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We prove that if $L$ is a linear map such that $L(\\mathcal{S}) \\subseteq \\mathcal{S}$, then $\\rho(L)$ (the spectral radius of $L$) is at most $1$ and when $L(\\mathcal{S}) = \\mathcal{S}$, we have $\\rho(L) = 1$. 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