Let $H$ be a hilbertspace, $y$ and $z$ are fixed but randomly chosen in $H.$ Then define the operator: $ T = \langle x,z\rangle y$.
Calculate $\|T\|$.
How should I do this? So far I have $\|T\|^2 = \sup |\langle x,z\rangle|^2 \|y\|^2$ where the $\sup$ is over all vectors $x$ with norm $1.$
Edit: also having trouble calculating $A^*$, I have $A^* = \overline{\langle y,x\rangle} z$ ,but I'm not certain about this.
Hint: You can try to find $c$ such that $\|T\|\leq c$ and $\|T\|\geq c$. Sometimes it is easier to show the two inequalities.
For $\|T\|\leq c$ you can use the Cauchy-Schwarz-inequality $|\langle x,z\rangle|\leq \|x\|\cdot\|z\|$.
For $z\neq 0$ the vector $\frac1{\|z\|}z$ has length $1$ and you can use this for $\|T\|\geq c$.