Page 292 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 27: The Adjoint of an Operator

Notes

Note The relation Tx,y x,T * y can be equivalently written as

T * x,y x,Ty since

T * x,y y,T * x = Ty,x = x,Ty = x,Ty

T * x,y x,Ty .

Example: Find adjoint of T if T is defined on  as Tx 0,x ,x ,... for every x x n  .
2
2
1
2
Let T* be the adjoint of T. Using inner product in  , we have
2
T * x,y = x,Ty
since Ty 0,y ,y ,... , we have
2
1
T * x,y x,Ty = x y Sx,y ,
n+1 n
n=1
where S x x ,x ,...
2
3
Hence T * x,y Sx,y for every x in  .
2
Since T* is unique, T*=S so that we have

T * x x ,x ,x ,... .
2 3 4
Theorem 2: Let H be the given Hilbert space and T* be adjoint of the operator T. Then T* is a
bounded linear transformation and T determine T* uniquely.
Proof: T* is linear.

Let y ,y 2 H and , be scalars. Then for x H, we have
1
x,T * y y Tx, y y
1 2 1 2

But Tx, y 1 y 2 Tx,y 1 Tx,y 2

Tx,y 1 x,T * y 2

x, T * y 1 x, T * y .
2

Hence for any x H,
x,T * y y x, T * y x, T * y
1 2 1 2

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