The classical Dirichlet space 
is the Hilbert space of holomorphic functions 
on the unit
disk for which
 |
This norm equals the squared Hardy space 
norm plus 
,
where 
is planar area measure (El-Fallah et al. 2014).
Multiplication by 
defines the Dirichlet shift. A function 
is a cyclic vector for this
operator iff its polynomial multiples are dense
in 
. The Brown-Shields
conjecture characterizes this cyclicity using outer functions and the logarithmic
capacity of a boundary zero set.
