{
  "id": "1010.4189",
  "version": "v1",
  "published": "2010-10-20T13:11:28.000Z",
  "updated": "2010-10-20T13:11:28.000Z",
  "title": "Numerical shadows: measures and densities on the numerical range",
  "authors": [
    "Charles F. Dunkl",
    "Piotr Gawron",
    "John A. Holbrook",
    "Zbigniew Puchała",
    "Karol Życzkowski"
  ],
  "comment": "37 pages, 8 figures",
  "journal": "Linear Algebra Appl., Vol. 434, pp. 327-342, (2011)",
  "doi": "10.1016/j.laa.2010.12.003",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For any operator $M$ acting on an $N$-dimensional Hilbert space $H_N$ we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of $M$. The shadow of $M$ at point $z$ is defined as the probability that the inner product $(Mu,u)$ is equal to $z$, where $u$ stands for a random complex vector from $H_N$, satisfying $||u||=1$. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian $M$ its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional $B$-spline. In the case of a normal $M$ the numerical shadow corresponds to a shadow of a transparent solid simplex in $R^{N-1}$ onto the complex plane. Numerical shadow is found explicitly for Jordan matrices $J_N$, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-20T13:11:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "numerical range",
      "complex plane",
      "dimensional hilbert space",
      "probability",
      "shadow measures play"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.4189D"
    }
  }
}