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The purpose of the present work is to answer an open problem which was raised by G. Costakis and A. Manoussos in their paper "J-class operators and hypercyclicity", J. Operator Theory 67 (2012), 101-119. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space.doi:10.1090/s0002-9939-2014-11891-0 fatcat:g4u3kwgionaz7l3brr2pwivew4