Toeplitz算子作用在不同Banach函数空间的Hardy型子空间之间的Coburn-Simonenko定理
The Coburn-Simonenko theorem for Toeplitz operators acting between Hardy type subspaces of different Banach function spaces
摘要：
设$Γ$是一条可求长的Jordan曲线，$X$和$Y$是$Γ$上的两个自反Banach函数空间，使得Cauchy奇异积分算子$S$在它们每一个上有界，$M(X,Y)$表示从$X$到$Y$的逐点乘子空间.考虑Riesz投影$P=(I+S)/2$，对应的Hardy型子空间$PX$和$PY$，以及由符号$a\in M(X,Y)$的$T(a)f=P(af)$定义的Toeplitz算子$T(a):PX\to PY$。我们证明了如果$X\hookrightarrow Y$和$a\in M(X,Y)\setminus\{0\}$，那么$T(a)\in\mathcal{L}(PX,PY)$在$PX$中有平凡核，在$PY$中有稠密像.特别地，如果$1<q\le p<\infty$、$1/r=1/q-1/p$和$a\in L^{r}\equiv M(L^p,L^q)$是非零函数，则Toeplitz算子$T(a)$，从Hardy空间$H^p$作用到Hardy空间$H^q$，在$H^p$中具有平凡核或在$H^q$中具有稠密像。
英文摘要：
Let $Γ$ be a rectifiable Jordan curve, let $X$ and $Y$ be two reflexive Banach function spaces over $Γ$ 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 $H^q$.
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