The Coburn-Simonenko theorem for Toeplitz operators acting between Hardy type subspaces of different Banach function spaces

Let $\Gamma$ be a rectifiable Jordan curve, let $X$ and $Y$ be two reflexive Banach function spaces over $\Gamma$ such that the Cauchy singular integral operator $S$ is bounded on each of them, and let $M(X,Y)$ denote the space of pointwise multipliers from $X$ to $Y$. Consider the Riesz projection $P=(I+S)/2$, the corresponding Hardy type subspaces $PX$ and $PY$, and the Toeplitz operator $T(a):PX\to PY$ defined by $T(a)f=P(af)$ for a symbol $a\in M(X,Y)$. We show that if $X\hookrightarrow Y$ and $a\in M(X,Y)\setminus\{0\}$, then $T(a)\in\mathcal{L}(PX,PY)$ has a trivial kernel in $PX$ or a dense image in $PY$. In particular, if $1<q\le p<\infty$, $1/r=1/q-1/p$, and $a\in L^{r}\equiv M(L^p,L^q)$ is a nonzero function, then the Toeplitz operator $T(a)$, acting from the Hardy space $H^p$ to the Hardy space $H^q$, has a trivial kernel in $H^p$ or a dense image in $H^q$.