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We define the (convex) joint numerical range for an infinite family ofcompact operators in a Hilbert space H. We use this set to determine whether aself-adjoint compact operator A with {||A||, -||A||} in its spectrum is minimalrespect 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 setm_S = conv{ |v|^2 : v in S and ||v|| = 1 } of a subspace S of H in terms ofjoint numerical ranges and obtain equivalences between the intersection ofmoments of two subspaces and of its two related joint numerical ranges.Moreover, we relate the condition of minimality of A or the intersection of themoments of the eigenspaces of ||A|| and -||A|| to the intersection of the jointnumerical ranges of two finite families of certain finite hermitian matrices.We also study geometric properties of the set m_S such as extremal curvesrelated with the basis E. All these conditions are directly related with thedescription of minimal self-adjoint compact operators.

Minimal self-adjoint compact operators, moment of a subspace and joint numerical range