Page 295 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
Now we check whether there is T* which is adjoint of T. Now e ,T * e T * e .e ,
n where the
n 1 1
R.H.S. gives the component wise inner product. Since T * e 1 M,T * e .e cannot be equal to
n
1
1
n 1,2,...
n
there is no T* on M such that

T e ,e e ,T * e
n 1 n 1
Hence completeness assumption cannot be ignored from the hypothesis.

Notes

1. The mapping T T * is called the adjoint operation on H .
2. From Theorem (2), we see that the adjoint operation is mapping T T * on H
into itself.

Theorem 3: The adjoint operation T T* on (H) has the following properties:

(i) T 1 T * T * 1 T * 2 (preserve addition)
2
(ii) T T * T * T * (reverses the product)
1 2 2 1
(iii) T * T * (conjugate linear)

(iv) T * T

(v) T * T T 2

Proof: (i) For every x, y H, we have

(x, (T + T )*y = ((T + T ) x, y) (By def. of adjoint)
1 2 1 2
= (T x + T x, y)
1 2
= (T x, y) + (T x, y)
1 2
= (x, T *y) + (x, T *y)
1 2
= (x, T *y + T *y)
1 2
= (x, (T * + T *)y)
1 2
(T +T )* = T * + T * by uniqueness of adjoint operator
1 2 1 2
(ii) For every x,y H, we have

x T T * y T T x,y
1
2
1
2
T T x ,y
2
1
288 LOVELY PROFESSIONAL UNIVERSITY