Extremal functions as divisors for kernels of Toeplitz operators
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- 作者：Hartmann ; Andreas ; Seip ; Kristian
- 关键词：30E05 ; 46E20 ; Toeplitz operators ; Extremal functions ; Invariant subspaces with respect to the backward shift ; Nearly invariant subspaces ; Carleson measures
- 刊名：Journal of Functional Analysis
- 出版年：2003
- 出版时间：August 20, 2003
- 年：2003
- 卷：202
- 期：2
- 页码：342-362
- 全文大小：243 K
文摘
In this paper, an extremal function of a Banach space of analytic functions in the unit disk (not all functions vanishing at 0) is a function solving the extremal problem supRef(0) for functions f of norm 1. We study extremal functions of kernels of Toeplitz operators on Hardy spaces Hp, 1<p<∞. Such kernels are special cases of so-called nearly invariant subspaces with respect to the backward shift, for which Hitt proved that when p=2, extremal functions act as isometric divisors. We show that the extremal function is still a contractive divisor when p<2 and an expansive divisor when p>2 (modulo p-dependent multiplicative constants). We give examples showing that the extremal function may fail to be a contractive divisor when p>2 and also fail to be an expansive divisor when p<2. We discuss to what extent these results characterize the Toeplitz operators via invariant subspaces for the backward shift.