Brown–Shields conjecture on cyclic vectors in the Dirichlet space

Determine whether, for every function f in the classical Dirichlet space D, the function f is cyclic for the shift operator S:f(z)→zf(z) on D if and only if f is outer and its boundary zero set Z(f)⊂T has logarithmic capacity zero.

Background

The cyclicity of the shift operator S:f(z)→zf(z) is completely characterized in the Hardy space H^2 by Beurling’s theorem, but remains largely unresolved in other analytic function spaces. The Dirichlet space D is a central setting where a full characterization is unknown.

Brown and Shields formulated a conjecture asserting that cyclicity in the Dirichlet space should be equivalent to being outer together with a capacity condition on the boundary zero set. Despite significant advances, a definitive resolution of this conjecture has not been achieved, and it continues to guide research on invariant subspaces and cyclic vectors in Dirichlet-type settings.