Operators on Hilbert Spaces - user web page - AIMS

Section 3.4. Hilbert-adjoint and self-adjoint operators Page 19
Example 3.4.6 A linear map on R n with matrix A is a self-adjoint if and only if A is symmetric
(A = A T ). A linear map on C n with matrix A is self-adjoint if and only if A is Hermitian
(A = A ∗ ).
Remark 3.4.7 Self-adjoint operators on Hilbert spaces are used in quantum mechanics to represent
physical observables such as position, momentum, angular momentum and spin. An
important example is the Hamiltonian operator given by
Hψ = − 2
2m ∇2 ψ + V ψ.
Definition 3.4.8 ([Kre78], 3·10-1 (Unitary operator)) A bounded linear operator T : H → H
on a Hilbert space H is said to be unitary if T is bijective and
Hence
TT ∗ = T ∗ T. (3.32)
T ∗ = T −1 . (3.33)
Definition 3.4.9 ([Kre78], 3·10-1 (Normal operators)) A bounded linear operator T : H → H
on a Hilbert space H is said to be normal if
TT ∗ = T ∗ T. (3.34)
Remark 3.4.10 If T is self-adjoint or unitary, then T is normal, but the converse is not generally
true. For example if I : H → H is the identity operator, then T = 2iI is normal since T ∗ = −2iI,
so that TT ∗ = T ∗ T = 4I but T ∗ = T and T ∗ = T −1 = − 1
2 iI.
Theorem 3.4.11 ([Kre78], 3·10-3 Theorem (Self-Adjointness)) Let T : H → H be a bounded
linear operator on a Hilbert space H. Then
1. If T is self-adjoint, then 〈Tx, x〉 ∈ R for all x ∈ H.
2. If H is complex and 〈Tx, x〉 ∈ R for all x ∈ H, then the operator T is self-adjoint.
Proof.
1. If T is self-adjoint, then for all x,
〈Tx, x〉 = 〈x, Tx〉. (3.35)
By definition, 〈Tx, y〉 = 〈x, T ∗ y〉 and since T is self-adjoint, we have
Combining equations (3.35) and (3.36) gives
〈Tx, x〉 = 〈x, Tx〉. (3.36)
〈Tx, x〉 = 〈Tx, x〉.
Hence 〈Tx, x〉 is equal to its complex conjugate which implies that it is real.