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Sturmian theorem are established for weakly coupled elliptic systems generated in a bounded domain by the expressions /,« = -Au + Au, l2w =■ -Am) + Bw, and Dirichlet boundary conditions. Here A denotes the Laplace operator, and A, B are m X m matrices. We do not assume that A, B are symmetric, but instead essentially require B irreducible and b¡j < 0 if / ¥= j. Estimates on the real eigenvalue of /, with a positive eigenvector are then obtained as applications. Our results are motivated bydoi:10.1090/s0002-9939-1985-0784181-8 fatcat:ynfg4xva2zei7k3qdfiui45lom