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I have defined a continuous functional calculus on the bounded self-adjoint linear operators for functions continuous on the spectrum, and are defined on an interval. How do I deal with wanting to define $f(T)$ for functions defined on the whole real line? Can I simply extend it without any technical arguments?

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    $\begingroup$ The point is of course that you can uniformly approximate $f$ on the spectrum of $T$ by polynomials $p_n$ and put $f(T) = \lim p_n (T)$. This expression doesn't bother whether $f$ is defined on a larger subset of $f$ than the spectrum of $T$ as mac points out below. I strongly recommend having a look at the proof of the spectral theorem, for instance in Pedersen, Analysis now, section 4.4. $\endgroup$ – t.b. Jun 14 '11 at 10:45
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If $f:\mathbb{R}\to\mathbb{C}$ is continuous and $T$ is bounded and self-adjoint, then $\sigma(T)$, the spectrum of $T$, is a compact subset of $\mathbb{R}$. So $g=f|_{\sigma(T)}$, the restriction of $f$ to $\sigma(T)$, is continuous, and you've already defined $g(T)$.

It's natural enough to simply define $f(T)$ to be $g(T)$. The map of "evaluation at $T$" will then be a nice homomorphism from the continuous functions $\mathbb{R}\to\mathbb{C}$ into the bounded linear operators, which is essentially what you want from a functional calculus.

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