Page 296 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 27: The Adjoint of an Operator

Notes
T x,T * y
2 1
*
*
x,T , T y
2
1
*
*
x, T T ,y
2
1
*
*
Therefore from the uniqueness of adjoint operator, we have T T * T T .
2
1
2
1
(iii) For every x,y H, we have
x, T * y T x,y Tx ,y
Tx,y
x,T * y x, T * y

x T * y .
Therefore from the uniqueness of adjoint operator, we have

T * T *.
(iv) For every y H we have

2
T * y T * y,T * y
TT * y,y

 T * y 2 TT * y,y is a real number 0
TT * y,y

TT * y y By Schwarz inequality

T T * y y  Tx T x

2
Thus T * y T T * y y y H
T * y T y y H. ...(1)

Now T * Sup T * y : y 1

from (1), we see that if y 1 then T * y T

T * T ...(2)

Now applying (2) from the operator T* in place of operator T, we get

T * * T *

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