The nonstandard two-disk problem plays a fundamental role in robust feedback optimization. Here, it is shown via Banach space duality theory that its solutions satisfy an extremal identity, and may be viewed as a dual extremal kernel of a particular L-1-optimization problem. A novel operator theoretic framework to characterize explicitly its solutions is developed, in particular, the two-disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non-Hilbert version of a vector valued H-2 space. Moreover, this operator is shown to be a combination of multiplication and Toeplitz operators. Under certain conditions, existence of maximal vectors is established leading to an explicit formula for the optimal controller. An "infinite matrix" representation with respect to a canonical basis is derived, together with an algorithm to compute it. The norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi-definite programming problems involving the computation of a matrix projective tensor norm.