On the construction of “new” semiorthogonal decompositions and Yoneda embeddings for some subcategories of the derived category of quasicoherent sheaves

The talk is devoted to the construction of new admissible subcategories and semiorthogonal decompositions from original ones.
If $(\mathcal{LA},\mathcal{RA})$ is a semi-orthogonal decomposition of the category $\underline{C}_0$ of compact objects in a compactly generated triangulated category $\underline{C}$ (respectively, $(\mathcal{LA},\mathcal{RA})$ is a $t$-structure and a weight structure invariant under $[1]$, the subcategory $\mathcal{LA}$ is left admissible in $\underline{C}_0$, and $\mathcal{RA}$ is right admissible), then there exists an “orthogonal” semi-orthogonal decomposition $(\mathcal{LA}{{}^{\perp}},\mathcal{RA}{{}^{\perp}}$) of $\underline{C}$ itself; the classes $\mathcal{RA}{{}^{\perp}}$ and $\mathcal{RA}{{}^{\perp}}$ are characterized by the absence of nonzero morphisms into them from elements of $\mathcal{RA}$ and $\mathcal{RA}$, respectively. Further, if $\underline{C}'$ is a subcategory of $\underline{C}$ that can be characterized in terms of morphisms from $\underline{C}_0$, then this semi-orthogonal decomposition is restricted to $\underline{C}'$. If $\underline{C}'$ is in some sense dual to $\underline{C}_0$, then we obtain a bijection between the semiorthogonal decompositions of $\underline{C}_0$ and $ \underline{C}'$.
These statements easily generalize to semiorthogonal decompositions of arbitrary length. Recent results of A. Ne'eman (and the speaker) allow us to apply them to various derived categories of quasicoherent sheaves on a scheme $X$ that is proper over the spectrum of a Noetherian ring $R$. This gives a one-to-one correspondence between the semiorthogonal decompositions of the categories $D_{perf}(X)$ and $D^b(\operatorname{coh}(X))$; The latter extend to $D^-(\operatorname{coh}(X))$, $D^+_{coh}(X)$, $D_{coh}(X)$, and $D_{qcoh} (X)$ $D^+_{coh}(\operatorname{Qcoh}(X))$, $D_{coh}(\operatorname{Qcoh}(X))$, and $D(\operatorname{Qcoh}(X))$ (if very weak additional assumptions are satisfied).
If time allows, I will explain that the transition from $(\mathcal{LA},\mathcal{RA})$ to ($\mathcal{LA}{{}^{\perp}},\mathcal{RA}{{}^{\perp}}$) is a special case of the orthogonality of weight and $t$- structures, which is precisely what inspired these arguments in the talk.