Boundedness of Higher order Hankel Forms, Factorization in Potential Spaces and Derivations

We prove a bounded decomposition for higher order Hankel formsand characterize the first order Hochschild cohomology groupsof the disk algebra with coefficients in the space of boundedHankel forms of some fixed order. Although these groups arenon-trivial, we prove that every bounded derivation is innerand necessarily implemented by a Hankel form of order one higher.In terms of operators, this result extends the similarity resultof Aleksandrov and Peller. Both of the main structural theoremshere rely on estimates involving multilinear maps on the n-foldproduct of the disk algebra and we obtain several higher orderanalogues of the factorization results due to Aleksandrov andPeller. 2000 Mathematics Subject Classification: 47B35, 46E15,46E25.