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Truncated Hamburger moment problem for an operator measure with compact support
Author(s) -
Arlinskiĭ Yury
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200028
Subject(s) - mathematics , holomorphic function , operator (biology) , bounded function , uniqueness , separable space , moment (physics) , measure (data warehouse) , bounded operator , pure mathematics , holomorphic functional calculus , hilbert space , transformation (genetics) , combinatorics , mathematical analysis , finite rank operator , banach space , biochemistry , chemistry , physics , repressor , classical mechanics , computer science , transcription factor , gene , database
Given bounded selfadjoint operators \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$F_1=I_{\mathfrak N}$\end{document} , F 2 , …, F 2 n + 1 in a separable Hilbert space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathfrak N$\end{document} , we consider the operator truncated Hamburger moment problem of finding a Herglotz‐Nevanlinna operator‐valued function M holomorphic in the neighborhood of infinity and having the form\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ M(z)=-\sum \limits _{k=1}^{2n+1}z^{-k} F_k +o\big (z^{-2n-1}\big ), \quad z\longrightarrow \infty . $$ \end{document} Criteria of the solvability and uniqueness of the solution are established and a description of all solutions is obtained. Our approach is based on the Schur transformation, the Schur parameters, and the special block operator Jacobi matrices.