Local characterizations for the matrix monotonicity and convexity of fixed order
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- by Otte Heinävaara
- Proc. Amer. Math. Soc. 146 (2018), 3791-3799
- DOI: https://doi.org/10.1090/proc/13674
- Published electronically: May 24, 2018
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Abstract:
We establish local characterizations of matrix monotonicity and convexity of fixed order by giving integral representations connecting the Loewner and Kraus matrices, previously known to characterize these properties, to respective Hankel matrices. Our results are new already in the general case of matrix convexity, and our approach significantly simplifies the corresponding work on matrix monotonicity. We also obtain an extension of the original characterization for matrix convexity by Kraus and tighten the relationship between monotonicity and convexity.References
- Julius Bendat and Seymour Sherman, Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), 58–71. MR 82655, DOI 10.1090/S0002-9947-1955-0082655-4
- Pattrawut Chansangiam, A survey on operator monotonicity, operator convexity, and operator means, Int. J. Anal. , posted on (2015), Art. ID 649839, 8. MR 3426646, DOI 10.1155/2015/649839
- Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062, DOI 10.1007/978-1-4612-6333-3
- Carl de Boor, Divided differences, Surv. Approx. Theory 1 (2005), 46–69. MR 2221566
- Otto Dobsch, Matrixfunktionen beschränkter Schwankung, Math. Z. 43 (1938), no. 1, 353–388 (German). MR 1545729, DOI 10.1007/BF01181100
- William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556, DOI 10.1007/978-3-642-65755-9
- Uwe Franz, Fumio Hiai, and Éric Ricard, Higher order extension of Löwner’s theory: operator $k$-tone functions, Trans. Amer. Math. Soc. 366 (2014), no. 6, 3043–3074. MR 3180739, DOI 10.1090/S0002-9947-2014-05942-4
- Frank Hansen, The fast track to Löwner’s theorem, Linear Algebra Appl. 438 (2013), no. 11, 4557–4571. MR 3034551, DOI 10.1016/j.laa.2013.01.022
- Frank Hansen and Jun Tomiyama, Differential analysis of matrix convex functions, Linear Algebra Appl. 420 (2007), no. 1, 102–116. MR 2277632, DOI 10.1016/j.laa.2006.06.018
- Frank Hansen and Jun Tomiyama, Differential analysis of matrix convex functions. II, JIPAM. J. Inequal. Pure Appl. Math. 10 (2009), no. 2, Article 32, 5. MR 2511925
- Fritz Kraus, Über konvexe Matrixfunktionen, Math. Z. 41 (1936), no. 1, 18–42 (German). MR 1545602, DOI 10.1007/BF01180403
- Karl Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216 (German). MR 1545446, DOI 10.1007/BF01170633
- The Sage Developers, SageMath, the Sage Mathematics Software System (Version 7.1), 2016, http://www.sagemath.org.
Bibliographic Information
- Otte Heinävaara
- Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland
- Email: otte.heinavaara@helsinki.fi
- Received by editor(s): September 15, 2016
- Received by editor(s) in revised form: January 26, 2017
- Published electronically: May 24, 2018
- Communicated by: Stephan Ramon Garcia
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3791-3799
- MSC (2010): Primary 26A48; Secondary 26A51, 47A63
- DOI: https://doi.org/10.1090/proc/13674
- MathSciNet review: 3825834