A Dirac delta operator

Abstract

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 $\lambda \mapsto \delta \left(\lambda I-T\right)$ at $T$. When $T$ is a bounded operator, then $\delta \left(\lambda I-T\right)$ is an operator-valued distribution. If $T$ is unbounded, $\delta \left(\lambda I-T\right)$ is a more general object that still retains some properties of distributions. We derive various operative formulas involving $\delta \left(\lambda I-T\right)$ and give several applications of its usage.