The joint essential maximal numerical range. (Q1421279)

The author studies elementary properties of the joint essential numerical range, defined for an \(n\)-tuple \(T=(T_1,\ldots,T_n)\) of Hilbert space (linear and bounded) operators by the formula \[ \begin{multlined}\text{JtMax}V(T)=\{(f(t_1),\ldots,f(T_n)): f\in {\mathcal B}(H)^\ast, \;f(I)=1=\| f\|, \\ {}\qquad \text{and} \;f(T_j^\ast T_j)=\| T_j\| ^2_e, \;j=1,\ldots,n\}, \end{multlined} \] where \(\| T_j\|_e\) denotes the norm of the coset of \(T_j\) in the Calkin algebra. The obtained results are exactly the same as those in [\textit{M. Z. Abd-Alla} and \textit{T. M. El-Adawy}, J. Egypt Math. Soc. 9, No. 1, 49--58 (2001; Zbl 1047.47002)]. In that paper, an equivalent (formally different) definition of the joint essential maximal numerical range is used.