Fundamental Properties of Composition Operators

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  • Fundamental Properties of Composition Operators
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154 4 Hankel Operators

                 T Hei   Hei  T  Hei   Hdei .

                 2 , it follows that Hdei  0, and thus that

Since dei is in ei H

                        T Hei   Hei  T  0.

Also, since c, we have Hei Hcei ei , and therefore

                   T Hei   Hcei T  Hei  T  0.

                 2 , Hcei  0. Therefore

Since cei is in ei H

                           T Hei   Hei  T .

It follows that if H is a multiple of Hei , then H commutes with T . To prove the converse, suppose that HT T H. Multiplying T H on the right by U , using the fact...

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