an equivalent condition for $B(H,K)$ to be a Hilbert space

Question

How can I prove that $B(H,K)$ with operator norm is a Hilbert space if and only if $\dim H=1$ or $\dim K=1$, where $H$ and $K$ are Hilbert spaces?

Any hint is appreciated.

Answer 1

For a normed space to be a Hilbert space, it has to satisfy the parallelogram identity.

If $\dim H=1$ (so $B(H,K)=\mathbb C$) or $\dim K=1$ (so $B(H,K)=H^*\simeq H$), then $B(H,K)$ is a Hilbert space. So we want to prove the converse. That is, we want to prove that if $\dim H\geq2$ and $\dim K\geq2$ then $B(H,K)$ is not a Hilbert space.

So take orthonormal bases $\{e_1,e_2\}\cup\{e_j'\}$ of $H$ and $\{f_1,f_2\}\cup\{f_j'\}$ of $K$. Define operators $T,S\in B(H,K)$ by $Te_1=f_1$, $T|_{(\mathbb C e_1)^\perp}=0$, and $Se_2=f_2$, $S|_{(\mathbb C e_2)^\perp}=0$.

Then $\|T\|=\|S\|=1$, and also $\|T+S\|=\|T-S\|=1$. Thus $$ \|T+S\|^2+\|T-S\|^2=2<4=2\|T\|^2+2\|S\|^2. $$