Representations by Commuting Operators

Abstract

As has been explained in the introduction to Chapter 3, by using the spectral projection theorem, one can obtain the spectral representations for the families A = (A _x ) _x∼X of commuting normal operators in H ₀ satisfying certain relations. Recall that, according to this theorem, the following equality holds

$${A_x} = \int\limits_{g(A)} {\lambda (x)dE(\lambda ( \cdot ))(x \in X)} $$

(0.1)

where g (A) is a generalized spectrum of the family. Assume that the operators A(x) are not arbitrary but satisfy certain relations which can be written in the form F(A.) = 0. More exactly, this means that a certain functional or operator F(f(·)) is defined on the space of complex-valued functions f(x) of the variable x∼X, and this functional is such that the formal substitution of A _x for f(x) in the corresponding expression is meaningful (for example,

$$X = {R^1}andF(f( \cdot )) = f({x_1}) + f({x_2})({x_1},{x_2} \in {R^1});F(f( \cdot )) = - f''( \cdot ) + q( \cdot )f( \cdot )$$

where q ∼ C(ℝ¹) etc.). Then, by using the fact that ξ is a generalized joint eigenvector of the family A with the eigenvalue λ(·) (this means that A _x ξ = λ(x)ξ, for any x ∼ X), one can usually conclude that the equality F(λ(·)) = 0 holds (for this purpose, we “convey” the vector ξ through the expression for F). In other words, every eigenvalue from g(A) satisfies the equation F(λ(·)) = 0 and, therefore, the region of integration in (0.1) may be replaced by a collection of all the solutions of this equation. If, in addition, this collection can be described somehow, then the functional integral from (0.1) may be transformed into a simpler and analyzable subject.