谱表示

<正> Stone M.对Hilbert空间中一个具有简单谱的自伴算子建立了谱表示定理,即有实轴上的有限Borel测度μ,使得同构于L~2(μ),同时变A为乘以自变量λ的算子.Jauch等([2])讨论了一列交换的自伴算子完全集谱表示定理,但要求一个关于测度绝对连续性的假定.此外,依据约化理论([3])可知,如果A是可分Hilbert空间的自伴

THE SPECTRAL REPRESENTATION

We give the following results.Theorem 1 Let {A_1|l∈A} be a commutative set of self-adjoint operators in Hilbert space with a sequence of cyclic vectors, where the index set A has arbitrary cardinal. Let all real numbers), S be the σ-Bool algebra generated by rectangular parallelepipeds of X. Then there existthe unique (in equivalent sense) bounded measure ν on S, measurable field of Hilbert spaces λ→(λ) on the Borel space (X, S), and a unitary isomorphism U from onto such thatU A_l U~(-1)=T_l, l ∈, where T_l is the operator multiplying λ_l in (λ= (λ_l.)_(l′∈)∈X).Theorem 2 we keep the notions of theoreml. Let T be a closed dense linear operator in such that TB BT for every bounded linear operator B satisfying BE~((l)) = E_l~((l)) B then there exists a unique (p.p.ν) S-measurable function X such that where E(·) is the spectral measure generated by {E_l~((l)0 |l∈A, t real}, and T_l is the operator multiplying f(λ) in.These results are the generalization of works [1], [2] and others.