The Coburn–Simonenko Theorem for Toeplitz Operators Acting Between Hardy Type Subspaces of Different Banach Function Spaces

Abstract

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 $T(a):PX\rightarrow PY$ 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 $H^q$.