Page 298 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 27: The Adjoint of an Operator

Notes
Cor: If T is a sequence of bounded linear operators on a Hilbert space and
n
T n T, then T * n T *.
We have

T * n T * T n T *

T n T (By properties of T*)

Since T T as n
n
*
T * T as n .
n
Theorem 4: The adjoint operation on H is one-to-one and onto. If T is a non-singular operator
on H, then T* is also non-singular and

1 1
T * T *.

Proof: Let : H H is defined by

T T * for every T H .

To show is one-to-one, let T ,T 2 H . Then we shall show that T 1 T 2 T 1 T .
2
1
Now T T
1 2

T * T *
1 2
T * * T * * (using Theorem 4. prop (iv))
1
2
T T
1 2
is one-to-one.
is onto:

For T* H ,we have on using Theorem 4 (iv),

T * = T* * =T.

Thus for every T* H , there is a T* H such that

T * T is onto.
1
Next let T be non-singular operator on H. Then its inverse T exists on H and
1
TT 1 T T I.

LOVELY PROFESSIONAL UNIVERSITY 291