Page 279 - DMTH502_LINEAR_ALGEBRA

Unit 25: Linear Functional and Adjoints of Inner Product Space

Notes
Example 6: In Example 5, we saw that some linear operators on an infinite-dimensional
inner product space do have an adjoint. As we commented earlier, some do not. Let V be the
inner product space of Example 6, and let D be the differentiation operator on C[x]. Integration
by parts shows that
(Df g) = f(1)g(1) f(0) g(0) (f Dg).
Let us fix g and inquire when there is a polynomial D*g such that (Df g) = (f D*g) for all f. If such
a D*g exists, we shall have
(f D*g) = f(1) g(1) f (0) g(0) (f Dg)
or
(f D*g + Dg) = f (1) g(1) f (0) g(0).

With g fixed, L(f) = f(1) g(1) f(0) g(0) is a linear functional of the type considered in Example 1 and
cannot be of the form L( f ) = (f h) unless L = 0. If D*g exists, then with h = D*g + Dg we do have L( f)
= (f h), and so g(0) = g(1) = 0. The existence of a suitable polynomial D*g implies g(0) = g(1) = 0.
Conversely, if g(0) = g(1) = 0, the polynomial D*g = – Dg satisfies (Df g) = (f D*g) for all f. If we
choose any g for which g(0) 0 or g(1) 0, we cannot suitable define D*g, and so we conclude that
D has no adjoint.
We hope that these examples enhance the reader’s understanding of the adjoint of a linear
operator. We see that the adjoint operation, passing from T to T*, behaves somewhat like
conjugation on complex numbers. The following theorem strengthens the analogy.

Theorem 4: Let V be a finite-dimensional inner product space. If T and U are linear operators on
V and c is a scalar,
(i) (T + U)* = T* + U*;

(ii) (cT)* = c T *;

(iii) (TU)* = U*T*;
(iv) (T*) = T.
Proof: To prove (i), let and be any vectors in V.
Then

((T + U) ) = (T + U )
= (T ) + (U )
= ( T* ) + ( U* )
= ( T* + U* )

= ( (T* + U*) )
From the uniqueness of the adjoint we have (T + U)* = T* + U*. We leave the proof of (ii) to the
reader. We obtain (iii) and (iv) from the relations

(TU ) = (U T* ) = ( U*T* )
(T* ) = ( T * ) (T / ) ( T ).

Theorem 4 is often phrased as follows: the mapping T T* is a conjugate-linear anti-isomorphism
of period 2. The analogy with complex conjugation which we mentioned above is, of course,

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