The relationship between $K_u^2\cap v{H}^2$ and inner functions

Yang, Xiaoyuan. "The relationship between $K_u^2\cap vH^2$ and inner functions." Czechoslovak Mathematical Journal 74.4 (2024): 1221-1240. <http://eudml.org/doc/299626>.

@article{Yang2024,
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\ne u)$. If $K_u^2\cap vH^2\ne \lbrace 0\rbrace $, 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 $K_B^2\cap uH^2\ne \lbrace 0\rbrace .$ In particular, we discuss the finite-dimensional intersection $K_u^2 \cap vH^2$. Moreover, we investigate connections between minimal Toeplitz kernels and $K_u^2\cap vH^2$.},
author = {Yang, Xiaoyuan},
journal = {Czechoslovak Mathematical Journal},
keywords = {model space; invariant subspace of the unilateral shift operator; Toeplitz kernel; inner function},
language = {eng},
number = {4},
pages = {1221-1240},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The relationship between $K_u^2\cap vH^2$ and inner functions},
url = {http://eudml.org/doc/299626},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Yang, Xiaoyuan
TI - The relationship between $K_u^2\cap vH^2$ and inner functions
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 4
SP - 1221
EP - 1240
AB - 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\ne u)$. If $K_u^2\cap vH^2\ne \lbrace 0\rbrace $, 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 $K_B^2\cap uH^2\ne \lbrace 0\rbrace .$ In particular, we discuss the finite-dimensional intersection $K_u^2 \cap vH^2$. Moreover, we investigate connections between minimal Toeplitz kernels and $K_u^2\cap vH^2$.
LA - eng
KW - model space; invariant subspace of the unilateral shift operator; Toeplitz kernel; inner function
UR - http://eudml.org/doc/299626
ER -