Czechoslovak Mathematical Journal, Vol. 74, No. 4, pp. 1221-1240, 2024
The relationship between $K_u^2\cap vH^2$ and inner functions
Xiaoyuan Yang
Received May 6, 2024. Published online November 7, 2024.
Abstract: 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 $K_u^2\cap vH^2$.
Keywords: model space; invariant subspace of the unilateral shift operator; Toeplitz kernel; inner function
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Affiliations: Xiaoyuan Yang, School of Science, Jiangsu Ocean University, 59 Cangwu Rd, Haizhou, Lianyungang, Jiangsu, 222005, P. R. China, e-mail: Yangxyxy@163.com
