On some notions of convergence for n‐tuples of operators. (12th September 2013)

• Record Type: Journal Article
• Authors: Alpay, Daniel; Colombo, F.; Sabadini, I.
• Is Part Of: Mathematical methods in the applied sciences. Volume 37:Number 16(2014:Nov. 15)
• Page Start: 2363
• Page End: 2371
• Publication Date: 2013-09-12
• Subjects: Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.2982
• Languages: English
• ISSNs: 0170-4214

Abstract

<abstract abstract-type="main" id="mma2982-abs-0001"> <title> <x xml:space="preserve">Abstract</x> </title> <p id="mma2982-para-0001">The aim of this paper is to show that we can extend the notion of <italic>convergence in the norm‐resolvent sense</italic> to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of <italic>S</italic>‐resolvent operator. With this notion, we can define bounded functions of unbounded operators using the <italic>S</italic>‐functional calculus for <italic>n</italic>‐tuples of noncommuting operators. The same notion can be extended to the case of the <italic>F</italic>‐resolvent operator, which is the basis of the <italic>F</italic>‐functional calculus, a monogenic functional calculus for <italic>n</italic>‐tuples of commuting operators. We also prove some properties of the <italic>F</italic>‐functional calculus, which are of independent interest. Copyright © 2013 John Wiley &amp; Sons, Ltd.</p> </abstract>