Here is a statement about "complete system of projectors" :
Let $E$ a $\mathbb{K}-$vector space and $p_1,...p_n$, $n$ linear maps. For all $i,j\in\{1,...,n\}$ : $p_i \circ p_j=0$ with $i\neq j$, for all $i\in\{1,...,n\}, \ p_i^2=p_i$ and $p_1+...+p_n=id_E$ iff $E= \bigoplus\limits_{i=1}^{n} \text{Im}(p_i)$.
What are the abstract or concrete applications of this result ?
Maybe it is linked with the kernel's lemma and spectral projectors in reduction's theory.
It is also probably linked with quadratic forms and symmetric endomorphisms. There are Fisher-Cochran's results in probability/statistics.
Maybe we can find similar properties in functional analysis.
And finally I was wondering if there are applications of that statement in finite or infinite groups representation theory ? For instance we have the Reynolds operator.
Thanks in advance !
Certainly the idea that (closed) subspaces of Hilbert spaces have (orthogonal) projectors to them (an immediate corollary of the minimization theorem, a true Hilbert-space analogue of the literally false pseudo-Dirichlet Principle in Banach spaces), is important throughout mathematics.
Yes, representation theory, yes, spectral theory, yes ... everything. In representation theory, e.g., of compact topological groups (so, certainly of finite groups on complex vector spaces), mutually non-isomorphic subrepresentations have orthogonal projectors. This is Schur orthogonality relations (which is a non-trivial fact even for finite groups).