{
  "id": "1102.0748",
  "version": "v1",
  "published": "2011-02-03T18:43:04.000Z",
  "updated": "2011-02-03T18:43:04.000Z",
  "title": "On the norm of the $q$-circular operator",
  "authors": [
    "Natasha Blitvić"
  ],
  "categories": [
    "math-ph",
    "math.CO",
    "math.MP",
    "math.OA"
  ],
  "abstract": "The $q$-commutation relations, formulated in the setting of the $q$-Fock space of Bo\\.zjeko and Speicher, interpolate between the classical commutation relations (CCR) and the classical anti-commutation relations (CAR) defined on the classical bosonic and fermionic Fock spaces, respectively. Interpreting the $q$-Fock space as an algebra of \"random variables\" exhibiting a specific commutativity structure, one can construct the so-called $q$-semicircular and $q$-circular operators acting as $q$-deformations of the classical Gaussian and complex Gaussian random variables, respectively. While the $q$-semicircular operator is generally well understood, many basic properties of the $q$-circular operator (in particular, a tractable expression for its norm) remain elusive. Inspired by the combinatorial approach to free probability, we revist the combinatorial formulations of these operators. We point out that a finite alternating-sum expression for $2n$-norm of the $q$-semicircular is available via generating functions of chord-crossing diagrams developed by Touchard in the 1950s and distilled by Riordan in 1974. Extending these norms as a function in $q$ onto the complex unit ball and taking the $n\\to\\infty$ limit, we recover the familiar expression for the norm of the $q$-semicircular and show that the convergence is uniform on the compact subsets of the unit ball. In contrast, the $2n$-norms of the $q$-circular are encoded by chord-crossing diagrams that are parity-reversing, which have not yet been characterized in the combinatorial literature. We derive certain combinatorial properties of these objects, including closed-form expressions for the number of such diagrams of any size with up to eleven crossings. These properties enable us to conclude that the $2n$-norms of the $q$-circular operator are significantly less well behaved than those of the $q$-semicircular operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-03T18:43:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "commutation relations",
      "semicircular operator",
      "combinatorial",
      "complex gaussian random variables",
      "fermionic fock spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.0748B"
    }
  }
}