10. Adjoint operators

Question 1

Define the adjoint operator/dual operator of a normed space

Answer 1

Let X and Y be normed spaces and let T ∈ B(X,Y). The map T′ :Y∗ →X∗ defined by
is said to be the adjoint operator (or dual operator) of T.
(T′φ)(x) = φ(Tx), φ ∈ Y∗, x ∈ X,

Question 2

Define the adjoint operator of a hilbert space

Answer 2

Let X and Y be Hilbert spaces and suppose that T ∈ B(X, Y ). There exists a unique operator T∗ ∈ B(Y,X) such that
(10.1) ⟨Tx,y⟩Y = ⟨x,T∗y⟩X, x ∈ X, y ∈ Y.
Moreover, ∥T∗∥ = ∥T∥.