Relations Between Hankel and Toeplitz Operators

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  • Relations Between Hankel and Toeplitz Operators
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The Hankel operators whose matrices with respect to the standard basis for H have only a nite number of entries dierent from 0 can be described in very explicitly. Note that H has this property whenever e for any H2. It is also obvious that if H has this property, then ein is in H for some nonnegative integer n. The following sharpening of this statement is less obvious. Theorem . If a Hankel operator has a matrix with respect to the stan- dard basis for H that has only a finite number of entries different from 0, and if c is the norm of the operator, then there exists a finite Blaschke product B and a nonnegative integer n such that cein B is a symbol for the Hankel operator.

Proof. The theorem is trivial if the operator is 0; in all...

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