Borel functional calculus

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In functional analysis, the Borel functional calculus is a functional calculus (i.e. an assignment of operators to functions defined on the real line), which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function ss2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator -Δ or the exponential

<math> e^{it \Delta}. \quad <math>

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.

More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function.

Contents

Motivation

If T is a self-adjoint operator on a finite dimensional inner product space H, H has an orthonormal basis

<math> \{e_k\}_{1 \leq k \leq \ell} <math>

consisting of eigenvectors of T, that is

<math> T e_k = \lambda_k e_k \quad 1 \leq k \leq \ell. \quad <math>

Thus, for any positive integer n,

<math> T^n e_k = \lambda_k^n e_k. \quad <math>

In this case, given a Borel function h, we can define an operator h(T) by specifying its behavior on the basis:

<math> h(T) e_k = h(\lambda_k) e_k. \quad <math>

In general, 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

<math> [T \psi](x) = f(x) \psi(x) \quad <math>

acting on L2 of some measure space. The domain of T consists of those functions for which the above expression is in L2. In this case, we can define analogously

<math> [h(T) \psi](x) = [h \circ f](x) \psi(x). <math>

For many technical purposes, the preceding formulation is good enough. However, it is desirable to formulate the functional calculus in a way in which it is clear that it does not depend on the particular representation of T as a multiplication operator. This 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 bounded complex-valued Borel functions f on the real line,

<math> \pi_T: h \mapsto h(T) <math>

such that the following conditions hold

  • 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
<math> \nu_\xi:E \mapsto \langle \pi_T(\mathbf{1}_E) \xi, \xi \rangle <math>
is a countably additive measure on the Borel sets of R. In the above formula 1E denotes the indicator function of E. These measures νξ are called the spectral measures of T.
  • <math> \pi_T([\eta +i]^{-1}) = [T + i]^{-1} \quad <math>
where η denotes the mapping zz on C.

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

<math> U_t = e^{i t A} \quad t \in \mathbb{R} <math>

is a 1-parameter strongly continuous unitary group whose infinitesimal generator is i A.

As an application, we consider the Schrödinger equation, or equivalently, the dynamics of a quantum mechanical system. In non-relativistic quantum mechanics, the Hamiltonian operator H models the total energy observable of a quantum mechanical system S. The unitary group generated by i H corresponds to the time evolution of S.

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.

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

  • <math> \operatorname{dom} S = \left\{\xi \in H: h \in L^2_{\nu_\xi}(\mathbb{R}) \right\}<math>
  • <math> \langle S \xi, \xi \rangle = \int_{\mathbb{R}} h(t) \ d\nu_{\xi} (t), \quad \mbox{for} \quad \xi \in \operatorname{dom} S<math>

The operator S of the previous theorem is denoted h(T).

References

  • R. Kadison and J. Ringrose, Fundamentals of the Theory of Operator Algebras vol 1, Academic Press, 1983
  • M. Reed and B. Simon, Methods of Mathematical Physics, Academic Press, 1972.
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