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I'm a physicist who needs mathematical advice: Let $A= \sum_{i=1}^{\infty} a_i P_{\phi_i}$ be a self-adjoint operator with projectors $P_{\phi_i}$ on the orthonormal eigenbasis $(\phi_i).$ Let $$B= \int_{\sigma(B)} \lambda dP(\lambda)$$ be another self-adjoint operator.

Then, we define $H = A \otimes B.$

Now, for finite density matrices one knows how to define the partial trace and I was wondering whether this is also possible in this case, i.e.: Let $\rho = \rho_A \otimes P_{\psi}$ for some normalized $\psi.$

Is there a way to define $\rho_{j,k}(t)= \langle \phi_j, Tr_{2}(e^{- iHt} \rho e^{iHt}) \phi_k \rangle $ where we trace-out the basis of the second operator? In physics textbooks I could only find the standard definition for the case that the spectral measure of $B$ is discrete.

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