Toeplitz kernels and the backward shift

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:492

Article

O'Loughlin, Ryan¹ 

[1] Univ Leeds, Sch Math, Leeds LS2 9JT, W Yorkshire, England

关键词: Vector-valued Hardy space;    Toeplitz operator;    Backward shift operator;   

DOI  :  10.1016/j.jmaa.2020.124489

来源: Elsevier

【 摘 要 】

In this paper we study the kernels of Toeplitz operators on both the scalar and the vector-valued Hardy space for 1 < p < infinity. We show existence of a minimal kernel for any element of the vector-valued Hardy space and we determine a symbol for the corresponding Toeplitz operator. In the scalar case we give an explicit description of a maximal function for a given Toeplitz kernel which has been decomposed in to a certain form. In the vectorial case we show not all Toeplitz kernels have a maximal function and in the case of p = 2 we find the exact conditions for when a Toeplitz kernel has a maximal function. For both the scalar and vector-valued Hardy space we study the minimal Toeplitz kernel containing multiple elements of the Hardy space, which in turn allows us to deduce an equivalent condition for a function in the Smirnov class to be cyclic for the backward shift. (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free