= Solution
In a globally static region, choose the time coordinate along the future timelike <Killing vector field> $K=\partial_t$. A <positive-frequency solution> of the <Klein-Gordon equation> has $i\mathcal L_Kp_i=\omega_i p_i$, with $\omega_i>0$. Assume appropriate boundary conditions and a <Cauchy surface> so that the conserved <Klein-Gordon inner product> is
$$
(f,g)=i\int_\Sigma d\Sigma^a\bigl(f^*\nabla_ag-g\nabla_af^*\bigr).
$$
For a real scalar, take the negative-frequency modes to be $n_i=p_i^*$. Choose a complete normalized basis with
$$
(p_i,p_j)=\delta_{ij},\qquad(n_i,n_j)=-\delta_{ij},\qquad(p_i,n_j)=0.
$$
The negative norm of the conjugate modes is crucial; this is not a positive-definite inner product on all classical solutions.
Promote the real <Klein-Gordon field> to the operator
$$
\widehat\phi=\sum_i(p_i a_i+n_i a_i^\dagger),\qquad
[a_i,a_j^\dagger]=\delta_{ij},\quad[a_i,a_j]=0.
$$
These <canonical commutation relations> implement the canonical field and conjugate-momentum commutator. The static <vacuum state in a stationary spacetime> satisfies $a_i|0\rangle=0$, and the <Hamiltonian> is a sum of oscillators, $\sum_i\omega_i(a_i^\dagger a_i+1/2)$ before zero-point regularization. Continuous labels replace sums by correctly normalized integrals. For a complex charged scalar there are separate particle and antiparticle oscillators, with the same mode-mixing argument in each sector.
Choose independently normalized past and future bases, and use the matrix indices of the question. Complex conjugation of the future positive-frequency expansion immediately gives
$$
\boxed{n_i^+=\sum_j\bigl(n_j^- A_{ji}^*+p_j^- B_{ji}^*\bigr).}
$$
Conservation of the <Klein-Gordon inner product> gives $A^\dagger A-B^\dagger B=I$ and $A^\dagger B^*-B^\dagger A^*=0$ in this column-index convention. These ensure preservation of the <canonical commutation relations>.
Let $a_j$ annihilate past modes and $b_i$ annihilate future modes. Extracting the future coefficient from the field gives the <Bogoliubov transformation>
$$
b_i=(p_i^+,\widehat\phi)=\sum_j\bigl(A_{ji}^*a_j-B_{ji}^*a_j^\dagger\bigr).
$$
The minus sign follows from $(n_j^-,n_k^-)=-\delta_{jk}$. In the <in-vacuum>, only $\langle0_{\rm in}|a_j a_k^\dagger|0_{\rm in}\rangle=\delta_{jk}$ contributes to the expectation of the future <number operator>. Therefore
$$
\boxed{\langle0_{\rm in}|b_i^\dagger b_i|0_{\rm in}\rangle=\sum_j|B_{ji}|^2=(B^\dagger B)_{ii}.}
$$
This is <particle number from Bogoliubov coefficients>: the intervening time-dependent geometry makes a past positive-frequency mode contain a future negative-frequency component. In infinite volume, localized <wave packets> avoid delta-function normalization artifacts; a common Fock-space implementation needs the usual square-summability condition on the creation-mixing coefficients.
For gravitational collapse, the initial vacuum can be defined before the hole forms, while late outgoing modes are defined using the asymptotic stationary time at <future null infinity>. The complete future basis also includes modes falling through the <event horizon>; exterior outgoing modes alone are not a complete basis of all propagated data. Tracing an outgoing mode backwards near a nonextremal horizon gives the exponential ray relation $v=v_H-Ce^{-\kappa u}$, where $u$ is late retarded time and $\kappa$ is the <surface gravity>. An outgoing factor $e^{-i\omega u}$ becomes proportional to $(v_H-v)^{i\omega/\kappa}$ in the early advanced coordinate, and contains both frequency signs when Fourier decomposed there. Their squared Bogoliubov-coefficient ratio is $e^{-2\pi\omega/\kappa}$; the canonical normalization then gives a Bose factor $(e^{2\pi\omega/\kappa}-1)^{-1}$, multiplied by the <greybody factor>, the transmission probability through the exterior potential. Thus <Hawking radiation> has temperature $T_H=\kappa/(2\pi)$ in units $\hbar=k_B=c=1$, with <greybody factor> modifications to the spectrum. This is a free-field calculation on a fixed or slowly evolving background; eventual backreaction is an additional problem, and the nonextremal exponential map must not be blindly used when $\kappa=0$.
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