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> Let $T$ be a normal operator on a complex inner product space. Then $T$ is a self adjoint operator if and only if 
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> 1) $T$ has distinct eigen values
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> 2) $T$ has repeated eigen values
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> 3) All eigen values of $T$ are real
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> 4) $T$ has atleast one real eigen value
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If $T$ is a self adjoint operator then all it's eigen values are real. 

This rules out 1, 2 and 4. 

But how to prove the converse? i.e. If all the eigen values of a normal operator $T$ are real then it is self adjoint.