Article
| Title: | The relationship between $K_u^2\cap vH^2$ and inner functions (English) |
| Author: | Yang, Xiaoyuan |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 4 |
| Year: | 2024 |
| Pages: | 1221-1240 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $u$ be an inner function and $K_u^2$ be the corresponding model space. For an inner function $v$, the subspace $vH^2$ is an invariant subspace of the unilateral shift operator on $H^2$. In this article, using the structure of a Toeplitz kernel ${\rm ker} T_{\overline {u}v}$, we study the intersection $K_u^2\cap vH^2$ by properties of inner functions $u$ and $v$ $(v\neq u)$. If $K_u^2\cap vH^2\neq \{0\}$, then there exists a triple $(B,b,g)$ such that $$\overline {u}v=\frac {\lambda b\overline {BO_g}}{g},$$ where the triple $(B,b,g)$ means that $B$ and $b$ are Blaschke products, $g$ is an invertible function in $H^\infty $, $O_g$ denotes the outer factor of $g$, and $\lambda $ is some constant with $|\lambda |=1.$ Furthermore, for any nonconstant inner function $u$, there exists a Blaschke product $B$ such that $K_B^2\cap uH^2\neq \{0\}.$ In particular, we discuss the finite-dimensional intersection $K_u^2 \cap vH^2$. Moreover, we investigate connections between minimal Toeplitz kernels and $K_u^2\cap vH^2$. (English) |
| Keyword: | model space |
| Keyword: | invariant subspace of the unilateral shift operator |
| Keyword: | Toeplitz kernel |
| Keyword: | inner function |
| MSC: | 30J05 |
| MSC: | 47A15 |
| MSC: | 47B35 |
| DOI: | 10.21136/CMJ.2024.0175-24 |
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| Date available: | 2024-12-15T06:40:24Z |
| Last updated: | 2024-12-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152698 |
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