Page 308 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 28: Self Adjoint Operators

Notes
A x,x Ax,x A x,x Ax,x x H.
1 2
A 1 A x,x A 1 A x,x x H.
2
A 1 A 2 A 2 A, by def. of .
(b) We have A A A x,x A x,x x H
1 2 1 2
A x,x A x,x x H  0
1 2
A x,x A x,x x H
2
1
A x,x A x,x x in H
2
1
A 1 A ,by def. of ' '.
2
This completes the proof of the theorem.
28.1.2 Definition – Positive Operator

A self adjoint operator on H is said to be positive if A 0 in the order relation. That is
if Ax,x 0 x H.

We note the following properties from the above definition.
(i) Identity operator I and the zero operator O are positive operators.
Since I and O are self adjoint and

2
Ix,x x,x x 0
also (Ox, x) = (0, x) = 0
I,O are positive operators.

(ii) For any arbitrary T on H, both TT* and T*T are positive operators. For, we have
TT * * T * * T* TT *

TT * is self adjoint
Also T * T * T * T * * T * T
T * T is self adjoint
Further we see that

2
TT * x,x T * x,T * x T * x 0
and T * Tx,x Tx,T * *x (Tx,Tx) Tx 2 0

Therefore by definition both TT* and T*T are positive operators.
Theorem 8: If T is a positive operator on a Hilbert space H, then I+T is non-singular.
Proof: To show I+T is non-singular, we are to show that I+T is one-one and onto as a mapping of
H onto itself.
I+T is one-one.

First we show I T x 0 x 0

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