Trace-norm-small perturbations preserve Hermitian spectral distributions

Let $H_n$ be Hermitian $n \times n$ matrices whose eigenvalues are distributed,
in the Weyl sense, as a real measurable function $f$ on $[0, 1]$, and let $E_n$
be complex $n \times n$ matrices with trace norm $o(n)$. The manuscript proves
that the eigenvalues of $H_n + E_n$ are also distributed as $f$, with no bound
on spectral norms and no normality assumption, as Barbarino and
Serra-Capizzano conjectured. In a unitary Schur form $T$ of $H_n + E_n$, the
strictly upper triangular part is determined by the imaginary part of $T$,
whose trace norm is at most that of $E_n$; a published dimension-free
weak-type estimate for triangular truncation (Randrianantoanina; Dodds, Dodds,
de Pagter and Sukochev) bounds it. Singular-value counting and Hermitian
rank-perturbation bounds then show that the real parts of the diagonal of $T$
are distributed like the eigenvalues of $H_n$, while the trace norm controls
their imaginary parts.