 The quotient C*-algebra of "L"("H") modulo the compact operators is called the Calkin algebra, in which one can consider properties of an operator up to compact perturbation.A bounded operator "T" on a Hilbert space "H" is said to be self-adjoint if "T" = "T*", or equivalently,

It follows that  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".