Joint k-numerical ranges of operators

Let ℋ be a complex Hilbert space and ℬ(ℋ) the algebra of all bounded linear operators on ℋ . For a positive integer k less than the dimension of ℋ and A=(A_1,...,AM)∈ℬ(ℋ)^m , the joint k-numerical range W_k( A) is the set of (α_1,..,α_m)∈ℂ^𝕞 such that α_i=Σ^k_j=1⟨ A_ixj,xj ⟩ for an orthonormal set {x_1,...,x_k} in ℋ . The geometrical properties of W_k( A) and their relations with the algebraic properties of {A_1,...,A_m} are investigated in this paper. First, conditions for W_k( A) to be convex are studied, and some results for finite dimensional operators are extended to the infinite dimensional case. An example of A is constructed such that W_k( A) is not convex, but W_r( A) is convex for all positive integer r not equal to k. Second, descriptions are given for the closure of W_k( A) and the closure of conv W_k( A) in terms of the joint essential numerical range of A for infinite dimensional operators A_1,...,A_m . These lead to characterizations of W_k( A) Wk(A) or conv W_k( A) to be closed. Moreover, it is shown that conv W_k( A) is closed whenever W_k+1( A) or conv W_k+1( A) is. These results are used to study the connection between the geometric properties of W_k( A) and algebraic properties of A_1,...,A_m . For instance, W_k( A) is a polyhedral set, i.e., the convex hull of a finite set, if and only if A_1,...,A_k have a common reducing subspace V of finite dimension such that the compression of A_1,...,A_m on the subspace V are diagonal operators D_1,...,D_m and W_k(A)=W_k(D_1,...,D_m) . Characterization is also given to A such that the closure of W_k( A) is polyhedral. For finite rank operators the following two condition are equivalent: (1) A_1,...,A_m is a commuting family of normal operators. (2) W_k(A)=W_k(A_1,...,A_m) is polyhedral for every positive integer k less than dim ℋ . However, the two conditions are not equivalent for compact operators. Characterizations are given for compact operators A_1,...,A_m satisfying (1) and (2), respectively. Results are also obtained for general non-compact operators. Open problems and future research topics are presented.