Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T06:20:50.804Z Has data issue: false hasContentIssue false
  • English
  • Français

INVERSION OF OPERATOR PENCILS ON HILBERT SPACE

Part of: General theory of linear operators Special classes of linear operators Linear spaces and algebras of 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 [email protected]
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 [email protected]
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 [email protected]
*
✉[email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albrecht, A., Howlett, P. and Pearce, C., ‘The fundamental equations for inversion of operator pencils on Banach space’, J. Math. Anal. Appl. 413 (2014), 411–421.Google Scholar
Albrecht, A. R., Howlett, P. G. and Pearce, C. E. M., ‘Necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space using Laurent series’, J. Math. Anal. Appl. 383(1) (2011), 95–110.Google Scholar
Avrachenkov, K. E., Filar, J. A. and Howlett, P. G., Analytic Perturbation Theory and its Applications, Other Titles in Applied Mathematics, 135 (SIAM, Philadelphia, PA, 2013).Google Scholar
Avrachenkov, K. E., Haviv, M. and Howlett, P. G., ‘Inversion of analytic matrix functions that are singular at the origin’, SIAM J. Matrix Anal. Appl. 22(4) (2001), 1175–1189.Google Scholar
Avrachenkov, K. E. and Lasserre, J.-B., ‘The fundamental matrix of singularly perturbed Markov chains’, Adv. Appl. Probab. 31(3) (1999), 679–697.Google Scholar
Dunford, N. and Schwartz, J., Linear Operators, Part I: General Theory, Wiley Classics (John Wiley, New York, 1988).Google Scholar
Franchi, M. and Paruolo, P., ‘Inverting a matrix function around a singularity via local rank factorization’, SIAM J. Matrix Anal. Appl. 37(2) (2016), 774–797.Google Scholar
Gohberg, I., Goldberg, S. and Kaashoek, M. A., Classes of Linear Operators, Vol. 1, Operator Theory: Advances and Applications, 49 (Birkhauser, Basel, 1990).Google Scholar
Howlett, P. G., ‘Input retrieval in finite dimensional linear systems’, ANZIAM J. (formerly J. Aust. Math. Soc. Ser. B) 23 (1982), 357–382.Google Scholar
Howlett, P., Albrecht, A. and Pearce, C., ‘Laurent series for inversion of linearly perturbed bounded linear operators on Banach space’, J. Math. Anal. Appl. 366(1) (2010), 112–123.Google Scholar
Howlett, P., Avrachenkov, K., Pearce, C. and Ejov, V., ‘Inversion of analytically perturbed linear operators that are singular at the origin’, J. Math. Anal. Appl. 353(1) (2009), 68–84.Google Scholar
Kato, T., Perturbation Theory for Linear Operators, Classics in Mathematics (Springer, Berlin, 1995).Google Scholar
Langenhop, C. E., ‘The Laurent expansion for a nearly singular matrix’, Linear Algebra Appl. 4 (1971), 329–340.Google Scholar
Langenhop, C. E., ‘On the invertibility of a nearly singular matrix’, Linear Algebra Appl. 7 (1973), 361–365.Google Scholar
Luenberger, D. G., Optimization by Vector Space Methods (Wiley, New York, 1969).Google Scholar
Sain, M. K. and Massey, J. L., ‘Invertibility of linear time invariant dynamical systems’, IEEE Trans. Automat. Control AC‐14 (1969), 141–149.Google Scholar
Schweitzer, P. and Stewart, G. W., ‘The Laurent expansion of pencils that are singular at the origin’, Linear Algebra Appl. 183 (1993), 237–254.Google Scholar
Stummel, F., ‘Diskrete Konvergenz linearer Operatoren. I’, Math. Ann. 190 (1970), 45–92.Google Scholar
Van Dooren, P., ‘The computation of Kronecker’s canonical form of a singular pencil’, Linear Algebra Appl. 27 (1979), 103–140.Google Scholar
Wilkening, J., ‘An algorithm for computing Jordan chains and inverting analytic matrix functions’, Linear Algebra Appl. 427(1) (2007), 6–25.Google Scholar
Yosida, K., Functional Analysis, 5th edn, Classics in Mathematics (Springer, Berlin–Heidelberg–New York, 1978).Google Scholar