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We consider the problem of identifying a linear deterministic operator from its response to a given probing signal. For a large class of linear operators, we show that stable identifiability is possible if the total support area of the operator's spreading function satisfies Δ ≤ 1/2. This result holds for an arbitrary (possibly fragmented) support region of the spreading function, does not impose limitations on the total extent of the support region, and, most importantly, does not require the support region to be known prior to identification. Furthermore, we prove that stable identifiability of almost all operators is possible if Δ <; 1. This result is surprising as it says that there is no penalty for not knowing the support region of the spreading function prior to identification. Algorithms that provably recover all operators with Δ ≤ 1/2, and almost all operators with Δ <; 1 are presented.