Timeline for Spectrum of the Normal Operator associated to compact supported spectral measures

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Dec 20, 2012 at 15:26	comment	added	Robert Rauch		Although I have already looked over your reference a couple of days ago, I did not realize that a nice answer to my question is hidden therein. Thanks a lot pointing that out!
Dec 20, 2012 at 14:15	comment	added	Branimir Ćaćić		Yes, you're absolutely right. I imagine that the result you want is precisely the theorem on Page 7 of math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/RaskinSam.pdf, namely, that the support of a compactly-supported projection-valued measure is one and the same as the spectrum of the associated normal operator.
Dec 20, 2012 at 13:23	comment	added	Robert Rauch		Yes, I mean a projection valued measure. By definition, $E(\mathbb{C})=id_H$, and also if $E$ is defined as the spectral measure associated to a given normal operator $A$, this is essentially true by definition. However, in our situation we start with a generic spectral measure $E$ from wich it is only known that there is some compact $K\subset\mathbb{C}$ such that $E(K)=id_H$ and define $A$ by the above formula. One can then show that the spectral measure associated to this $A$ is exactly $E$, but this is not obvious and actually the crucial point is to show that $E(spec A)=id_H$
Dec 20, 2012 at 12:32	comment	added	Branimir Ćaćić		By "spectral" do you mean a projection-valued measure? Because if so, then doesn't this hold just by definition?
Dec 20, 2012 at 12:27	history	asked	Robert Rauch	CC BY-SA 3.0