Functions of several self-adjoint operators

Abstract

Hermann Weyl, in [6], defined a functional calculus to deal with the unbounded selfadjoint operators of differentiation and multiplication by a position coordinate.In this paper we examine this calculus in the case of bounded operators.1We let x be a vector (xi, • • • , xn) in Rn, dx = dxi ■ ■ ■ dxn, <X, x)=XiXi+ • ■ • +X"x".We let A he an M-tuple of bounded selfadjoint operators iA\, • • • , A") and (X, A) = Xi^4i+ • • • +\nA".Define exp(i(X, A)) by the usual power series, which converges in norm since (X, A) is bounded.If /(X) is the Fourier transform of /, /(X) = (2x)_n/2//(x)exp(i(X,x))dx, and if both / and / are in D, we define/(^) = (2x)-"'2//(X)exp(-z(X, A))d\.This is the

Locations

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