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On some notions of convergence for n ‐tuples of operators
Author(s) -
Alpay Daniel,
Colombo F.,
Sabadini I.
Publication year - 2013
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2982
Subject(s) - mathematics , resolvent , functional calculus , tuple , operator (biology) , operator norm , operator theory , norm (philosophy) , bounded function , algebra over a field , convergence (economics) , calculus (dental) , pure mathematics , discrete mathematics , mathematical analysis , medicine , biochemistry , chemistry , dentistry , repressor , economic growth , political science , transcription factor , law , economics , gene
The aim of this paper is to show that we can extend the notion of convergence in the norm‐resolvent sense to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of S ‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the S ‐functional calculus for n ‐tuples of noncommuting operators. The same notion can be extended to the case of the F ‐resolvent operator, which is the basis of the F ‐functional calculus, a monogenic functional calculus for n ‐tuples of commuting operators. We also prove some properties of the F ‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.