Let $\mathcal H$ be a complex Hilbert space and $T \in \mathcal L (\mathcal H).$ If $\mathcal H = (0)$ what can we say about $\sigma (T)\ $?
Clearly in this case $T = 0$ and $I = 0.$ So for all $\lambda \in \mathbb C$ we have $T - \lambda I = 0,$ which is trivially invertible in $\mathcal L (\mathcal H) = (0).$ Hence I think $\sigma (T) = \mathbb C.$ But we know that spectrum of a bounded operator is always compact but here we find something non-compact as a spectrum. Doesn't it give us a contradiction?