The forward maximal regularity operator commutes with the backward

Given a generator of a bounded analytic semigroup on a Hilbert space, we show that the corresponding forward maximal regularity operator commutes with the backward. In particular, for self-adjoint generators the images under the two maximal regularity operators have equal unweighted Hilbert space norms.
Comment: fixed a typo: "apply Theorem 2.1 or solely (3.2)" to "apply Theorem 2.1 or solely (2.2)"