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In this paper, we give an explicit characterization of the orthogonal complement of the "set of stabilizing inputs" for an unstable convolution operator. This characterization involves the right-coprimne factorization of the transfer function matrix of the convolution operator. Using this characterization, we then explicitly demonstrate specific inputs that would cause -instability in a given large-scale interconnected system, using the technique of orthogonal decomposition of the input space. Finally, using the technique of exponential weighting, we give a general result that can be used to establish the -instability of a given system by proving that an associated system is -unstable. In this way, one can obtain -instability criteria corresponding to virtually every known criterion for -instability that uses the technique of orthogonal decomposition.