Basic Properties of Toeplitz Operators

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  • Basic Properties of Toeplitz Operators
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88 2 The Unilateral Shift and Factorization of Functions

 Now we can nally prove that functions of the above form are outer.

Theorem . If f is in H 2 and F is dened by

 2 i                      
                         1     e z
        F (z)  exp                   log f (ei
                                                  ) d   ,
                        2 0 ei  z

then F is outer. Proof. It was shown above (Theorem ) that F is in H 2 . Thus F G for an inner function and G an outer function (Theorem ). It suces i ) 1 a.e., it follows that to show that is a constant function. Since (e F(ei ) ...

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