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INVERSION OF OPERATOR PENCILS ON HILBERT SPACE

Part of: General theory of linear operators Linear spaces and algebras of operators Special classes of linear operators

Published online by Cambridge University Press:  21 December 2018

AMIE ALBRECHT Open the ORCID record for AMIE ALBRECHT [Opens in a new window] ,
PHIL HOWLETT Open the ORCID record for PHIL HOWLETT [Opens in a new window]  and
GEETIKA VERMA
AMIE ALBRECHT
Affiliation:
Scheduling and Control Group, Centre for Industrial and Applied Mathematics, School of Information Technology and Mathematical Sciences, University of South Australia, Australia email amie.albrecht@unisa.edu.au
PHIL HOWLETT*
Affiliation:
Scheduling and Control Group, Centre for Industrial and Applied Mathematics, School of Information Technology and Mathematical Sciences, University of South Australia, Australia email phil.howlett@unisa.edu.au
GEETIKA VERMA
Affiliation:
Scheduling and Control Group, Centre for Industrial and Applied Mathematics, School of Information Technology and Mathematical Sciences, University of South Australia, Australia email geetika.verma@unisa.edu.au
*
✉phil.howlett@unisa.edu.au
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Abstract

We consider a linear operator pencil with complex parameter mapping one Hilbert space onto another. It is known that the resolvent is analytic in an open annular region of the complex plane centred at the origin if and only if the coefficients of the Laurent series satisfy a doubly-infinite set of left and right fundamental equations and are suitably bounded. If the resolvent has an isolated singularity at the origin we propose a recursive orthogonal decomposition of the domain and range spaces that enables us to construct the key nonorthogonal projections that separate the singular and regular components of the resolvent and subsequently allows us to find a formula for the basic solution to the fundamental equations. We show that each Laurent series coefficient in the singular part of the resolvent can be approximated by a weakly convergent sequence of finite-dimensional matrix operators and we show how our analysis can be extended to find a global expression for the resolvent of a linear pencil in the case where the resolvent has only a finite number of isolated singularities.

Keywords

linear operator pencilsperturbation theoryresolvent operatorsLaurent seriesfundamental equations

MSC classification

Primary: 47A10: Spectrum, resolvent
Secondary: 47B40: Spectral operators, decomposable operators, well-bounded operators, etc. 47L30: Abstract operator algebras on Hilbert spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 108 , Issue 2 , April 2020 , pp. 145 - 176
Copyright
© 2018 Australian Mathematical Publishing Association Inc.

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