Toeplitz Kernel Question

$$newcommand{kern}{operatorname{Ker}}$$When looking at a Toeplitz Operator (on $mathbb{H}^2(mathbb{C^+}))$ with unimodular symbol, $phi=e^{i psi}$, what can we say about $kern T_{phi}$ when $$limsup_{x rightarrow -infty} psi (x) – liminf_{x rightarrow infty} psi (x) < pi$$ and $$limsup_{x rightarrow -infty} -psi (x) – liminf_{x rightarrow infty} -psi (x) < pi?$$ Page 16 of this paper says this is a basic property that implies that $kern T_{phi}$ and $kern T_{bar phi}$ are trivial; however, I cannot find a simple explanation for this. What I feel is a possible explanation is Theorem 5 (part 4) of this work of Hruschev, Nikolski and Pavlov
but this hardly seems "basic".