Page 297 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
T * * T *

T T *  T * * T ...(3)
From (2) and (3) it follows that

T T * .

(v) We have T * T T * T

T T  T * T

T 2 ...(4)

Further for every x H, we have

Tx 2 Tx,Tx

T * Tx,x

T * T x,x

T * T x x (By Schwarz inequality)

2
T * T x
Then we have

2 2
Tx T * T x x H ...(5)
Now T sup Tx : x 1

2 2
T sup Tx : x 1
2
sup Tx : x 1
From (5) we see that

if x 1 , then Tx 2 T * T .

2
Therefore, Sup Tx : x 1 T * T
2
T T * T . ...(6)
From (5) and (6) it follows that

2
T * T T .
This completes the proof of the theorem.

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