Page 305 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes T O i.e. zero operator.
This completes the proof of the theorem.
Theorem 5: If T is an operator on a Hilbert space H, then

Tx,x 0 x in H T O.

Proof: Let T = O. Then for all x in H, we have

Tx,x Ox,x 0,x 0.

Conversely, let Tx,x 0 x,y H. Then we show that T is the zero operator on H.

If , any two scalars and x,y are any vectors in H, then

T x y , x y Tx Ty, x y

T x , x y T y , x y

Tx,x Tx,y Ty,x Ty,x

2 2
Tx,x Tx,y Ty,x Ty,x
2 2
T x y , x y Tx,x Ty,y Tx,y Ty,x ...(1)

But by hypothesis Tx,x 0 x H.
L.H.S. of (1) is zero, consequently the R.H.S. of (1) is also zero. Thus we have

Tx,y Ty,x 0 ...(2)

for all scalars , and x,y H.

Putting 1, 1 in (2) we get

Tx,y Ty,x 0 ...(3)
Again putting i, 1 in (2) we obtain

i Tx,y i Ty,x 0 ...(4)

Multiply (3) by (i) and adding to (4) we get

2i Tx,y 0 x,y H

Tx,y 0 x,y H

Tx,Tx 0 x,y H (Taking y = Tx)

Tx 0 x,y H

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