quantum mechanics - Is there a deterministic observable that has only single eigenvalue?

Is there an observable in quantum mechanics which has only one eigenvalue and an eigenspace associated with that single eigenvalue? This observable is deterministic in the sense that it gives same measurement value all the time. But the final state would be any of the wave functions living in its eigenspace corresponding to the single eigenvector, with different probabilities.

What would that mean practically, to quantum mechanics?

Answer

If I understand the question and the comments correctly, what is needed is an everywhere defined operator that preserves norms and has only a single point in the spectrum. The first condition forces the operator to be a partial isometry, while the second forces it to be a multiple of the identity. The intersection is then any operator $zI$, where $z$ is a complex number of norm one and $I$ is the identity operator.