Banach 23

The lecture: banach23

Excerpts

Definitions

  1. External direct sums of Hilbert spaces
  2. Internal direct sums of Hilbert subspaces
  3. Unitary operator between Hilbert spaces

Results

  1. T \in \mathscr{L}(H) implies ||T||^2 = ||T*T||
  2. \mathscr{L}(H) is C^* algebra with taking adjoint as involution.
  3. <Tx,x> = 0 for all x implies T = 0. (in the pf its supposed to say <Tx,y> = etc., and are using sequilinear innerproduct <x,y>' := <Tx,y>.
  4. P \in \mathscr{L}(H) idempotent is orthogonal projection iff P = P^*
  5. T \in \mathscr{L}(H) is self adjoint iff <Tx,x> \in \mathbb{R} \forall x.
  6. U \in \mathscr{L}(H_1, H_2) is unitary iff it is invertible and isometric.
  7. T \in \mathscr{L}(H) is normal iff ||Tx|| = ||T^*x|| \forall x.
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