Normal operators constitute the most important tractable class of operators known; the most important statement about them is the spectral theorem. Students of operator theory generally agree that the finite-dimensional version of the spectral theorem has to do with diagonal forms. (Every finite normal matrix is unitarily equivalent to a diagonal one.) The general version, applicable to infinite-dimensional spaces, does not have a universally accepted formulation. Sometimes bounded operator representations of function algebras play the central role, and sometimes Stieltjes integrals with unorthodox multiplicative properties. There is a short, simple, and powerful statement that does not attain maximal generality (it applies to only one operator at a time, not to algebras of operators), but that does have all classical formulations of the spectral theorem as easy corollaries, and that has the advantage of being a straightforward generalization of the familiar statement about diagonal forms. That statement will be called the spectral theorem in what follows; it says that every normal operator is unitarily equivalent to a multiplication. The statement can be proved by exactly the same techniques as are usually needed for the spectral theorem; see , [40, pp. 911–912].
KeywordsHilbert Space Functional Calculus Hermitian Operator Cardinal Number Partial Isometry
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