A Hilbert Space Problem Book pp 66-73 | Cite as

# Partial Isometries

## Abstract

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 [56], [40, pp. 911–912].

## Keywords

Hilbert Space Functional Calculus Hermitian Operator Cardinal Number Partial Isometry## Preview

Unable to display preview. Download preview PDF.