",
      "orthogonality",
      "derivation",
      "complex separable hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}