$\DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\Lim}{Lim^1} \DeclareMathOperator{\diag}{diag} \DeclareMathOperator{\trace}{Tr} \DeclareMathOperator{\spans}{span} \DeclareMathOperator{\spec}{\sigma} \DeclareMathOperator{\essspec}{\sigma_{\mathrm{ess}}} \DeclareMathOperator{\ind}{ind} \newcommand{\Hil}{\mathcal{H}} \newcommand{\vmid}{\,\middle\vert\,} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\emptyset}{\varnothing} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert } \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\angles}[1]{\left\langle #1 \right\rangle} \newcommand{\snorm}[1]{\lVert #1 \rVert } \newcommand{\sabs}[1]{\lvert #1 \rvert} \newcommand{\sangles}[1]{\langle #1 \rangle} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$