Generalized Hankel operators on the Fock space

Abstract

Key words Hankel operator, Fock space MSC (2000) 47B35, 32A36In this paper we study generalized Hankel operators of the form H l z k :The investigations in this article extend the ones in [11] and [6], where the special cases l = 0 and l = 1 are considered, respectively. The main result is that the operators H l z k are not bounded for l < k − 1. The proof relies on a combinatoric argument and a generalization to general conjugate holomorphic L 2 symbols, generalizing arguments from [6], seems possible and is planned for future work.

Generalized Hankel operators on the Fock space II

“… Abstract In this paper we study boundedness of generalized Hankel operators of the form \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}${\rm H}_{{\overline{z}}^k}^l: {\mathscr F}^2\big (|z|^2\big )\rightarrow L^2\big (|z|^2\big )$\end{document} and thereby extend our results from 10. Here, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\rm H_{{\overline{z}}^k}^l(f):=(\rm Id-\rm P_l)\big (\overline{z}^k f\big )$\end{document} and P l is the projection onto \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A^2_l\big (\mathbb C,| z|^2\big ):=\rm cl\big (\rm span\lbrace \overline{z} ^mz^n\,|\,m,n\in \mathbb N,\,m\le l\rbrace \big )$\end{document}.…”

“…However, it should be mentioned that the remaining case is much more difficult to prove, as already mentioned in 10. One of the two main purposes of the present paper is to close this gap ( k = l + 1).…”

“…In this section, we remind the reader to some ideas from 10. Remember that the set \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace z^n/c_n\,|\,n\in \mathbb N\rbrace$\end{document} is a complete orthonormal system for the Fock space \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}${\mathscr F}^2(| z|^2)$\end{document}, where the scalars c n = ‖ z n ‖ 2 are called moments.…”

“…It is obvious by definition that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\rm H_{{\overline{z}}^k}^l=0$\end{document} for l ≥ k . In all other cases boundedness of the operator \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\rm H_{{\overline{z}}^k}^l$\end{document} is not obvious and does not hold for l < k − 1 (See below and for details 10).…”

“…The above theorem is generalized in 5 for generalized Fock spaces. The operators \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^{l}_{\overline{z}^k}$\end{document} have been studied in 10 for l ≤ k − 2. There, the following theorem can be found:…”