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 $$H^q$$ .