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

TL;DR

This paper extends the Coburn-Simonenko theorem to Toeplitz operators between Hardy-type subspaces of different Banach function spaces, establishing conditions for trivial kernels or dense images.

Contribution

It generalizes the Coburn-Simonenko theorem to Toeplitz operators acting between Hardy subspaces of different Banach function spaces, under specific embedding conditions.

Findings

If $X$ embeds into $Y$ and $a$ is a nonzero multiplier, then $T(a)$ has a trivial kernel or dense image.

For $L^p$ and $L^q$ spaces with $1<q extless p<\infty$, nonzero symbols induce Toeplitz operators with trivial kernel or dense image.

The results apply to Toeplitz operators between Hardy spaces on Jordan curves with bounded Cauchy singular integral operators.

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