There’s no problem with your time-reversal symmetry definition or you observation about the Hermiticity of H and its diagonalizability.

Which brings us to your choice of $U_T = J^t J$:

a) it does satisfy $U_T^† H^* U_T = H$, but not necessarily all the conditions for a time-reversal operator.

b) to be specific - the condition $U_T^* U_T = ±1$ isn’t guaranteed by this construction.

Time-reversal symmetry imposes additional constraints on the structure of the Hamiltonian beyond just Hermiticity.

Not all Hermitian Hamiltonians are time-reversal symmetric. A system with a magnetic field breaks time-reversal symmetry for example.

The existence of a time-reversal symmetry often implies degeneracies in the energy spectrum (Kramers degeneracy for half-integer spin systems), which not all Hamiltonians possess. So, while all time-reversal symmetric Hamiltonians are Hermitian, not all Hermitian Hamiltonians are time-reversal symmetric.

There’s no problem with your time-reversal symmetry definition or your observation about the Hermiticity of $H$ and its diagonalizability.

Which brings us to your choice of $U_T = J^t J$:

a) it does satisfy $U_T^† H^* U_T = H$, but not necessarily all the conditions for a time-reversal operator.

b) to be specific - the condition $U_T^* U_T = \pm 1$ isn’t guaranteed by this construction.

Time-reversal symmetry imposes additional constraints on the structure of the Hamiltonian beyond just Hermiticity.

Not all Hermitian Hamiltonians are time-reversal symmetric. A system with a magnetic field breaks time-reversal symmetry for example.

The existence of a time-reversal symmetry often implies degeneracies in the energy spectrum (Kramers degeneracy for half-integer spin systems), which not all Hamiltonians possess. So, while all time-reversal symmetric Hamiltonians are Hermitian, not all Hermitian Hamiltonians are time-reversal symmetric.

There’s no problem with your time-reversal symmetry definition or you observation about the Hermiticity of H and its diagonalizability.

Which brings us to your choice of U_T = J^t J:

a) it does satisfy U_T^† H^* U_T = H, but not necessarily all the conditions for a time-reversal operator.

b) to be specific - the condition U_T^* U_T = ±1 isn’t guaranteed by this construction.

Time-reversal symmetry imposes additional constraints on the structure of the Hamiltonian beyond just Hermiticity.

Not all Hermitian Hamiltonians are time-reversal symmetric. A system with a magnetic field breaks time-reversal symmetry for example.

The existence of a time-reversal symmetry often implies degeneracies in the energy spectrum (Kramers degeneracy for half-integer spin systems), which not all Hamiltonians possess. So, while all time-reversal symmetric Hamiltonians are Hermitian, not all Hermitian Hamiltonians are time-reversal symmetric.

There’s no problem with your time-reversal symmetry definition or you observation about the Hermiticity of H and its diagonalizability.

Which brings us to your choice of $U_T = J^t J$:

a) it does satisfy $U_T^† H^* U_T = H$, but not necessarily all the conditions for a time-reversal operator.

b) to be specific - the condition $U_T^* U_T = ±1$ isn’t guaranteed by this construction.

Time-reversal symmetry imposes additional constraints on the structure of the Hamiltonian beyond just Hermiticity.

Not all Hermitian Hamiltonians are time-reversal symmetric. A system with a magnetic field breaks time-reversal symmetry for example.

The existence of a time-reversal symmetry often implies degeneracies in the energy spectrum (Kramers degeneracy for half-integer spin systems), which not all Hamiltonians possess. So, while all time-reversal symmetric Hamiltonians are Hermitian, not all Hermitian Hamiltonians are time-reversal symmetric.