Category:Operator with Zero Numerical Range is Zero Operator

This category contains pages concerning Operator with Zero Numerical Range is Zero Operator:

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $\norm {\, \cdot \,}$ be the inner product norm on $\struct {\HH, \innerprod \cdot \cdot}$.

Let $\struct {\map D T, T}$ be a densely-defined linear operator on $\HH$ such that:

$\map W T = \set 0$

where $\map W T$ is the numerical range of $T$.

Then $T = 0$.