Spectral essential range in Borel functional calculus ($\sigma (f(T))=\{\lambda :E(|f-\lambda |<\epsilon )\ne 0\text{ for all }\epsilon >0\}$)

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 $E(|f|>\epsilon )=0$ for every $\epsilon >0$, and its norm is the corresponding essential supremum. Ordinary pointwise range is insufficient: for multiplication by $t$ on $L^2[0,1]$, $1_{\{0\}}(T)=0$ even though the function takes the value $1$.