# 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) $||T|| = \sup_{||x|| = 1} \{||\}$
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.