Let $L$ be a self-adjoint operator with discrete spectrum $S=\{\lambda_1 < \lambda_2 < \dots \}$ on a Hilbert space $H$ such that the spectral theorem holds, i.e. for any $F \in H$ we have the orthogonal decomposition
$$ F = \sum_{k=1}^{\infty} F_k,$$
where $F_k \in \operatorname{Ker}(L - \lambda_k \operatorname{Id})$.
Is there a standard name and some standard notation for the operators $T_k$ "taking out" a projection, defined as $T_k = \operatorname{Id} - \pi_k$, where $\pi_k$ denotes the orthogonal projection onto $\operatorname{Ker}(L - \lambda_k \operatorname{Id})$?
If $P$ is a projection, the operator $I-P$ is called its complementary projection. It is sometimes denoted $P^\perp$.
(So named because they project onto complementary subspaces, and/or because of the parallel with set complements: $I=P+P^\perp$ is similar to $X=A\cup A^c$.)