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Let $A$ be a bounded linear operator on an infinite dimensional Hilbert space. How would you define $A^{Tr}$, the transpose of $A$ with respect to fixed but arbitrary orthonormal basis of $A$?

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  • $\begingroup$ en.wikipedia.org/wiki/Riesz_representation_theorem which implies the existence and uniqueness of the adjoint of an operator. $\endgroup$ – Vim Jun 15 '17 at 7:14
  • $\begingroup$ are the adjoint of an operator and the transpose of an operator the same thing? $\endgroup$ – Dong North Jun 15 '17 at 7:18
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    $\begingroup$ I think so. In fact we usually call the "transpose" adjoint in the infinite dimensional case. $\endgroup$ – Vim Jun 15 '17 at 7:19
  • $\begingroup$ but in a finite Hilbert space H, are the transpose of an operator A on H different from the adjoin of an operator on H? $\endgroup$ – Dong North Jun 15 '17 at 7:25
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    $\begingroup$ In the usual terminology, matrices have a transpose and operators have an adjoint. It is somewhat unusual to talk about the adjoint of a matrix. It is extremely unusual to talk about the transpose of an operator. $\endgroup$ – Omnomnomnom Jun 15 '17 at 7:35

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