{
  "id": "2001.01342",
  "version": "v1",
  "published": "2020-01-06T00:37:00.000Z",
  "updated": "2020-01-06T00:37:00.000Z",
  "title": "Some remarks on Tsallis relative operator entropy",
  "authors": [
    "Shigeru Furuichi",
    "Hamid Reza Moradi"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "This paper intends to give some new estimates for Tsallis relative operator entropy ${{T}_{v}}\\left( A|B \\right)=\\frac{A{{\\natural}_{v}}B-A}{v}$. Let $A$ and $B$ be two positive invertible operators with the spectra contained in the interval $J \\subset (0,\\infty)$. We prove for any $v\\in \\left[ -1,0 \\right)\\cup \\left( 0,1 \\right]$, $$ (\\ln_v t)A+\\left( A{{\\natural}_{v}}B+tA{{\\natural}_{v-1}}B \\right)\\le {{T}_{v}}\\left( A|B \\right) \\le (\\ln_v s)A+{{s}^{v-1}}\\left( B-sA \\right) $$ where $s,t\\in J$. Especially, the upper bound for Tsallis relative operator entropy is a non-trivial new result. Meanwhile, some related and new results are also established. In particular, the monotonicity for Tsallis relative operator entropy is improved. Furthermore, we introduce the exponential type relative operator entropies which are special cases of the perspective and we give inequalities among them and usual relative operator entropies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-06T00:37:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A63",
      "46L05",
      "47A60"
    ],
    "keywords": [
      "tsallis relative operator entropy",
      "exponential type relative operator entropies",
      "usual relative operator entropies",
      "upper bound",
      "paper intends"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}