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Let $V$ be a $n$ dimensional vector space spanned by $\{e_{i}\}_{i=1}^{n}$.

Let $T:V\to V$ be a linear operator with matrix transformation $A$. Is there any relationship between the dual operator $T^{*}:V^{*}\to V^{*}$, and the complex conjugate $A^{*}$ of $A$?

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    @lovinglifein2012: Yes. The difference originates from the fact that the inner product has conjugate symmetry and Riesz representation theorem gives us an anti-isomorphism.2012-11-09

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See Stone and Goldbart, Mathematics for Physics, page 753 for a brief explanation. The dual operator is linear, so it does not have a simple relationship to the complex conjugate of an operator.