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 is self adjoint and positive definite and the other 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 separable Hilbert space, $B_t$ is a cylindrical Brownian motion, while $a$ 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 $x_k(t)= \langle X_t,\phi_k\rangle$, $A\phi_k=\lambda_k\phi_k$ and $\langle \phi_k,\sqrt{a}\phi_j\rangle=\sqrt{a_k}\delta_{jk}$.