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Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk ${}^N$ or on the Segal-Bargmann space over ${ℂ}^N$. Even in the case N = 1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large “finite section”, that is, by the compression $A_n$ of A to the linear span of the monomials $z_1^{k_1}...z_N^{k_N}:0\le k_j\le n$. Unfortunately, in general the spectrum of $A_n$ does not mimic the spectrum of A as...