Give one example of each of the following. Also justify your choice of example.

i) A self-adjoint operator on .

ii) A normal operator on a Hilbert space which is not unitary.

i) A self-adjoint operator:
An example of a self-adjoint operator on a Hilbert space is the position operator x^\hat{x} in L2(R)L^2(\mathbb{R}), the space of square-integrable functions. The position operator x^\hat{x} acts on a function f(x)∈L2(R)f(x) \in L^2(\mathbb{R}) by x^f(x)=xf(x)\hat{x} f(x) = x f(x).

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