Operator Identities in Linear Algebra

Question

Assume that $U$ is a unitary operator and $ | a \rangle $ is its eigenvector. Then is the following true:

$$ \langle a | U = (U | a \rangle )^{\dagger} = (a | a \rangle )^{\dagger} = \langle a | a^{*} $$

Answer 1

First of all, what you were trying to say was a bit unclear, especially to the users of this site who are mostly unfamiliar with the conventions of QM.

Second, your statement is false. Here's my version.

Let $U$ be a unitary operator, and suppose that $|a\rangle$ is an eigenvector of $U$ associated with eigenvalue $a \in \Bbb C$. Note that $a$ must satisfy $|a| = 1$, so that $a^{-1} = a^*$. The following is true:

$$ \langle a |\;U = \left( U^\dagger \;|a \rangle\right)^\dagger = \left( U^{-1} \;|a \rangle\right)^\dagger = \left( a^{-1} \;|a \rangle\right)^\dagger = \left( a^* \;|a \rangle\right)^\dagger = a \;\langle a | $$

Note: $\dagger$ here denotes the operator adjoint, and $*$ denotes the complex conjugate.