For two commuting contractions $T_1,T_2$ on a Hilbert space $H$ with $T=T_1T_2$ Show that $D_T>D_{T_i}$ for $i=1,2$ where $D_T=(I-T^*T)^{1/2}$.

Question

Let $T_1, T_2$ be commuting contractions on a Hilbert space $H$, that is $\|T_1\|\leq 1$ and $\|T_2\|\leq 1$, $T_1T_2=T_2T_1$. Let $T=T_1T_2$. Clearly $I-T^*T$ and $I-T_i^*T_i$ for $i=1,2$ are positive operators hence let $D_T=(I-T^*T)^{1/2}$ and $D_{T_i}=(I-T_i^*T_i)^{1/2}$. Then we wish to prove that $D_T\geq D_{T_i}$, that is $D_T-D_{T_i}$ is a positive operator.

Note that, $I-T^*T-I+T_1^*T_1=T_1^*(I-T_2^*T_2)T_1\geq 0$ as $T_2$ is a contraction.

Hence we have $D_{T}^2\geq D_{T_1}^2$. Similarly we can prove that $D_{T}^2\geq D_{T_2}^2$. How should we proceed further?