Banach 25

This lecture ( Banach25) was given on 10/23/09.

Excerpts

Definitions

  1. \mu a positive, finite, compactly supported Borel measure on \mathbb{C}, then define T_\mu \in \mathscr{L}(L^2(\mu)) via (T_\mu f)(z) = zf(z).  T_\mu is normal and called a canonical multiplication operator.
  2. T \in \mathscr{L}(H) is cyclic if \exists x such that images of x under polynomials in T,T^* are dense in H, in this case x is a cyclic vector for T (e.g. T = T_\mu and x = 1)
  3. T_i \in \mathscr{L}(H_i) are unitarily equivalent, if \exists U \in \mathscr{L}(H_2,H_1) that is unitary such that T_2 = U^{-1}TU.

Results

  1. (Spectral Thm for compact normal operators) T \in \mathscr{L}(H) compt. normal.  Let \lambda_1, \lambda_2,... be its nonzero eigenvalues and M_1, M_2, ... corresponding eigenspaces; M_0 = \ker T.  Then H = \sum_{n\ge 0} \oplus M_n (internal direct sum).
  2. Thm III.3.1 Every cyclic normal operator is unitarily equivalent to a canonical multiplication operator.
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