Let \({\mathcal C}\) be the collection of all closed densely defined linear maps commuting with the adjoint of the compression of the standard unilateral shift on the Hardy space of the unit disc to a shift coinvariant subspace. A linear map \(T\) with the domain \(D(T)\) dense in a Hilbert space is said to be dissipative on \(D(T)\) if \(\langle Tx,x\rangle\) has nonnegative imaginary part for all \(x\in D(T)\). In the paper under review, the author proves necessary and sufficient conditions for a linear map in \({\mathcal C}\) to be dissipative using a characterization of linear maps comprising \({\mathcal C}\), due to \textit{D. Suárez} [Pac. J. Math. 179, 371--396 (1997; Zbl 0895.47021)] and a characterization of dissipative operators as the infinitesimal generators of \(C_0\)-semigroups of contraction operators, due to \textit{R. S. Phillips} [Trans. Am. Math. Soc. 90, 193-254 (1959; Zbl 0093.10001)].