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We identify subsets of the joint numerical range of an operator tuple in terms of its joint spectrum. This result helps us to transfer weak convergence of operator orbits into certain approximation and interpolation properties for powers in the uniform operator topology. This is a far‐reaching generalization of one of the main results in our recent paper [Müller and Tomilov,  J. Funct. Anal . 274 (2018) 433–460]. Moreover, it yields an essential (but partial) generalization of Bourin's ‘pinching’ theorem from [Bourin,  J. Operator Theory  50 (2003) 211–220]. It also allows us to revisit several basic results on joint numerical ranges, provide them with new proofs and find a number of new results.

Joint numerical ranges and compressions of powers of operators