Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-25T20:48:46.096Z Has data issue: false hasContentIssue false
  • English
  • Français

Holomorphic functional calculi and sums of commuting opertors

Published online by Cambridge University Press:  17 April 2009

David Albrecht ,
Edwin Franks  and
Alan McIntosh
David Albrecht
Affiliation:
Computer Science Department, Monash University, Claytin Vic 3168, Australia, e-mail: dwa@cs.monash.edu.au
Edwin Franks
Affiliation:
Computer Science Department, Monash University, Claytin Vic 3168, Australia, e-mail: dwa@cs.monash.edu.au
Alan McIntosh
Affiliation:
Department of Mathematics, Macquarie University, New South Wales 2109, Australia, e-mail: edwin@mpce.mq.edu.au, alan@mpce.mq.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let S and T be commuting operators of type ω and type ϖ in a Banach space X. Then the pair has a joint holomorphic functional calculus in the sense that it is possible to define operators f(S, T) in a consistent manner, when f is a suitable holomorphic function defined on a product of sectors. In particular, this gives a way to define the sum S + T when ω + ϖ < π. We show that this operator is always of type μ where μ = max{ω, ϖ}. We explore when bounds on the individual functional calculi of S and T imply bounds on the functional calculus of the pair (S, T), and some implications for the regularity problem of when ∥(S + T)u∥ is equivalent to ∥Su∥ + ∥Tu∥.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 2 , October 1998 , pp. 291 - 305
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Albrecht, D., Functional calculi of commuting unbounded operators, Ph.D. thesis (Monash University, Melbourne, Australia, 1994).Google Scholar
[2]Auscher, P., McIntosh, A. and Nahmod, A., ‘Holomorphic functional calculi of operators, quadratic estimates and interpolation’, Indiana Univ. Math. J. 46 (1997), 375–403.CrossRefGoogle Scholar
[3]Baillon, J.B. and Clement, Ph., ‘Examples of unbounded imaginary powers of operators’, J. Fund. Anal. 100 (1991), 419–434.Google Scholar
[4]Cowling, M.G., Doust, I., McIntosh, A. and Yagi, A., ‘Banach space operators with a bounded H ∞ functional calculus’, J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.Google Scholar
[5]deLaubenfels, R., ‘Unbounded holomorphic functional calculus and abstract Cauchy problems for operators with polynomial bounded resolvents’, J. Fund. Anal. 114 (1993), 348–394.Google Scholar
[6]deLaubenfels, R., Existence families, functional calculi, and evolution, Lecture notes in mathematics 1570 (Springer-Verlag, Berlin, Heidelberg, New York, 1994).CrossRefGoogle Scholar
[7]Dore, G. and Venni, A., ‘On the closedness of the sum of two closed operators’, Math. Z. 196 (1987), 189–201.CrossRefGoogle Scholar
[8]Franks, E., ‘Modified Cauchy kernels and functional calculus for operators on Banach space’, J. Austral. Math. Soc. Ser. A 63 (1997), 91–99.Google Scholar
[9]Franks, E., ‘Operators of type ω and strip type’, preprint (1997).Google Scholar
[10]Franks, E. and McIntosh, A., ‘Discrete quadratic estimates and holomorphic functional calculi of operators in Banach spaces’, Macqarie Mathematics Report 95/192 (1995).Google Scholar
[11]Giga, M., Giga, Y. and Sohr, H., ‘L p estimates for the Stokes system’, in Functional Analysis and Related Topics (Berlin), (Komatsu, H., Editor), Lecture notes in mathematics 1540 (Springer-Verlag, Berlin, Heidelberg, New York, 1991), pp. 55–67.Google Scholar
[12]Giga, Y. and Sohr, H., ‘Abstract L p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains’, J. Fund. Anal. 102 (1991), 72–94.Google Scholar
[13]Kato, T., Perturbation Theory for Linear Operators, (2nd edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1976).Google Scholar
[14]Lancien, F., Lancien, G. and Le Merdy, C., ‘A joint functional calculus for sectorial operators with commuting resolvents’, Proc. London Math. Soc. (to appear).Google Scholar
[15]Lancien, G. and Le Merdy, C., ‘A generalized H ∞ functional calculus for operators on subspaces of L p and application to maximal regularity’, (preprint).Google Scholar
[16]Li, C. and McIntosh, A., ‘Clifford algebras and H ∞ functional calculi of commuting operators’, in Clifford algebras in analysis and related topics, (Ryan, John, Editor) (CRC Press, Boca Raton, FL., 1995), pp. 89–101.Google Scholar
[17]McIntosh, A., ‘Operators which have an H ∞ functional calculus’, in Miniconference on Operator Theory and Partial Differential Equations, Proc. Centre Math. Analysis 14 (A.N.U., Canberra, Australia, 1986), pp. 210–231.Google Scholar
[18]McIntosh, A. and Qian, T., ‘Convolution singular integrals operators on Lipschitz curves’, in Proceedings of the Special Year on Harmonic Analysis at Nankai Institute of Mathematics, Tianjin, China, Lecture Notes in Mathematics 1494 (Springer-Verlag, Berlin, Heidelberg, New York, 1991), pp. 142–162.Google Scholar
[19]McIntosh, A. and Yagi, A., ‘Operators of type-ω without a bounded H ∞ functional calculus’, in Miniconference on Operators in Analysis, Proc. Centre. Math. Analysis 24 (A.N.U., Canberra, Australia, 1989), pp. 159–172.Google Scholar
[20]Monniaux, S., ‘A new approach to the Dore-Venni theorem’, Math. Nachr. (to appear).Google Scholar
[21]Prüss, J. and Sohr, H., ‘On operators with bounded imaginary powers in Banach spaces’, Math. Z. 203 (1990), 429–452.Google Scholar
[22]Seeley, R., ‘Norms and domains of the complex powers ’ Amer. J. Math. 93 (1971), 299–309.CrossRefGoogle Scholar
[23]Stein, E.M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, N.J., 1970).Google Scholar