= Solution
Use the <Fourier transform> convention $\widehat w(E)=\int_{\mathbb R}w(t)e^{itE}dt$. Since $w\geq0$, it is real, so $\widehat w(-E)=\widehat w(E)^*$. The positive-frequency cutoff therefore also eliminates frequencies at or below $-\Delta$. Normalization gives $\int w=\widehat w(0)=1$.
For <spectral filtering of Hamiltonian terms>, choose
$$
\boxed{A^{(Z)}=\int_{\mathbb R}w(t)e^{itH}h_Ze^{-itH}dt.}
$$
In an energy <eigenbasis>, its <matrix elements> are
$$
\langle\phi_i|A^{(Z)}|\phi_j\rangle=\widehat w(E_i-E_j)\langle\phi_i|h_Z|\phi_j\rangle.
$$
At $i=j=0$ the frequency is zero, so the normalization preserves the ground-state expectation:
$$
\boxed{\langle\phi_0|A^{(Z)}|\phi_0\rangle=\langle\phi_0|h_Z|\phi_0\rangle.}
$$
The <spectral filter> is a positive weighted average of <unitary conjugations>; in particular the integral is bounded in <operator norm> by $\|h_Z\|$. We can choose it even without an extra assumed tail bound: the <evenization of a nonnegative bandlimited filter> proved in part (e) produces another admissible <spectral filter> with the same type of positive-time decay. Make that choice consistently in all the filtered terms and shell definitions below.
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