Banach Lecture 8

This lecture (banach8) was given on 9/14/09.

Excerpts

This lecture presented many examples of Gelfond Theory.  

1. Consider again B = l^1(\mathbb{Z}).  It was shown \mathscr{M}(B) can be identified with \mathbb{T}.  Writing x \in B as x = \sum_{n \in \mathbb{Z}} x(n)e_n it follows that

\hat x (e^{it}) = \sum_{n \in \mathbb{Z}} x(n)e^{int}

conclusion: \hat x is a function on the unit circle with absolutely convergent Fourier series.  An so define the Weiner Algebra 

W = \widehat{l^1(\mathbb{Z})}

it become a banach algebra.

Thm I.8.1 (D.Levy) Let f \in W and h holom. on open set containing f(\mathbb{T}), then h\circ f \in W.

proof: apply thm I.7.2.

Cor. (N Weiner) If f \in W, then 1/f \in W.

Rmk: this is hard result to prove without Gelfond theory.

Some more examples included determining \mathscr{M}(B) for B equal

  1. l^1(\mathbb{Z})
  2. l^1(\mathbb{Z}_{\ge 0})
  3. L^1(\mathbb{T})
  4. the Disk algebra = A = \{ f \in C(\overline{\mathbb{D}})| f|_{\mathbb{D}} \mbox{ is holom.} \}, norm is sup norm.

Thm. I.8.2 Let f_1, ..., f_m \in A have no common zero in the closed unit disk.  Then there exists g_1, ..., g_m \in A s.t \sum_i f_ig_i = 1.

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