Page 313 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes
Example: Let H be any Hilbert space and I : H H be the identity operator.
Define T = 2iI. Then T is normal operator, but not self-adjoint.
Solution: Since I is an adjoint operator and the adjoint operation is conjugate linear,
T* = –2iI* = –2iI so that

TT* =T*T =4I.
T is a normal operator on H.
But T = T* T is not self-adjoint.

Note If T H is normal, then T* is normal.

since if T* is the adjoint of T; then T**= T.
T is normal TT*=T*T
Hence T*T**=T*T=TT*=T**T* so that T*T**=T**T

T* is normal if T H .

Theorem 1: The limit T of any convergent sequence (T ) of normal operators is normal.
k
Proof: Now T * T * T T * T T
k k k
T * T * as k since T k T as k .
k
Now we prove TT* T * T so that T is normal.

*
TT * T * T TT * T T * T T * T * T T *T TT * TT * T T * T T * T T
k k k k k k k k k k k k k
T * T TT *
k k
TT * T * T TT * T T * k T * T k TT * ...(1)
k
k
 T is normal i.e. T T* T * T
k k k k k
since T k T as T * k T*, R.H.S. of (1) 0

TT * T * T 0

TT* T * T
T is normal.
This completes the proof of the theorem.

Theorem 2: The set of all normal operators on a Hilbert space H is a closed subspace of H
which contains the set of all set-adjoint operators and is closed under scalar multiplication.
Proof: Let M be the set of all normal operators on a Hilbert space H. First we shall show that M
is closed subset of H .

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