Page 291 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Let y be a vector in H and f its corresponding functional in H*.
y
Let us define

T* : H into H * by

T* : f f ...(3)
y z
f H*. Thus
Under the natural correspondence between H and H*, let z H corresponding to z
starting with a vector y in H, we arrive at a vector z in H in the following manner:

y f y T * f y f z z, ...(4)

where T* : H* H * and y f and z f are on H to H*
y z
under the natural correspondence. The product of the above three mappings exists and it is
denoted by T*.
Then T* is a mapping on H into H such that
T * y z.

We define this T* to be the adjoint of T. We note that if we identify H and H* by the natural
correspondence y f ,
y then the conjugate of T and the adjoint of T are one and the same.
After establishing, the existence of T*, we now show (1). For x H, by the definition of the
conjugate T* on an operator T,

T * f x f Tx ...(5)
y y
By Riesz representation theorem,

y f so that
y
f Tx Tx,y ...(6)
y
Since T* is defined on H*, we get

T * f x f x x,z ...(7)
y z
But we have from our definition T*y = z ...(8)
From (5) and (6) it follows that
T * f x Tx,y ...(9)
y
From (7) and (8) it follows that

T * f x x,T * y ...(10)
y
From (9) and (10), we thus obtain

Tx,y x,T * y x,y H.

This completes the proof of the theorem.

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