Page 306 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 28: Self Adjoint Operators

T = 0 (zero operator) Notes
This completes the proof of the theorem.

Theorem 6: An operator T on a Hilbert space H is self-adjoint.
Tx,x is real for all x.
Proof: Let T* =T (i.e. T is self adjoint operator)

Then for every x H,we have
Tx,Tx x,T * x x,Tx Tx,x

Tx,x equals its own conjugate and is therefore real.

Conversely, let Tx,x is real x H.We claim that T is self adjoint i.e. T*=T.
since Tx,x is real x H,

Tx,x Tx,x x,T * x T * x,x

Tx,x T * x,x 0 x H
Tx T * x,x 0 x H

T T * x,x 0 x H
T – T* = 0 [ if (Tx, x) = 0 T = 0]

T = T*
T is self adjoint.
This completes the proof of the theorem.
Cor. If H is real Hilbert space, then A is self adjoint

Ax,y Ay,x x,y H.

A is self adjoint for any x,y H.
Ax,y x,A * y A * y,x .

since H is real Hilbert space A * y,x A * y,x so that Ax,y Ay,x  A* A

Theorem 7: The real Banach space of all self-adjoint operators on a Hilbert space H is a partially
ordered set whose linear and order structures are related by the following properties:

(a) If A 1 A then A +A A +A for every A S;
2
2
1
(b) If A A and 0, then A A .
1 2 1 2
Proof: Let S represent the set of all self-adjoint operators on H. We define a relation on S as
follows:
If A A 2 S, we write A 1 A if A x,x A ,x x in H.
1
2
2
1
We shall show that ' ' is a partial order relation on S. ' ' is reflexive.

LOVELY PROFESSIONAL UNIVERSITY 299