Banach Lecture 16,17

These lectures ( Banach16&17) were given on 10/2/09, 10/5/09


In these lecture the following result was proved.

Thm: Let X is an infinite dimensional Banach space, K \in \mathscr{L}(X) is compact.  If sp(K) - 0 is nonempty then the set consists of eigenvalues of finite multiplicity (meaning \dim \ker (K - \lambda I) is finite dimensional ) with no limit other than zero.  This will follow from the following properties.  With M_j = (K-I)^jX, N_j = \ker (K-I)^j, then

  1. \ker (K - I) is finite dimensional
  2. (K - I)X is closed and of finite codimension.
  3. \exists j,i such that M_j = M_{j+1}, N_i = N_{i+1}
  4. i = j and X = M_i \oplus N_i
  5. (K - I)|_{M_i} is invertible and (K - I)|_{N_i} is nilpotent or order i

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