Page 320 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 29: Normal and Unitary Operators

Notes
T * T I x,x O x H

T * T I O
T * T I

This completes the proof of the theorem.

29.1.3 Isometric Operator
Definition: An operator T on H is said to be isometric if Tx Ty x y x,y H.

Since T is linear, the condition is equivalent to Tx x for every x H.

For example: let e ,e ,...,e ,... be an orthonormal basis for a separable Hilbert space H and
1 2 n
T H be defined as T x e x e ... x e x e ... where x x .
1 1 2 2 1 2 2 3 n

Then Tx 2 x n 2 x 2
n 1
T is an isometric operator.
The operator T defined is called the right shift operator given by Te = e .
n n+1
Theorem 9: If T is any arbitrary operator on a Hilbert space H then H is unitary it is an
isometric isomorphism of H onto itself.
Proof: Let T is a unitary operator on H. Then T is invertible and therefore T is onto.

Further TT* = I.
Hence Tx x for every x H. [By Theorem (7)]

T preserves norms and so T is an isometric isomorphism of H onto itself.
Conversely, let T is an isometric isomorphism of H onto itself. Then T is one-one and onto.
–1
Therefore T exists. Also T is an isometric isomorphism.
Tx x x
T*T = I [By Theorem (7)]

T * T T 1 IT 1

T * TT 1 T 1

T * I T 1

TT* I T * T and so T is unitary.
This completes the proof of the theorem.

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