Borel functional calculus
In
The 'scope' here means the kind of function of an operator which is allowed. The Borel functional calculus is more general than the continuous functional calculus, and its focus is different than the holomorphic functional calculus.
More precisely, the Borel functional calculus allows for applying an arbitrary
Motivation
If T is a self-adjoint operator on a finite-dimensional
Thus, for any positive integer n,
If only polynomials in T are considered, then one gets the
Generally, any self-adjoint operator T is unitarily equivalent to a multiplication operator; this means that for many purposes, T can be considered as an operator
For many technical purposes, the previous formulation is good enough. However, it is desirable to formulate the functional calculus in a way that does not depend on the particular representation of T as a multiplication operator. That's what we do in the next section.
The bounded functional calculus
Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,
- πT is an involution-preserving and unit-preserving homomorphism from the ring of complex-valued bounded measurable functions on R.
- If ξ is an element of H, then is acountably additive measure on the Borel sets E of R. In the above formula 1E denotes the indicator functionof E. These measures νξ are called the spectral measures of T.
- If η denotes the mapping z → z on C, then:
Theorem — Any self-adjoint operator T has a unique Borel functional calculus.
This defines the functional calculus for bounded functions applied to possibly unbounded self-adjoint operators. Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups:
Theorem — If A is a self-adjoint operator, then
As an application, we consider the
We can also use the Borel functional calculus to abstractly solve some linear initial value problems such as the heat equation, or Maxwell's equations.
Existence of a functional calculus
The existence of a mapping with the properties of a functional calculus requires proof. For the case of a bounded self-adjoint operator T, the existence of a Borel functional calculus can be shown in an elementary way as follows:
First pass from polynomial to continuous functional calculus by using the Stone–Weierstrass theorem. The crucial fact here is that, for a bounded self adjoint operator T and a polynomial p,
Consequently, the mapping
Alternatively, the continuous calculus can be obtained via the
Given an operator T, the range of the continuous functional calculus h → h(T) is the (abelian) C*-algebra C(T) generated by T. The Borel functional calculus has a larger range, that is the closure of C(T) in the weak operator topology, a (still abelian) von Neumann algebra.
The general functional calculus
We can also define the functional calculus for not necessarily bounded Borel functions h; the result is an operator which in general fails to be bounded. Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by the composition of h with f.
Theorem — Let T be a self-adjoint operator on H, h a real-valued Borel function on R. There is a unique operator S such that
The operator S of the previous theorem is denoted h(T).
More generally, a Borel functional calculus also exists for (bounded) normal operators.
Resolution of the identity
Let be a self-adjoint operator. If is a Borel subset of R, and is the indicator function of E, then is a self-adjoint projection on H. Then mapping
- .
Stone's formula[3] expresses the spectral measure in terms of the resolvent :
Depending on the source, the resolution of the identity is defined, either as a projection-valued measure ,[4] or as a one-parameter family of projection-valued measures with .[5]
In the case of a discrete measure (in particular, when H is finite-dimensional), can be written as
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
References
- ISBN 0-8218-0819-2.
- ISBN 0-12-585050-6.
- doi:10.1070/RM9917.
- ISBN 978-0-07-054236-5.
- ISBN 0-273-08496-8.