A Dirac delta operator
If $T$ is a (densely defined) self-adjoint operator acting on a complex Hilbert space $\mathcal{H}$ and $I$ stands for the identity operator, we introduce the delta function operator $λ\mapsto δ\left(λI-T\right) $ at $T$. When $T$ is a bounded operator, then $δ\left(λI-T\right) $ is an operator-valued distribution. If $T$ is unbounded, $δ\left(λI-T\right) $ is a more general object that still retains some properties of distributions. We derive various operative formulas involving $δ\left(λI-T\right) $ and give several applications of its usage.
math.FA↗