Operators on Hilbert Spaces - user web page - AIMS

Section 3.4. Hilbert-adjoint and self-adjoint operators Page 15
Now by Cauchy-Schwarz inequality,
so that
|〈Sx, y〉|
sup
x=0 xy
y=0
Sxy
≤ sup
x=0 xy
y=0
= S, (3.16)
h ≤ S. (3.17)
Combining equations (3.15) and (3.17) gives h = S.
(c) Let us assume that T : H1 → H2 is a linear operator such that for all x ∈ H1 and y ∈ H2,
we have
h(x, y) = 〈Sx, y〉 = 〈Tx, y〉.
Then, by Lemma 3.3.4, Sx = Tx for all x ∈ H1. Hence S = T by definition.
Remark 3.3.6 An important consequence of the Riesz representation theorem is the existence
of the adjoint of a bounded operator on a Hilbert space.
3.4 Hilbert-adjoint and self-adjoint operators
Definition 3.4.1 ([Kre78], 3·9-1 Definition (Hilbert-adjoint operator, T ∗ ))
Given two Hilbert spaces, H1 and H2, let T : H1 → H2 be a bounded linear operator. Then the
Hilbert-adjoint operator T ∗ of T is the operator T ∗ : H2 → H1 such that for all x ∈ H1 and
y ∈ H2,
〈Tx, y〉 = 〈x, T ∗ y〉. (3.18)
Let us show that there exists such an operator T ∗ .
Theorem 3.4.2 ([Kre78], 3·9-2 Theorem (Existence)) The Hilbert-adjoint operator T ∗ of T
exist, is unique and is a bounded linear operator with norm
Proof.
Consider
T ∗ = T. (3.19)
h(y, x) = 〈y, Tx〉. (3.20)