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 1E is the indicator function of E, then 1E(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.

Read more about this topic:  Borel Functional Calculus

Famous quotes containing the words resolution and/or identity:

    I had crossed de line of which I had so long been dreaming. I was free; but dere was no one to welcome me to de land of freedom. I was a stranger in a strange land, and my home after all was down in de old cabin quarter, wid de ole folks, and my brudders and sisters. But to dis solemn resolution I came; I was free, and dey should be free also; I would make a home for dem in de North, and de Lord helping me, I would bring dem all dere.
    Harriet Tubman (c. 1820–1913)

    The “female culture” has shifted more rapidly than the “male culture”; the image of the go-get ‘em woman has yet to be fully matched by the image of the let’s take-care-of-the-kids- together man. More important, over the last thirty years, men’s underlying feelings about taking responsibility at home have changed much less than women’s feelings have changed about forging some kind of identity at work.
    Arlie Hochschild (20th century)