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Let $(X,B,\mu)$ be a probability space and let $U$ be a unitary operator on $L_2(X,B,\mu)$. Suppose that $g_n$ is a convergent sequence in $L_2(X,B,\mu)$, $g_n\rightarrow g$. Suppose also that there is some increasing sequence of integers $m_k$ such that for every $n$, $\lim_{k\rightarrow \infty}U^{m_k}g_n= g_n$. Would it be correct to say: $$\lim_{k\rightarrow\infty}\lim_{n\rightarrow\infty}U^{m_k}g_n=\lim_{n\rightarrow\infty}\lim_{k\rightarrow\infty}U^{m_k}g_n=g,$$ i.e., is it ok to switch the limits?

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up vote 4 down vote accepted

It is OK. Since $\lim_{n\to\infty} U^{m_k} g_n=U^{m_k}g$, it suffices to show that $\lim_{k\to\infty}U^{m_k}g=g$. Note that

$$\|U^{m_k} g-g\|\le \|U^{m_k} (g-g_n)\|+\|U^{m_k} g_n-g_n\| +\|g-g_n\|=\|U^{m_k} g_n-g_n\| +2\|g-g_n\|,$$ where the last step is because $U^{m_k}$ is unitary. Letting $k\to\infty$, it follows that $$\limsup_{k\to\infty} \|U^{m_k} g-g\|\le 2\|g-g_n\|.$$ Letting $n\to\infty$, the conclusion follows.

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