Abstract
In this chapter, we use the spectral theorem of Chapter XVIII to give a finer theory, making sense of the expression f(A) when f is not continuous. Ultimately, one wants to use very general functions f in the context of measure theory, namely bounded measurable functions, as a corollary of what was done in Chapter XVIII. For our purposes here, we deal with an intermediate category of functions, essentially characteristic functions of intervals. These give rise to projection operators, whose formalism is important for its own sake. We also want to deal with unbounded operators as an application.
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© 1993 Springer-Verlag Berlin Heidelberg
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Lang, S. (1993). Further Spectral Theorems. In: Real and Functional Analysis. Graduate Texts in Mathematics, vol 142. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0897-6_19
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DOI: https://doi.org/10.1007/978-1-4612-0897-6_19
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-6938-0
Online ISBN: 978-1-4612-0897-6
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