I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously diagonalised? In the paper, one operator, $A$, is self adjoint and positive definite and the other, $a$ is bounded and positive.

Thanks.

Context: The problem comes from the generalised Ornstein-Uhlenbeck equation,$dX_t=-AX_t dt + \sqrt{2a}dB_t$ where $A$ is a constant self-adjoint positive definite operator on $H$, a seperable Hilbert space, and $B_t$ is a cylindrical Brownian motion. $a$ is a constant, positive operator . The author then diagonalises the system to become $dX_k(t)=-\lambda_kX_k(t) dt + \sqrt{2a_k}dB_k(t)$ where $x_k(t)= \langle X_t,\phi_k\rangle$ and with $A\phi_k=\lambda_k\phi_k$ and $\langle \phi_k,\sqrt{a}\phi_j\rangle=\sqrt{a_k}\delta_{jk}$.