Page 242 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 21: The Conjugate of an Operator

Notes
= [ T * (T* (f))] (x)
1
= [( T * T*) (f)] (x)
1
Hence, we get
(T T )* = T * T* so that
1 1
(T T ) = (T T )* = T * T.
1 1 1
reverses the product.
Lastly if I is the identity operator on N, then
[I* (f)] (x) = f (I (x)) = f (x) = (I f) (x).
I* = I so that (I) = I* = I
preserves the identity transformation.

This completes the proof of the theorem.
Theorem 3: Let T be an operator on a normal linear space N. If N N* in the natural imbedding,
then T** is an extension of T. If N is reflexive, then T** = T.

Proof: By definition, we have
(T*)* = T**
Using theorem 2, we have T* = T .

Hence T** = T* = T .
By definition of conjugate of an operator
T : N N, T* : N* N*, T** : N** N**.
Let J ; x F be the natural imbedding of N onto N** so that
x
Fx (f) = f (x) and J (x) = F .
x
Further, since T** is the conjugate operator of T*, we get
T** (x ) x = x (T* (x )) where x N* and x N**
T** (x ) x = T** (J (x)) x .

Using the definition of conjugate, we get
T** (J (x)) x = J (x) (T* (x )).
By definition of canonical imbedding
J (x) (T* (x )) = T* (x ) x.
Again T* (x ) (x) = x (T (x)) (By definition of conjugate)

Now x (T (x)) = J (T (x))x (By natural imbedding)
Hence T** (J (x))x = J (T (x))x .
T** . J = J T

and so T** is the norm preserving extension of T. If N is reflexive, N = N** and so T** coincides
with T.

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