The essential spectrum

  • Alexandru Aleman
  • William T. Ross
  • Nathan S. Feldman
Part of the Frontiers in Mathematics book series (FM)


If B(H) is the algebra of bounded linear operators on a Hilbert space (H) and K is the ideal of compact operators on (H), one forms the Calkin algebra B(H)/K and the natural map π: B(H)B(H)/K. Recall that AB(H) is Fredholm if π(A) is invertible in B(H)/K. A well-known theorem [19, p. 356] says that A is Fredholm precisely when Rng A is closed and both ker A and H/RangA are finite dimensional. An operator A is semi-Fredholm if π(A) is either right or left invertible in B(H)/K. Equivalently. A is semi-Fredholm if and only if RngA is closed and either ker(A) or H/RngA is finite dimensional. We also use the notation
$$ \sigma (A): = \{ \lambda \in \mathbb{C}:\lambda I - A is not invertible\} (spectrum of A),$$
$$ \sigma (A): = \{ \lambda \in \mathbb{C}:\lambda I - A is not Fredholm\} (essential spectrum of A).$$
Note that σ e (A) ⊂ σ(A). For a semi-Fredholm operator A let
$$ ind(A): = dim ker A - dim (H/Rng A)$$
be the index of A. When the set ℤ∪{±∞| is endowed with the discrete topology, the map A→ ind(A) (from the set of semi-Fredholm operators to ℤ∪{±∞| is continuous [19, p. 361].


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Copyright information

© Birkhäuser Verlag AG 2009

Authors and Affiliations

  • Alexandru Aleman
    • 1
  • William T. Ross
    • 2
  • Nathan S. Feldman
    • 3
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Department of Mathematics and Computer ScienceUniversity of RichmondRichmondUSA
  3. 3.Department of MathematicsWashington & Lee UniversityLexingtonUSA

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