# Banach 25

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

### Excerpts

Definitions

- a positive, finite, compactly supported Borel measure on , then define via . is normal and called a canonical multiplication operator.
- is cyclic if such that images of under polynomials in are dense in , in this case is a cyclic vector for (e.g. and )
- are unitarily equivalent, if that is unitary such that .

Results

- (Spectral Thm for compact normal operators) compt. normal. Let be its nonzero eigenvalues and corresponding eigenspaces; . Then (internal direct sum).
- Thm III.3.1 Every cyclic normal operator is unitarily equivalent to a canonical multiplication operator.

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## About this entry

You’re currently reading “Banach 25,” an entry on Math Meandering

- Published:
- October 29, 2009 / 12:16 pm

- Category:
- Banach Algebras (Course)

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