Generalization of Hankel Operators

Abstract

We considerd and proved with an estimation the analogues of Brown- Halmos and Neahri's theorems on the norms of Hankel operators, acting on Subspaces of Hardy type reflexive rearrangement-invariant spaces with nontrivial indices. We studied invariant subspaces of the Bergman spaces on bounded symmetric domains and quasi-invariant subspaces of the Segal- Bergmann spaces. We completely characterized small Hankel operators with finite rank on their spaces and showed a general result of the degree. We showed the conditions that the product of two Hankel operators is also a Hankel operator. We derived new necessary and sufficient conditions for admissibility observation operators for certain Co-semigroups.We also showed a new sufficient criterion for admissibility for observation operators with infinite-dimensional out-put space on contraction semigroups. We found a sharp estimate of an infinite-time admissible observation operator that leads to convergent of a normed of power infinitesimal generator.