In their paper https://link.springer.com/chapter/10.1007%2F978-1-4684-7595-1_2 Birman and Solomyak consider transformers of the Hilbert space $\mathcal{S_2}$ of the Hilbert-Schmidt operators on a separable Hilbert space $\mathcal{H}$ that are defined by $\mathcal{F}_{\mu}:\mathcal{S}_2\rightarrow\mathcal{S}_2,T\mapsto F_{\mu}T$ and $\mathcal{E}_{\lambda}:\mathcal{S}_2\rightarrow\mathcal{S}_2,T\mapsto TE_{\lambda}$ where $E_{\lambda}$ and $F_{\mu}$ are said to be orthogonal expansions of unity and define for $\delta=[a,b)$ and $\partial=[c,d)$ and $\Delta=\delta\times\partial$ the additive function $$G(\Delta)=\mathcal{E}(\delta)\mathcal{F}(\partial)=(\mathcal{E}_b-\mathcal{E}_a)(\mathcal{F}_d-\mathcal{F}_c)$$. They say that $G$ can be continued to the orthogonal spectral measure $G(e)$ defined over a certain set class and say: Now let $\phi(\lambda,\mu)$ be a Borel function essentially bounded with respect to measure $G(e)$, $$(G)-\text{ess}\sup|\phi(\lambda,\mu)|<\infty.$$ They elaborate on this in these lecture notes: https://www.math.kth.se/spect/preprints02_03/birman_solomyak.pdf.
$\textbf{Question:}$ I assume that orthogonal expansion of unity means orthogonal projection. Is that right? Where does this term come from? If I understand the reminder on spectral measures (2.1) and what is done in section 3 of these last notes right, can I define this $(G)-\text{ess}\sup$ in the paper in the following way: $$(G)-\text{ess}\sup|\phi(\lambda,\mu)|=\inf\{s\geq 0:(G(\{(\lambda,\mu)\in\mathbb{R}^2:|\phi(\lambda,\mu)|>s\})T,T)_{\mathcal{S_2}}=0\forall T\in\mathcal{S}_2\}$$ where $(T,S)_{\mathcal{S}_2}=\text{tr}S^*T$ is the inner product in $\mathcal{S}_2$ and what can be said about the fact that no complex measures $(G(.)T,S)$ appear? Thank You very much in advance!