Banach 24

This lecture (Banach24 ) was given on 10/21/09

Excerpts

Definitions

  1. approximate eigenvalue
  2. Subspaces of a hilbert space that induce operators 

Results

  1. Prop III.2.3, T \in \mathscr{L}(H) normal.  Then (1) Tx = \lambda x implies T^*x = \bar \lambda x (2) All pts of sp(T) are approximate eigenvalues, (3) ||T|| = \sup_{||x|| = 1} \{|<Tx,x>|\}
  2. Cor: T \in \mathscr{L}(H) self adjoint, then the \sup in 1 is equal to \max \{\lambda \in sp(T)\} and similarly for \inf and \min.
  3. Prop. III.2.4 T \in \mathscr{L}(H) and M a subspace with TM \subset M and P \colon H \to M is the orthogonal projection.  Then (1) M is T inv. iff TP = PTP, (2) M is T inv iff M^\perp is T^* inv. (3) M induces T iff PT = TP.

There was a note about operator matrices.

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