Page 321 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes

Note If T is an operator on a Hilbert space H such that Tx x x H and T is definitely
an isometric isomorphism of H onto itself. But T need not be onto and so T need not be
unitary. The following example will make the point more clear.

Example: Let T be an operator on l defined by T x ,x ,... 0,x ,x ,...
2 1 2 1 2
Tx x x I .
2
T is an isometric isomorphism of l into itself.
2
However T is not onto. If y ,y ,... is a sequence in l such that y 0, then no sequence in l
1 2 2 1 2
whose T-image is y ,y ,... . Therefore T is not onto and so T is not unitary.
1 2

29.2 Summary

 An operator T on a Hilbert space H is said to be normal if it commutes with its adjoint i.e.
*
*
if TT = T T. Conclusively every self adjoint operator is normal.
 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.
 An operator U on a Hilbert space H is said to be unitary if UU* = U*U =I.

 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.

29.3 Keywords

Normal Operator: An operator T on a Hilbert space H is said to be normal if it commutes with
its adjoint i.e. if TT* = T*T.

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

29.4 Review Questions

1. If T is an operator on a Hilbert space H, then T is normal its real and imaginary part
commute.
2. An operator T on H is normal T * x Tx for every x.

3. The set of all normal operators on H is a closed subset of H which contains the set of all
self-adjoint operators and is closed under scalar multiplication.
4. If H is finite-dimensional, show that every isometric isomorphism of H into itself is
unitary.
5. Show that the unitary operators on H form a group.

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