# Weak Operator Topology

#### Mauritzvdworm

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

MHF Hall of Honor
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.

• Mauritzvdworm