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

Abstract

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‖ 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 mS=conv{|v|2:v∈S and ‖v‖=1} of a subspace S⊂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‖ 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 mS such as extremal curves related with the basis E. All these conditions are directly related with the description of minimal self-adjoint compact operators.