# 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?