# Dual operator relationship with complex conjugate.

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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–  wj32 Nov 9 '12 at 23:33
So in part (2) where they explain it, is it just the regular transpose that corresponds to $T^{*}$, not the complex conjugate? –  Kyle Schlitt Nov 9 '12 at 23:35
I've heard of "dual space" and "dual basis" but not "dual operator". Is there any link to it or short explanation? –  DonAntonio Nov 9 '12 at 23:36
@DonAntonio: Usually we define it as $(T^*f)(v)=fTv$. –  wj32 Nov 9 '12 at 23:38
Thanks, @wj32... –  DonAntonio Nov 9 '12 at 23:39

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.

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