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

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