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The relationship between $K_u^2\cap vH^2$ and inner functions
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 4, pp. 1221-1240
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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 $K_u^2\cap vH^2$.
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 $K_u^2\cap vH^2$.
DOI : 10.21136/CMJ.2024.0175-24
Classification : 30J05, 47A15, 47B35
Keywords: model space; invariant subspace of the unilateral shift operator; Toeplitz kernel; inner function 
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Yang, Xiaoyuan. The relationship between $K_u^2\cap vH^2$ and inner functions. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 4, pp. 1221-1240. doi: 10.21136/CMJ.2024.0175-24

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