  We define the (convex) joint numerical range for an infinite family of
compact operators in a Hilbert space H. We use this set to determine whether a
self-adjoint compact operator A with {||A||, -||A||} in its spectrum is minimal
respect to the set of diagonals in a fixed basis E of H in the operator norm,
that is ||A|| <= ||A+D||, for all diagonal D. We also describe the moment set
m_S = conv{ |v|^2 : v in S and ||v|| = 1 } of a subspace S of H in terms of
joint numerical ranges and obtain equivalences between the intersection of
moments of two subspaces and of its two related joint numerical ranges.
Moreover, we relate the condition of minimality of A or the intersection of the
moments of the eigenspaces of ||A|| and -||A|| to the intersection of the joint
numerical ranges of two finite families of certain finite hermitian matrices.
We also study geometric properties of the set m_S such as extremal curves
related with the basis E. All these conditions are directly related with the
description of minimal self-adjoint compact operators.
