On Certain (Nearly) Convex Joint Numerical Ranges

Abstract

Let H be a complex Hilbert space with scalar product denoted (··), and let L(H) denote the algebra of bounded linear operators on H. For a family r = {T _i : i ∈ J} ⊂ L(H) one defines the joint numerical range W(τ) by

$$ W\left( \tau \right) = \left\{ {\left\{ {\langle {{{\text{T}}}_{i}}x,x\rangle :x \in H,\left\| x \right\| = 1} \right.} \right\} $$