Contraction and operator norm

Question

Hi there I am stuck at this problem. Let $T\in M_2(\mathbb{C})$ be as follows: \begin{gather} T = \ \begin{bmatrix} a & c \\ 0 & a \end{bmatrix} \end{gather} Show that if $|a|<1$ and $|c|\le1-|a|^2$, then $T$ is a contraction.

Here is what I have so far: Since $|c|^2\le 1-2|a|^2+|a|^4$, it follows that

$$‖T‖^2≤(2|a|^2+|c|^2)\le1+a^4.$$

What steps should be taken to show that $‖T‖\le1$?

Answer 1

Given $(z_1,z_2)\in\mathbb{C}^2$,

$$||T\left(z_1,z_2\right)||^2=|az_1+cz_2|^2+|az_2|^2\leq|az_1|^2+|cz_2|^2+|az_2|^2$$

Now I suggest you use $|c|^2\leq 1-|a|^2$ first and see if you can simplify things.

Hope this helps.