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We introduce and discuss unitary equivalence of operators in the Cowen-Douglas class $B_n(\Omega)$. We prove that operators in the Cowen-Douglas class are unitarily equivalent if and only if there exists a holomorphic matrix $M$ which satisfies $\tilde K_\lambda(z)=M(z)K_\lambda(z)M(\lambda)^*$ for the spaces' kernels. We use this to establish that $\Delta_\lambda\log\lvert\lvert k_\lambda\rve
