A question about time inversion [Weinberg]

[Antarres]

Recently I've started reading Weinberg QFT, Vol 1, to introduce myself better to quantum field theory, after finishing courses in qft at the university. Reading through the second chapter, I've found myself confused by his treatment of degenerate multiplets in Appendix C. Here is the equations that confuses me along with some text around it.

...
To explore the general posibilities of time reversal, let us assume that on massive one-particle state it has the action
$$T\Psi_{\textbf{p},\sigma,n} = (-1)^{j-\sigma}\sum_m\mathcal{T}_{mn}\Psi_{-\textbf{p},-\sigma,m}$$
where p, j, σ are particle's momentum, spin and spin z-component, respectively, and n,m are indices labeling members of a degenerate multiplets of particle species. (The appearance of $(-1)^{j-\sigma}$ and the reversal of $\textbf{p}$ and $\sigma$ are deduced in the same way as in Section 2.6.) The matrix $\mathcal{T}_{mn}$ is unknown, except that since T is antiunitary $\mathcal{T}$ must be unitary.
...

This last statement doesn't seem obvious to me. As far as I know, antiunitary operators in quantum mechanics cannot have a matrix representation(because they are antilinear, unlike matrices), so it is customary to write them in the form $T = UK$, where $U$ is unitary, and $K$ is complex conjugation. This doesn't help here(at least I don't see it helping), since Weinberg chose to represent time reversal operator by it's action on states, from which he derives the $(-1)^{j-\sigma}$ factor and reversal of momentum and helicity, when acting on the one-particle states of massive particles. (It's derived in section 2.6. The derivation is relatively short.) So I don't see why $\mathcal{T}$ has to be unitary in the previous equation, I couldn't really approach the problem from any side so far, so some help would be appreciated. The analysis that proceeds after this seems relatively easy to understand, I just have problem with this beginning of the analysis.

P.S. I put this article here, judging by the theme of the book. Maybe it could be put in Homework and exercises, too, you're free to move it wherever it's more suitable. I wouldn't consider it homework or anything, since it's just self-learning, so I wanted to clarify this to myself.

 

[Antarres]

I've come to a conclusion to this problem that is based on my intuition. Basically, if antiunitary operator can be represented as composition of unitary action and conjugation, and if this conjugation aspect is described in a standard way by action on each component of a degenerate multiplet, resulting in the phase that is introduced, than I assume that the mixing of components should have no other option than to be unitary. So for that reason the $\mathcal{T}$ matrix that mixes the components is unitary.

This approach is just my intuition I've developed thinking about the problem, but it doesn't really seem rigorous to me, so I hoped with this question that there is some more direct(rigorous) reason for this conclusion. But maybe there is nothing more than that.