Preservation of the joint essential matricial range

Let $A = (A_1, dots, A_m)$ be an $m$-tuple of elements of a unital $C$*-algebra ${cal A}$ and let $M_q$ denote the set of $q times q$ complex matrices. The joint $q$-matricial range $W^q(A)$ is the set of $(B_1, dots, B_m) in M_q^m$ such that $B_j = Phi(A_j)$ for some unital completely positive linear map $Phi: {cal A} rightarrow M_q$. When ${cal A}= B(H)$, where $B(H)$ is the algebra of bounded linear operators on the Hilbert space $H$, the {bf joint spatial $q$-matricial range} $W^q_s(A)$ of $A$ is the set of $(B_1, dots, B_m) in M_q^m$ for which there is a $q$-dimensional $V$ of $H$ such that $B_j$ is a compression of $A_j$ to $V$ for $j=1,dots, m$. Suppose $K(H)$ is the set of compact operators in $B(H)$. The joint essential spatial $q$-matricial range is defined as $$W_{ess}^q(A) = cap { {bf cl}(W_s^q(A_1+K_1, dots, A_m+K_m)): K_1, dots, K_m in K(H) },$$ where ${bf cl}$ denotes the closure. Let $pi$ be the canonical surjection from $B(H)$ to the Calkin algebra $B(H)/K(H)$. We prove that $W_{ess}^q(A) =W^q(pi(A) $, where $pi(A) = (pi(A_1), dots, pi(A_m))$. Furthermore, for any positive integer $N$, we prove that there are self-adjoint compact operators $K_1, dots, K_m$ such that $${bf cl}(W^q_s(A_1+K_1, dots, A_m+K_m)) = W^q_{ess}(A) quad hbox{ for all } q in {1, dots, N}.$$ These results generalize those of Narcowich-Ward and Smith-Ward, obtained in the $m=1$ case, and also generalize a result of M{u}ller obtained in case $m ge 1$ and $q=1$. Furthermore, if $W_{ess}^1({bf A}) $ is a simplex in ${mathbb R}^m$, then we prove that there are self-adjoint $K_1, dots, K_m in K(H)$ such that ${bf cl}(W^q_s(A_1+K_1, dots, A_m+K_m)) = W^q_{ess}(A)$ for all positive integers $q$.