Weak Operator Topology

Aug 2009
122
19
Pretoria
Let \(\displaystyle T_{n}\) be an operator on the Hilbert space \(\displaystyle L^2(\partial\mathbb{D},dt)\) with \(\displaystyle \partial\mathbb{D}\) denoting the boundary of the open unit disk in the complex plane. This operator is defined by
\(\displaystyle T_{n}f(t)=e^{int}f(t)\) for \(\displaystyle f\in L^2(\partial\mathbb{D},dt)\)
Show that this operator converges to the zero operator in the Weak Operator Topology

(Simple question, I'm just doing something wrong)
 

Opalg

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Let \(\displaystyle T_{n}\) be an operator on the Hilbert space \(\displaystyle L^2(\partial\mathbb{D},dt)\) with \(\displaystyle \partial\mathbb{D}\) denoting the boundary of the open unit disk in the complex plane. This operator is defined by
\(\displaystyle T_{n}f(t)=e^{int}f(t)\) for \(\displaystyle f\in L^2(\partial\mathbb{D},dt)\)
Show that this operator converges to the zero operator in the Weak Operator Topology

(Simple question, I'm just doing something wrong)
The inner product of \(\displaystyle T_nf\) with \(\displaystyle g\) is given by \(\displaystyle \langle T_nf,g\rangle = \int_0^{2\pi}\!\!\!e^{int}f(t)\overline{g(t)}\,dt\). But \(\displaystyle f\overline{g}\in L^1(\partial\mathbb{D})\) (by Hölder's inequality), and \(\displaystyle \langle T_nf,g\rangle\) is the (–n)-th Fourier coefficient of that function, which goes to 0 as \(\displaystyle n\to\infty\), by the Riemann–Lebesgue lemma. That shows that \(\displaystyle T_n\to0\) in the WOT.
 
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