Super‐identical pseudospectra

The complex N × N matrices A and B are said to have super‐identical pseudospectra if, for each z ∈ ℂ, the singular values of A − zI are the same as those of B − zI . We explore this condition and its consequences. On the positive side, drawing on ideas from invariant theory, we prove that there exists an integer m = m ( N ) such that ‘almost every’ m ‐tuple of N × N matrices with super‐identical pseudospectra contains a pair that are unitarily equivalent. On the negative side, we present an example of a pair of non‐derogatory 4 × 4 matrices A and B with super‐identical pseudospectra such that || A 2 || ≠ || B 2 ||.