  The resolvent of an operator in a Banach space is defined on an open subset
of the complex plane and is holomorphic. It obeys the resolvent equation. A
generalization of this equation to Schwartz distributions is defined and a
Schwartz distribution, which satisfies that equation is called a resolvent
distribution. Its restriction to the subset, where it is continuous, is the
usual resolvent function. Its complex conjugate derivative is,but a factor, the
spectral Schwartz distribution, which is carried by a subset of the spectral
set of the operator. The spectral distribution yields a spectral decomposition.
The spectral distribution of a matrix and a unitary operator are given. If the
the operator is a self-adjoint operator on a Hilbert space, the spectral
distribution is the derivative of the spectral family. We calculate the
spectral distribution of the multiplication operator and some rank one
perturbations. These operators are not necessarily self adjoint and may have
discrete real or imaginary eigenvalues or a nontrivial Jordan decomposition.
