functional analysis – If $ T in mathcal {L} (H) $ is a contraction, then $ {h mid || h || = || Th || } $ is a subspace

Assume that $ H $ is a Hilbert space and $ T in mathcal {L} (H) $ is a contraction, that is to say $ || T || $ 1. Using the existence of square roots for a non-negative self-adjoint operator, we can easily show that $ H_0 = {h in H mid || h || = || Th || } $ is a subspace of $ H $since it's just the null space of the operator $ D_T = (I-T ^ * T) ^ {1/2} $.

Is there any basic evidence that $ H_0 $ is a subspace, that is, without using the existence of square roots of operators or functional calculus?