Page 319 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Since TT* = T*T, therefore R.H.S. of (1) and (2) are equal.
Hence their L.H.S. are also equal.
(T – I) (T – I)* = (T – I)* (T – I)
T – I is normal.

29.1.2 Unitary Operator

An operator U on a Hilbert space H is said to be unitary if UU* =U*U =I.

Notes
(i) Every unitary operator is normal.
(ii) U*= U i.e. an operator is unitary iff it is invertible and its inverse is precisely equal

-1
to its adjoint.
Theorem 8: If T is an operator on a Hilbert space H, then the following conditions are all
equivalent to one another.
(i) T*T = I.

(ii) (Tx,Ty) = (x,y) for all x,y H.

(iii) Tx x x H.

Proof: (i) (ii)

(Tx,Ty) = (x,T*Ty) = (x,Iy) = (x,y) x and y.
(ii) (iii)

We are given that

Tx,Ty x,y x,y H.
Taking y = x, we get

(Tx,Tx) = (x,x) Tx 2 x 2

Tx x x H.
(iii) (i)

Given Tx x x

Tx 2 x 2

Tx,Tx x,x

T * Tx,x x,x

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