{
  "id": "1510.08350",
  "version": "v1",
  "published": "2015-10-28T15:39:30.000Z",
  "updated": "2015-10-28T15:39:30.000Z",
  "title": "Tests for complete $K$-spectral sets",
  "authors": [
    "Michael A. Dritschel",
    "Daniel Estévez",
    "Dmitry Yakubovich"
  ],
  "comment": "32 pages, 1 figure",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Phi$ be a family of functions analytic in some neighborhood of a complex domain $\\Omega$, and let $T$ be a Hilbert space operator whose spectrum is contained in $\\overline\\Omega$. Our typical result shows that under some extra conditions, if the closed unit disc is complete $K'$-spectral for $\\phi(T)$ for every $\\phi\\in \\Phi$, then $\\overline\\Omega$ is complete $K$-spectral for $T$ for some constant $K$. In particular, we prove that under a geometric transversality condition, the intersection of finitely many $K'$-spectral sets for $T$ is again $K$-spectral for some $K\\ge K'$. These theorems generalize and complement results by Mascioni, Stessin, Stampfli, Badea-Beckerman-Crouzeix and others. We also extend to non-convex domains a result by Putinar and Sandberg on the existence of a skew dilation of $T$ to a normal operator with spectrum in $\\partial\\Omega$. As a key tool, we use the results from our previous paper on traces of analytic uniform algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-28T15:39:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A25",
      "30H50",
      "46J15",
      "47A12",
      "47A20"
    ],
    "keywords": [
      "spectral sets",
      "geometric transversality condition",
      "analytic uniform algebras",
      "hilbert space operator",
      "extra conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}