Q:

Is there a special meaning for this operator?

Does the operator
$$\prod_{j\neq i} \frac{\mathbf{\Lambda}-\lambda_i}{\lambda_j-\lambda_i}$$
have any special meaning that makes it useful in quantum mecahnics, given that $\{\lambda_i\}$ are the eigenvalues of the Hermitian non-degenerate operator $\mathbf\Lambda$?

A:

It is the projection operator onto the eigenspace to $\lambda_j$.

It evaluates to $1$ on $\mathrm{Eig}(\lambda_j)$.
Every other eigenspace is annihilated because of the nominator. Note that the product is over all but one eigenvalue.
It is self-adjoint (every factor is self-adjoint and they mutually commute) and
squares to itself.

Due to the product it is useful/practical to use in finite dimensions only I believe.
I have somewhere seen it utilized in the context of spin chains, where one would project composite spins onto a definite $\vec{S}^2$ subspace. It was a seminar talk, but maybe I can find reference.

