Let u be an inner function and K u 2 be the corresponding model space. For an inner function v, the subspace vH2 is an invariant subspace of the unilateral shift operator on H2. In this article, using the structure of a Toeplitz kernel \(\text{ker}\,T_{\overline{u}v}\) , we study the intersection K u 2 ⋂ vH2 by properties of inner functions u and v (v ≠ u). If K u 2 ⋂ vH2 ≠ {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∞, Og denotes the outer factor of g, and λ is some constant with ∣λ∣ = 1. Furthermore, for any nonconstant inner function u, there exists a Blaschke product B such that K B 2 ⋂uH2 ≠ {0}. In particular, we discuss the finite-dimensional intersection K u 2 ⋂vH2. Moreover, we investigate connections between minimal Toeplitz kernels and K u 2 ⋂ vH2.
