Page 302 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 28: Self Adjoint Operators

If T* is an adjoint operator T on H then we know from the definition that Notes
Tx,y x,T * y x,y H .

If T is self-adjoint, then T = T*.
Tx,y x,Ty x,y H.

Conversely, if Tx,y x,Ty x,y H then we show that T is self-adjoint.

If T* is adjoint of T then (Tx, y) = (x, T*y)

We have x,Ty x,T * y
x, T T * y 0 x,y H

But since x 0 T T * y 0 y H
T = T*

T is self adjoint.
(iii) For any T H ,T T * and T * T are self adjoint. By the property of self-adjoint operators,
we have

T T * * T * T * *

T * T

T T *

T T * * T T*,

and T * T * T * T * * T * T

T * T * * T * T.
Hence T T * and T * T are self adjoint.

Theorem 1: If A is a sequence of self-adjoint operators on a Hilbert space H and if A n
n
converges to an operator A, then A is self adjoint.
Proof: Let A be a sequence of self adjoint operators and let A n A.
n
A n is self adjoint A * n A for n 1,2,...
n
We claim that A A *

Now A A* A A n A n A * n A * A *
n
A A * A A A A * A A *
n n n n

A n A A n A n A n A  A * n A n

A A 0 A A
n n

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