Banach Journal of Mathematical Analysis

Multiplicative operator functions and abstract Cauchy problems

Felix Früchtl

Abstract

We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including $C_{0}$-groups and cosine operator functions, and more generally, Sturm–Liouville operator functions.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 347-373.

Dates
Received: 22 February 2017
Accepted: 12 May 2017
First available in Project Euclid: 8 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1515402092

Digital Object Identifier
doi:10.1215/17358787-2017-0042

Mathematical Reviews number (MathSciNet)
MR3779718

Zentralblatt MATH identifier
06873505

Citation

Früchtl, Felix. Multiplicative operator functions and abstract Cauchy problems. Banach J. Math. Anal. 12 (2018), no. 2, 347--373. doi:10.1215/17358787-2017-0042. https://projecteuclid.org/euclid.bjma/1515402092

References

• [1] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, 2nd ed., Monogr. Math. 96, Birkhäuser, Basel, 2011.
• [2] W. R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter Stud. Math. 20, de Gruyter, Berlin, 1995.
• [3] Y. A. Chapovsky, Existence of an invariant measure on a hypergroup, preprint, arXiv:1212.6571v1 [math.GR].
• [4] K. de Leeuw and I. Glicksberg, The decomposition of certain group representations, J. Anal. Math. 15 (1965), 135–192.
• [5] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys Monogr. 15, Amer. Math. Soc., Providence, 1977.
• [6] N. Dinculeanu, Vector Integration and Stochastic Integration in Banach Spaces, Pure Appl. Math. (Hoboken), Wiley, New York, 2000.
• [7] N. Dinculeanu, “Vector integration in Banach spaces and application to stochastic integration” in Handbook of Measure Theory, Vol. I, II, North-Holland, Amsterdam, 2002, 345–399.
• [8] K. Ey and R. Lasser, Facing linear difference equations through hypergroup methods, J. Difference Equ. Appl. 13 (2007), no. 10, 953–965.
• [9] G. Fischer and R. Lasser, Homogeneous Banach spaces with respect to Jacobi polynomials, Rend. Circ. Mat. Palermo (2) Suppl., No. 76 (2005), 331–353.
• [10] F. Früchtl, Sturm-Liouville hypergroups and asymptotics, preprint, to appear in Monatsh. Math.
• [11] F. Früchtl, Sturm-Liouville operator functions, in preparation.
• [12] F. Früchtl, Sturm-Liouville operator functions: A general concept of multiplicative operator functions on hypergroups, Ph.D. dissertation, Technische Universität München, Munich, Germany, 2016, http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:91-diss-20160525-1279001-1-0 (accessed 11 December 2017).
• [13] G. Gasper, Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math. (2) 95 (1972), no. 2, 261–280.
• [14] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, 2nd ed., Grundlehren Math. Wiss. 115, Springer, Berlin, 1979.
• [15] H. Heyer, Probability Measures on Locally Compact Groups, Ergeb. Math. Grenzgeb. 94, Springer, Berlin, 1977.
• [16] H. Heyer, “Random fields and hypergroups” in Real and Stochastic Analysis: Current Trends, World Scientific, Hackensack, NJ, 2014, 85–182.
• [17] R. I. Jewett, Spaces with an abstract convolution of measures, Adv. Math. 18 (1975), no. 1, 1–101.
• [18] Y. Katznelson, An Introduction to Harmonic Analysis, 3rd ed., Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 2004.
• [19] S. Kurepa, A cosine functional equation in Banach algebras, Acta Sci. Math. (Szeged) 23 (1962), 255–267.
• [20] R. Lasser, Orthogonal polynomials and hypergroups, Rend. Mat. (7) 3 (1983), no. 2, 185–209.
• [21] R. Lasser, Orthogonal polynomials and hypergroups, II: The symmetric case, Trans. Amer. Math. Soc. 341 (1994), no. 2, 749–770.
• [22] R. Lasser, Harmonic analysis on hypergroups, in preparation.
• [23] R. Lasser and M. Leitner, Stochastic processes indexed by hypergroups. I, J. Theoret. Probab. 2 (1989), no. 3, 301–311.
• [24] M. Leitner, Stochastic processes indexed by hypergroups, II, J. Theoret. Probab. 4 (1991), no. 2, 321–332.
• [25] K. Musiał, “Pettis integral” in Handbook of Measure Theory, Vol. I, II, North-Holland, Amsterdam, 2002, 531–586.
• [26] A. Nasr-Isfahani, Representations and positive definite functions on hypergroups, Serdica Math. J. 25 (1999), no. 4, 283–296.
• [27] M. S. Osborne, Locally Convex Spaces, Grad. Texts in Math. 269, Springer, Cham, 2014.
• [28] W. Rudin, Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991.
• [29] M. Sova, Cosine operator functions, Rozprawy Mat. 49 (1966), 47 pages.
• [30] A. Weinmann and R. Lasser, Lipschitz spaces with respect to Jacobi translation, Math. Nachr. 284 (2011), no. 17–18, 2312–2326.
• [31] H. Zeuner, One-dimensional hypergroups, Adv. Math. 76 (1989), no. 1, 1–18.
• [32] H. Zeuner, Moment functions and laws of large numbers on hypergroups, Math. Z. 211 (1992), no. 3, 369–407.