Let $\mathcal H$ be a separable, infinite-dimensional Hilbert space.
The proof of Corollary $3$ in this paper claims that a given compact Hermitian operator $T$ can be assumed to be of the form (by considering $-T$ if necessary) $$T = \operatorname{diag}\{p_1, p_2, \ldots\} \oplus A \tag{1}$$ with $p_j \geqslant 0$ and $A \leqslant 0$ (not excluding the possibility that $A$ is absent).
How is this proved?
By the spectral theorem, I know that $T$ is unitarily equivalent to $$T_0:= \operatorname{diag}\{d_1, d_2, \ldots\}$$ with $d_n \in \Bbb R$ and $d_n \to 0$, but I haven't been able to use this to get something like $(1)$.
Also, are they writing $T$ as $\operatorname{diag}\{p_1, p_2, \ldots\} \oplus A$ only up to unitary equivalence?
The expression in $(1)$ is obtained simply by separating the negative $d_j$ from the nonnegative ones. As in $$ \begin{bmatrix} 1&0&0&0\\ 0&-2&0&0\\ 0&0&-3&0\\ 0&0&0&4\end{bmatrix} \simeq \begin{bmatrix} 1&0&0&0\\ 0&4&0&0\\ 0&0&-2&0\\ 0&0&0&-3\end{bmatrix} . $$
As for unitary equivalence, every time you express an operator in coordinates you are choosing an orthonormal basis. So yes, it is "up to unitary equivalence" if you want for some reason to stick with a particular orthonormal basis.