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Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_{\\rm vor}$ in terms of zeros of the 2-modified Fredholm determinant $\\mathcal D(\\lambda,0)=\\det_{2}(I-K_{\\lambda}(0))$ associated with the Hilbert Schmidt operator $K_{\\lambda}(\\mu)$ for $\\mu=0$. 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Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_{\\rm vor}$ in terms of zeros of the 2-modified Fredholm determinant $\\mathcal D(\\lambda,0)=\\det_{2}(I-K_{\\lambda}(0))$ associated with the Hilbert Schmidt operator $K_{\\lambda}(\\mu)$ for $\\mu=0$. 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