Page 294 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 27: The Adjoint of an Operator

Notes
x, T' T * y =0 x H

T' T * y = 0 for every y H

Hence T'y = T * y for every y H.

T T *.
This completes the proof of the theorem.

Notes
1. We note that the zero operator and the identity operator I are adjoint operators. For,
(i) x,0 * y 0x,y 0,y 0 x,0 = x,0y

so from uniqueness of adjoint 0* = 0.
(ii) (x,Iy) = (Ix,y) = (x,y) = (x,Iy)
so from uniqueness of adjoint I*=I.
2. If H is only an inner product space which is not complete, the existence of T*
corresponding to T in the above theorem is not guaranteed as shown by the following
example.

Example: Let M be a subspace of L consisting of all real sequences, each one containing
2
only finitely many non-zero terms. M is an incomplete inner product space with the same inner
product for  given by
2

x,y = x y n ...(1)
n
n=1
For each x M, define

x
T x n ,0,0,...... ...(2)
n
n=1
Then from the definition, for x,y M,

x
T x,y y 1 n .
n=1 n

Now let e 0,0,...,1,0,... where 1 is in the n place.
th
n
Then using (3) we obtain
e (j) 1
T ,e 1. n 1. .
e n 1
j n

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