Let $u$ be an inner function and denote by $H^2$ the Hardy space on the open unit disc D. A model space $K_u$ associated to $u$ is a Hilbert space of the form $K_u=(uH^2)^⊥$ where ⊥ denotes the orthogonal complement in $H^2$. Does FLT still true in $K_u$ when u is an infinite Blaschke product ?
NB: (1) FLT for $K_u$ is stated as follows: Prove or disprove that there is no f,g,h in $K_u$ such that $$f^n+g^n=h^n$$ when $dim(span(f,g,h))>1$ and $n>2$.
(2) It was shown that FLT is true for $K_u$ when u is a finite Blaschke product ( cf. Stephan Ramon Garcia, Javad Mashreghi, William T. Ross - Introduction to Model Spaces and their Operators-Cambridge University Press (2016), p. 125. Here is a link https://libgen.is/search.php?req=Introduction+to+Model+Spaces+and+their+Operators&open=0&res=25&view=simple&phrase=1&column=def).
General question: What about the general case when $u$ is an arbitrary inner function ?
