Compact Operator On Hilbert Space - Compact Self Adjoint Operator

Compact Self Adjoint Operator

A bounded operator T on a Hilbert space H is said to be self-adjoint if T = T*, or equivalently,

It follows that <Tx, x> is real for every x ∈ H, thus eigenvalues of T, when they exist, are real. When a closed linear subspace L of H is invariant under T, then the restriction of T to L is a self-adjoint operator on L, and furthermore, the orthogonal complement L⊥ of L is also invariant under T. For example, the space H can be decomposed as orthogonal direct sum of two T–invariant closed linear subspaces: the kernel of T, and the orthogonal complement (ker T)⊥ of the kernel (which is equal to the closure of the range of T, for any bounded self-adjoint operator). These basic facts play an important role in the proof of the spectral theorem below.

The classification result for Hermitian n × n matrices is the spectral theorem: If M = M*, then M is unitarily diagonalizable and the diagonalization of M has real entries. Let T be a compact self adjoint operator on a Hilbert space H. We will prove the same statement for T: the operator T can be diagonalized by an orthonormal set of eigenvectors, each of which corresponds to a real eigenvalue.