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Literatur
R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras,Annals of Math.,56 (1952), pp. 494–503.
For two bounded self-adjoint operatorsA andB, A≧B means that (Au, u)≧(Bu, u), for allu ∈ lore.
For a self-adjoint operatorB:\(||B|| = \mathop {1.u.b.}\limits_{||u|| = 1} ||Bu|| = \mathop {1.u.b.|}\limits_{||u|| = ||v|| = 1} (Bu,v)| = \mathop {1.u.b.|}\limits_{||u|| = 1} (Bu,u)|;\) for anf(x)εC(X):\(||f|| = \mathop {\max }\limits_{x \in X} |f(x)|,\) this maximum being finite and attained sinceX is compact andf(x) is continuous.
М. А. Наимарк, Спектральные функции симметрического оператора, Известия Акад. Наук СССР, сер. мат.,4 (1940), pp. 277–309 (English summary pp. 309–318). Об одном представлении аддитивных операторных функций множеств, Доклады Акад. Наук СССР,41 (1943), pp. 373–375.
Of course, equation (9) is to be understood in the sense that the operators are equal when applied to the elements of the original space lore.
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Sz.-Nagy, B. A moment problem for self-adjoint operators. Acta Mathematica Academiae Scientiarum Hungaricae 3, 285–293 (1952). https://doi.org/10.1007/BF02027827
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DOI: https://doi.org/10.1007/BF02027827