Page 299 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Taking the adjoint on both sides of the above, we obtain
1
TT 1 * T T * I *.
By using Theorem 4 and note (2) under Theorem 2, we obtain

T 1 * T* T * T 1 * I.
T * is invertible and hence non-singular.
Further from the above, we conclude

T * 1 T 1 *.
This completes the proof of the theorem.

Note From the properties of the adjoint operation T T * on H discussed in Theorems
(3) and (4), we conclude that the adjoint operation T T* is one-to-one conjugate linear
mapping on H into itself.

Example: Show that the adjoint operation is one-to-one onto as a mapping of (H) into
itself.
Solution: Let : (H) (H) be defined

(T) = T* T (H)
We show is one-to-one and onto.
is one-one:

Let T , T (H). Then
1 2
(T ) = (T ) T * = T *
1 2 1 2
(T *)* = (T *)*
1 2
T ** = T **
1 2
T = T
1 2
is one-to-one.
is onto:

Let T be any arbitrary member of (H). Then T* (H) and we have (T*) = (T*)* = T** = T. Hence,
the mapping is onto.

27.2 Summary

 Let T be an operator on Hilbert Space H. Then there exists a unique operator T* on H such
that Tx,y x,T * y for all x,y H.The operator T* is called the adjoint of the operator T.

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