{
  "id": "2303.11533",
  "version": "v1",
  "published": "2023-03-21T01:35:05.000Z",
  "updated": "2023-03-21T01:35:05.000Z",
  "title": "The operator $(p, q)$-norm of some matrices",
  "authors": [
    "Imam Nugraha Albania",
    "Masaru Nagisa"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We compute the operator $(p,q)$-norm of some $n\\times n$ complex matrices, which can be seen as bounded linear operators from the $n$ dimensional Banach space $\\ell^p(n)$ to $\\ell^q(n)$. We have shown that a special matrix $A=\\begin{pmatrix} 8 & 1 & 6 \\\\ 3 & 5 & 7 \\\\ 4 & 9 & 2 \\end{pmatrix}$ which corresponds to a magic square has $\\|A\\|_{p,p} = \\max \\{\\|A\\xi\\|_p : \\xi\\in\\ell^p(n), \\|\\xi\\|_p=1\\} =15$ for any $p\\in [1,\\infty]$. In this paper, we extend this result and we compute $\\|A\\|_{p,q}$ for $1\\le q \\le p \\le \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-21T01:35:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A30",
      "15B05"
    ],
    "keywords": [
      "dimensional banach space",
      "bounded linear operators",
      "special matrix",
      "magic square",
      "complex matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}