American Institute of Mathematical Sciences

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2015, 2015(special): 965-973. doi: 10.3934/proc.2015.0965

Exact lumping of feller semigroups: A $C^{\star}$-algebras approach

Lavinia Roncoroni 1,

1. 

Max-Planck-Institut for Mathematics in the Sciences, Inselstrasse 22, Leipzig, D-04103, Germany

Received  September 2014 Revised  December 2014 Published  November 2015

In this note we analyze a particular exact lumping of Feller semigroups in the context of $C^{\star}$-algebras, in order to pass from a space of functions defined on a locally compact Hausdorff space ${X}$ to a space of functions defined on a closed subspace ${\mathscr{C}}\subset X$. We want our reduction to preserve the essential properties of the Feller semigroup.
Keywords: abstract cauchy problems, Gelfand transform., Exact lumping.
Mathematics Subject Classification: Primary: 34G10, 46J10, 47D06; Secondary: 34K3.
Citation: Lavinia Roncoroni. Exact lumping of feller semigroups: A $C^{\star}$-algebras approach. Conference Publications, 2015, 2015 (special) : 965-973. doi: 10.3934/proc.2015.0965
References:
[1]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986, x+460 pp.

[2]

F. Atay and L. Roncoroni, Exact Lumpability of Linear Evolution Equations, preprint, MPI-MIS, 109, 2013.

[3]

R. M. Blumenthal and R.k. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, 29, Academic Press, New York-London, 1968, x+313 pp.

[4]

H. Brezis, Analyse fonctionnelle, (French) [Functional analysis] Thorie et applications. [Theory and applications] Collection Mathmatiques Appliques pour la Matrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983, xiv+234 pp.

[5]

N. L. Carothers, A Short Course on Banach Space Theory, London Mathematical Society Student Texts, 64, Cambridge University Press, Cambridge, 2005, xii+184 pp.

[6]

J. A. Van Casteren, Markov Processes, Feller Semigroups and Evolution Equations, Series on Concrete and Applicable Mathematics, 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011, xviii+805 pp.

[7]

P. G. Coxson, Lumpability and Observability of Linear Systems, Journal of Mathematical Analysis and Applications, 99 (1984) 435-446.

[8]

K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolutions Equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000, xxii+586 pp.

[9]

L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear algebra and its Applications 404 (2005), 85-117.

[10]

E. Kaniuth, A Course in Commutative Banach algebras, Graduate Texts in Mathematics, 246, Springer, New York, 2009. xii+353 pp.

[11]

G. Li and H. Rabitz, A general analysis of exact lumping in chemical kinetics, Chemical Engineering Science, 44 No.6 (1989), 1413-1430.

[12]

G. K. Pedersen, Analysis Now, Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989. xiv+277 pp.

[13]

W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Dsseldorf-Johannesburg, 1973. xiii+397 pp.

[14]

J. Toth, G. Li, H. Rabitz and A. S. Tomlin, The effect of lumping and expanding on kinetic differential equations, SIAM J. Appl. Math. 57 No.6 (1997), 1531-1556.

[15]

J. Wei and J. C. W. Kuo, A Lumping Analysis in Monomolecular Reaction Systems, Ind. Eng. Chem. Fundamen., 8 (1969), 124-133 (DOI: 10.1021/i160029a020)

[16]

Z. Rózsa and J. Tóth, Exact linear lumping in abstract spaces, Proceedings of the 7th Colloquium on the Qualitative Theory of Differential Equations,

[17]

J. H. Zwart, Geometric Theory for Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, 115. Springer-Verlag, Berlin, 1989. viii+156 pp.

show all references

References:
[1]

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer-Verlag, Berlin, 1986, x+460 pp.

[2]

F. Atay and L. Roncoroni, Exact Lumpability of Linear Evolution Equations, preprint, MPI-MIS, 109, 2013.

[3]

R. M. Blumenthal and R.k. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, 29, Academic Press, New York-London, 1968, x+313 pp.

[4]

H. Brezis, Analyse fonctionnelle, (French) [Functional analysis] Thorie et applications. [Theory and applications] Collection Mathmatiques Appliques pour la Matrise. [Collection of Applied Mathematics for the Master's Degree] Masson, Paris, 1983, xiv+234 pp.

[5]

N. L. Carothers, A Short Course on Banach Space Theory, London Mathematical Society Student Texts, 64, Cambridge University Press, Cambridge, 2005, xii+184 pp.

[6]

J. A. Van Casteren, Markov Processes, Feller Semigroups and Evolution Equations, Series on Concrete and Applicable Mathematics, 12, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011, xviii+805 pp.

[7]

P. G. Coxson, Lumpability and Observability of Linear Systems, Journal of Mathematical Analysis and Applications, 99 (1984) 435-446.

[8]

K. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolutions Equations, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000, xxii+586 pp.

[9]

L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear algebra and its Applications 404 (2005), 85-117.

[10]

E. Kaniuth, A Course in Commutative Banach algebras, Graduate Texts in Mathematics, 246, Springer, New York, 2009. xii+353 pp.

[11]

G. Li and H. Rabitz, A general analysis of exact lumping in chemical kinetics, Chemical Engineering Science, 44 No.6 (1989), 1413-1430.

[12]

G. K. Pedersen, Analysis Now, Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989. xiv+277 pp.

[13]

W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Dsseldorf-Johannesburg, 1973. xiii+397 pp.

[14]

J. Toth, G. Li, H. Rabitz and A. S. Tomlin, The effect of lumping and expanding on kinetic differential equations, SIAM J. Appl. Math. 57 No.6 (1997), 1531-1556.

[15]

J. Wei and J. C. W. Kuo, A Lumping Analysis in Monomolecular Reaction Systems, Ind. Eng. Chem. Fundamen., 8 (1969), 124-133 (DOI: 10.1021/i160029a020)

[16]

Z. Rózsa and J. Tóth, Exact linear lumping in abstract spaces, Proceedings of the 7th Colloquium on the Qualitative Theory of Differential Equations,

[17]

J. H. Zwart, Geometric Theory for Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences, 115. Springer-Verlag, Berlin, 1989. viii+156 pp.

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