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Can anyone give me an example/application where a self-adjoint bounded operator on a Hilbert space are put through a function?

I know how to define it but I am struggling to find an application area for them.

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There's more than one flavour of "functional calculus" so you may need to be more specific. If you wanted to convince yourself that, at least, it is useful to define the exponential of a (normal) operator, then you could read about Stone's theorem on 1-parameter unitary groups. –  Mike F May 8 '11 at 21:40
if the function is $x\mapsto\exp(itx)$, you get a $1$-parameter group of unitary transformations (the parameter is $t$) –  user8268 May 8 '11 at 21:41
Echoing Mike, what sorts of functions, and in what context? Continuous, Borel, analytic? What versions of the spectral theorem have you learned? –  Jonas Meyer May 8 '11 at 23:10
I learnt it for continuous functions via Stone Weierstrass theorem -- the simplest kind I believe. So it uses the limit of some polynomials to define it. –  Chris May 9 '11 at 7:06
So would the multiplication operator (example from Wiki) suit as an example for an opreator, that you would like to put through a function? –  draks ... Apr 5 '12 at 18:08

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