Borel Functional Calculus - Resolution of The Identity

Resolution of The Identity

Let T be a self-adjoint operator. If E is a Borel subset of R, and 1_E is the indicator function of E, then 1_E(T) is a self-adjoint projection on H. Then mapping

is a projection-valued measure called the resolution of the identity for the self adjoint operator T. The measure of R with respect to Ω is the identity operator on H. In other words, the identity operator can be expressed as the spectral integral I = ∫ 1 dΩ. Sometimes the term "resolution of the identity" is also used to describe this representation of the identity operator as a spectral integral.

In the case of a discrete measure (in particular, when H is finite dimensional), I = ∫ 1 dΩ can be written as

in the Dirac notation, where each |i> is a normalized eigenvector of T. The set { |i〉 } is an orthonormal basis of H.

In physics literature, using the above as heuristic, one passes to the case when the spectral measure is no longer discrete and write the resolution of identity as

and speak of a "continuous basis", or "continuum of basis states", { |i〉 }. Mathematically, unless rigorous justifications are given, this expression is purely formal.