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The quantum mechanical time evolution operator governs how observables and/or states evolve during finite time steps, and is always unitary. Use this tag for questions about the time evolution operator, or the different equations of motion in the Schrödinger/Heisenberg/Dirac pictures. For time-independent Hamiltonians, the time evolution operator is simply exp(-iHt).

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Why I cannot write the time evolution operator $e^{-i(T+V)t}$ as the product of operators $e...

When we put the Hamiltonian $H$ in an exponential like this, we are not assuming the $H$ represents a scalar value. The exponential notation used in this case is just shorthand for the Taylor series …
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