= Spectral essential range in Borel functional calculus
{title2=$\sigma(f(T))=\{\lambda:E(|f-\lambda|<\varepsilon)\ne0\text{ for all }\varepsilon>0\}$}
For a bounded <Borel measurable function> $f$ and the <projection-valued measure> $E$ of a <normal operator>, the displayed set is the <spectrum of an element> of $f(T)$. Outside it, a bounded reciprocal away from a spectral-null set supplies the inverse. Inside it, unit vectors supported by the indicated arbitrarily small spectral sets make $f(T)-\lambda$ fail to be bounded below. The kernel of the calculus consists of functions satisfying $E(|f|>\varepsilon)=0$ for every $\varepsilon>0$, and its <norm> is the corresponding essential supremum. Ordinary pointwise range is insufficient: for multiplication by $t$ on $L^2[0,1]$, $\mathbf1_{\{0\}}(T)=0$ even though the function takes the value $1$.
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