论文标题
关于Toeplitz操作员的界限
On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions
论文作者
论文摘要
我们证明了toeplitz操作员的界限和紧凑性的足够条件，$ t_a $在加权的sup-normed Banach空间中$ h_v^\ infty $ holomorthic函数在开放单位光盘$ \ mathbb {d} $上定义了复杂平面的;重物$ v $和符号$ a $都假定为$ \ mathbb {d} $上的radial函数。在作者的较早作品中，表明存在有界的，谐波（因此是非radial）符号$ a $ a $，以至于$ t_a $在任何空间中都没有限制$ h_v^\ infty $，并带有可允许的重量$ v $。在这里，我们表明，在$ \ mathbb {d} $的边界处的径向符号$ a $的对数衰减率的额外假设可以保证$ t_a $的界限。 $ t_a $的有限性和紧凑性的足够条件，在许多变化中，源自作者最近发现的一般，抽象的必要条件和充分的条件。结果适用于满足所谓条件$（b）$的大量重量，除了标准重量类别外，还包括许多迅速降低重量。
We prove sufficient conditions for the boundedness and compactness of Toeplitz operators $T_a$ in weighted sup-normed Banach spaces $H_v^\infty$ of holomorphic functions defined on the open unit disc $\mathbb{D}$ of the complex plane; both the weights $v$ and symbols $a$ are assumed to be radial functions on $\mathbb{D}$. In an earlier work by the authors it was shown that there exists a bounded, harmonic (thus non-radial) symbol $a$ such that $T_a$ is not bounded in any space $H_v^\infty$ with an admissible weight $v$. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol $a$ at the boundary of $\mathbb{D} $ guarantees the boundedness of $T_a$. The sufficient conditions for the boundedness and compactness of $T_a$, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition$(B)$, which includes in addition to standard weight classes also many rapidly decreasing weights.
