= Singular Weyl sequence
{title2=$\|f_n\|=1,\quad f_n\rightharpoonup0,\quad(L-\lambda I)f_n\to0$}
A singular Weyl sequence for a bounded <self-adjoint operator> $L$ at $\lambda$ is a <spectral Weyl sequence> that also converges weakly to zero. Such a sequence exists exactly when $\lambda$ is in the <essential spectrum of a bounded self-adjoint operator>. An infinite-dimensional shifted kernel gives a weakly null <orthonormal sequence>. If that kernel is finite-dimensional but the shifted range is not closed, choose approximate null vectors in the kernel complement and extract a weakly convergent subsequence; its limit belongs to both kernel and complement, so is zero.
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