Let $H$ be a complex Hilbert space. Define what it means for a linear operator $T: H \rightarrow H$ to be self-adjoint. State a version of the spectral theorem for compact selfadjoint operators on a Hilbert space. Give an example of a Hilbert space $H$ and a compact self-adjoint operator on $H$ with infinite dimensional range. Define the notions spectrum, point spectrum, and resolvent set, and describe these in the case of the operator you wrote down. Justify your answers.