Let $\mathbb{D}$ be the open unit disk and let $H^\infty (\mathbb{D})$ be the Banach algebra of bounded holomorphic functions. Let $f_1,\ldots , f_n\in H^\infty (\mathbb{D})$. Now there is a greatest common divisor of $f_1,\ldots , f_n$ in the space of all holomorphic functions.
My question is if $f_1,\ldots, f_n$ has a greatest common divisor in $H^\infty (\mathbb{D})$. If the answer is no, then whether we can say that the topological closure of the ideal generated by $f_1,\ldots, f_n$ is equal to the closure of a principal ideal in $H^\infty (\mathbb{D})$.
Any answer or reference will be appreciated. Thank you.
Certainly if the functions were polynomials, then the notion of greatest common divisor is well-defined. Using the "inner-outer factorization", for functions in $H^2$, a "greatest common inner function" seems to make sense. But the notion of a "greatest common outer function" does not make as much sense, at least to me.
Let $f_1$ be the singular inner function defined from the measure having mass 1 at 1, and let $f_2$ be the singular inner function defined from the measure having mass 1 at -1. I expect that it can be shown that $\{ f_1,\ f_2 \}$ is a counterexample to your proposition. After all, what would be the generator of the principal ideal?