example 12.56 Multiplication by the coordinate: spectrum without eigenvectors

Rests on

• depends_on definition 12.50 Point, continuous and residual spectrum ¶
  • depends_on corollary 12.19 Double complement; the density criterion ¶
    • depends_on proposition 12.17 The complement is always a closed subspace ¶
      • depends_on corollary 12.5 Continuity of the norm and of orthogonality ¶
        • depends_on definition 5.24 Orthogonal vectors ¶
        • depends_on proposition 12.4 Cauchy–Schwarz and continuity of the inner product ¶
        • proves proof ch:10-hilbert-spaces@proof-2 ¶
      • depends_on definition 12.16 Orthogonal complement ¶
        • depends_on definition 5.24 Orthogonal vectors ¶ ↺
        • depends_on equation 5.52 eq:lin-orthogonal-complement ¶
      • proves proof ch:10-hilbert-spaces@proof-8 ¶
    • depends_on theorem 12.18 Projection theorem ¶
      • depends_on proposition 12.17 The complement is always a closed subspace ¶ ↺
      • depends_on theorem 12.14 Closest point in a closed convex set ¶
        • depends_on definition 12.13 Convex set ¶
        • depends_on definition 12.2 Hilbert space ¶
        • depends_on proposition 12.6 Parallelogram law and polarization ¶
        • proves proof ch:10-hilbert-spaces@proof-7 ¶
      • proves proof ch:10-hilbert-spaces@proof-9 ¶
    • proves proof ch:10-hilbert-spaces@proof-10 ¶
  • depends_on definition 12.49 Resolvent set; spectrum ¶
    • depends_on definition 12.35 Bounded operator; operator norm ¶
      • depends_on definition 5.37 Linear transformation ¶
        • depends_on definition 4.33 Vector space ¶
      • depends_on definition 5.19 Norm ¶
        • depends_on definition 4.33 Vector space ¶ ↺
    • depends_on definition 5.47 Inverse of a linear transformation ¶
      • depends_on definition 5.37 Linear transformation ¶ ↺
• depends_on example 12.11 The function space $L^{2}$ ¶
  • depends_on definition 12.2 Hilbert space ¶ ↺
  • depends_on definition 5.18 Inner product ¶
    • depends_on definition 4.33 Vector space ¶ ↺
• depends_on theorem 12.55 The spectrum of a self-adjoint operator is real ¶
  • depends_on corollary 12.19 Double complement; the density criterion ¶ ↺
  • depends_on definition 12.41 The operator classes ¶
    • depends_on definition 6.9 Compact set ¶
      • depends_on definition 6.5 Open cover ¶
        • depends_on definition 6.2 Open set ¶
        • depends_on equation 3.51 eq:set-indexed ¶
    • depends_on proposition 12.21 Characterization of orthogonal projections ¶
      • depends_on definition 12.20 Orthogonal projection operator ¶
        • depends_on theorem 12.18 Projection theorem ¶ ↺
      • depends_on theorem 12.18 Projection theorem ¶ ↺
      • proves proof ch:10-hilbert-spaces@proof-11 ¶
    • depends_on theorem 12.38 Existence and uniqueness of the adjoint ¶
      • depends_on definition 5.41 Adjoint ¶
        • depends_on definition 5.18 Inner product ¶ ↺
        • depends_on definition 5.37 Linear transformation ¶ ↺
      • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶
        • depends_on definition 12.35 Bounded operator; operator norm ¶ ↺
        • depends_on proposition 12.8 Absolutely convergent series test ¶
        • proves proof ch:10-hilbert-spaces@proof-19 ¶
      • depends_on theorem 12.46 Riesz representation ¶
        • depends_on definition 12.45 Continuous linear functional; the dual ¶
        • depends_on theorem 12.18 Projection theorem ¶ ↺
        • proves proof ch:10-hilbert-spaces@proof-24 ¶
      • proves proof ch:10-hilbert-spaces@proof-20 ¶
  • depends_on theorem 12.54 The spectrum is compact and non-empty ¶
    • depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶
      • depends_on definition 12.49 Resolvent set; spectrum ¶ ↺
      • depends_on proposition 12.37 $\mathcal{B}(\mathcal{H})$ is a Banach algebra ¶ ↺
      • proves proof ch:10-hilbert-spaces@proof-27 ¶
    • depends_on proposition 12.53 The resolvent set is open, the resolvent analytic ¶
      • depends_on definition 8.5 Complex derivative ¶
      • depends_on proposition 12.52 Neumann series; the spectrum is bounded ¶ ↺
      • proves proof ch:10-hilbert-spaces@proof-28 ¶
    • depends_on theorem 8.18 Liouville ¶
      • depends_on proposition 8.10 Fundamental theorem for contours ¶
        • depends_on equation 8.6 eq:cpx-contour-integral ¶
        • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • proves proof ch:06-complex-analysis@proof-5 ¶
      • depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
        • depends_on lemma 8.9 ML estimate ¶
        • depends_on theorem 8.16 Cauchy integral formula ¶
        • proves proof ch:06-complex-analysis@proof-11 ¶
      • proves proof ch:06-complex-analysis@proof-12 ¶
    • proves proof ch:10-hilbert-spaces@proof-29 ¶
  • proves proof ch:10-hilbert-spaces@proof-30 ¶
• proves proof ch:10-hilbert-spaces@proof-31 ¶

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