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I have an operator $A$ in a Hilbert space, and after numerically calculating its spectrum I can see that the eigenvalues are real and distinct. What can I say about this operator with this information?

Does it mean $A$ is compact and self adjoint? Are the eigenvectors of $A$ orthogonal?

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    You would need more information to determine if the operator is compact (for example, the only accumulation point can be $0$). And I don't think having real spectrum is sufficient to show the operator is self-adjoint. Do you have a particular operator in mind, or is this a general question? – 2017-01-20
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    I have a particular operator but it is very complicated (block matrix composed of several sub-operators each of which is quasi-periodic) so I was hoping it might possess some nice properties when viewed at the abstract level. – 2017-01-20

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