Trying to compute norm of an operator

Question

Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces and let $T_i: \mathcal{H} \to \mathcal {K}$ be bounded linear operators satisfying $\|T_1 \|=1$ and $\|T_2^*T_3 \|=1$. Consider $T: \mathcal{H} \oplus \mathcal{K} \to \mathcal{H}$ defined as $T(h, k) = T_1^*k+ T_2^*T_3h$. I am trying to compute $\| T \|$.

It is clear that $ 1 \leq \|T \| \leq 2$ but it I'm unable to compute the exact norm. Any ideas?

Answer 1

You cannot expect to compute the norm of an operator of the form $T(h,k)=Th+Sk$ without consideration of the particular form of $T$ and $S$.

Consider $H_1=H_2=\mathbb C$, and $T=S=1$ and fix $h,k\in\mathbb C$ with $|h|^2+|k|^2=1$. Then $$ \|T(h,k)\|=|h+k|\leq\sqrt2, $$ and we get equality if $h=k=\frac1{\sqrt2}$. Hence $\|T\|=\sqrt2$.

Consider instead $H_1=H_2=\mathbb C^2$, and $$ T=\begin{bmatrix} 1&0\\0&0\end{bmatrix},\qquad\qquad S=\begin{bmatrix} 0&0\\0&1\end{bmatrix}. $$ You have $\|T\|=\|S\|=1$, and for any $h,k\in\mathbb C^2$ we have $$ T(h,k)=\begin{bmatrix} h_1\\ k_1\end{bmatrix} $$ so $$ \|T(h,k)\|^2=|h_1|^2+|k_1|^2\leq \|h\|^2+\|k\|^2=\|(h,k)\|^2. $$ Thus $\|T\|\leq1$, and it is easy to see that $\|T\|=1$.