Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities
HTML articles powered by AMS MathViewer
- by Apoorva Khare PDF
- Trans. Amer. Math. Soc. 375 (2022), 2217-2236 Request permission
Abstract:
A special case of a fundamental result of Loewner and Horn [Trans. Amer. Math. Soc. 136 (1969), pp. 269–286] says that given an integer $n \geqslant 1$, if the entrywise application of a smooth function $f : (0,\infty ) \to \mathbb {R}$ preserves the set of $n \times n$ positive semidefinite matrices with positive entries, then $f$ and its first $n-1$ derivatives are non-negative on $(0,\infty )$. In a recent joint work with Belton–Guillot–Putinar [J. Eur. Math. Soc., in press], we proved a stronger version, and used it to strengthen the Schoenberg–Rudin characterization of dimension-free positivity preservers [Duke Math. J. 26 (1959), pp. 617–622; Duke Math. J. 9 (1942), pp. 96–108].
In recent works with Belton–Guillot–Putinar [Adv. Math. 298 (2016), pp. 325–368] and with Tao [Amer. J. Math. 143 (2021), pp. 1863-1929] we used local, real-analytic versions at the origin of the Horn–Loewner condition, and discovered unexpected connections between entrywise polynomials preserving positivity and Schur polynomials. In this paper, we unify these two stories via a Master Theorem (Theorem A) which (i) simultaneously unifies and extends all of the aforementioned variants; and (ii) proves the positivity of the first $n$ nonzero Taylor coefficients at individual points rather than on all of $(0,\infty )$.
A key step in the proof is a new determinantal / symmetric function calculation (Theorem B), which shows that Schur polynomials arise naturally from considering arbitrary entrywise maps that are sufficiently differentiable. Of independent interest may be the following application to symmetric function theory: we extend the Schur function expansion of Cauchy’s (1841) determinant (whose matrix entries are geometric series $1 / (1 - u_j v_k)$), as well as of a determinant of Frobenius [J. Reine Angew. Math. 93 (1882), pp. 53–68] (whose matrix entries are a sum of two geometric series), to arbitrary power series, and over all commutative rings.
References
- Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Putinar, Matrix positivity preservers in fixed dimension. I, Adv. Math. 298 (2016), 325–368. MR 3505743, DOI 10.1016/j.aim.2016.04.016
- Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Putinar, Schur polynomials and matrix positivity preservers, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), Discrete Math. Theor. Comput. Sci. Proc., BC, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, [2016] ©2016, pp. 155–166 (English, with English and French summaries). MR 4111341
- Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Putinar, A panorama of positivity. I: Dimension free, In: Analysis of Operators on Function Spaces (The Serguei Shimorin Memorial Volume; A. Aleman, H. Hedenmalm, D. Khavinson, M. Putinar, Eds.), pp. 117–165, Trends in Math., Birkhauser, 2019.
- Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Putinar, A panorama of positivity. II: fixed dimension, Complex analysis and spectral theory, Contemp. Math., vol. 743, Amer. Math. Soc., [Providence], RI, [2020] ©2020, pp. 109–150. MR 4061939, DOI 10.1090/conm/743/14958
- Alexander Belton, Dominique Guillot, Apoorva Khare, and Mihai Putinar, Moment-sequence transforms, DOI 10.4171/jems/1145 J. Eur. Math. Soc., Published online.
- R. P. Boas Jr. and D. V. Widder, Functions with positive differences, Duke Math. J. 7 (1940), 496–503. MR 3436
- Augustin-Louis Cauchy, Mémoire sur les fonctions alternées et sur les sommes alternées, Exercices Anal. et Phys. Math., 2:151–159, 1841.
- Carl H. FitzGerald and Roger A. Horn, On fractional Hadamard powers of positive definite matrices, J. Math. Anal. Appl. 61 (1977), no. 3, 633–642. MR 506356, DOI 10.1016/0022-247X(77)90167-6
- G. Frobenius, Ueber die elliptischen Functionen zweiter Art, J. Reine Angew. Math. 93 (1882), 53–68 (German). MR 1579913, DOI 10.1515/crll.1882.93.53
- Dominique Guillot, Apoorva Khare, and Bala Rajaratnam, Preserving positivity for rank-constrained matrices, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6105–6145. MR 3660215, DOI 10.1090/tran/6826
- Roger A. Horn, The theory of infinitely divisible matrices and kernels, Trans. Amer. Math. Soc. 136 (1969), 269–286. MR 264736, DOI 10.1090/S0002-9947-1969-0264736-5
- Masao Ishikawa, Masahiko Ito, and Soichi Okada, A compound determinant identity for rectangular matrices and determinants of Schur functions, Adv. in Appl. Math. 51 (2013), no. 5, 635–654. MR 3118549, DOI 10.1016/j.aam.2013.08.001
- Masao Ishikawa, Soichi Okada, Hiroyuki Tagawa, and Jiang Zeng, Generalizations of Cauchy’s determinant and Schur’s Pfaffian, Adv. in Appl. Math. 36 (2006), no. 3, 251–287. MR 2219947, DOI 10.1016/j.aam.2005.07.001
- Apoorva Khare and Terence Tao, Schur polynomials, entrywise positivity preservers, and weak majorization, Sém. Lothar. Combin. 80B (2018), Art. 14, 12. MR 3940589
- Apoorva Khare and Terence Tao, On the sign patterns of entrywise positivity preservers in fixed dimension, Amer. J. Math. 143 (2021), 1863-1929, DOI 10.1353/ajm.2021.0049
- C. Krattenthaler, Advanced determinant calculus, Sém. Lothar. Combin. 42 (1999), Art. B42q, 67. The Andrews Festschrift (Maratea, 1998). MR 1701596
- C. Krattenthaler, Advanced determinant calculus: a complement, Linear Algebra Appl. 411 (2005), 68–166. MR 2178686, DOI 10.1016/j.laa.2005.06.042
- Greg Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. (2) 156 (2002), no. 3, 835–866. MR 1954236, DOI 10.2307/3597283
- D. Laksov, A. Lascoux, and A. Thorup, On Giambelli’s theorem on complete correlations, Acta Math. 162 (1989), no. 3-4, 143–199. MR 989395, DOI 10.1007/BF02392836
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Georg Pólya and Gabor Szegő, Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie, Heidelberger Taschenbücher, Band 74, Springer-Verlag, Berlin-New York, 1971 (German). Vierte Auflage. MR 0344041, DOI 10.1007/978-3-642-61987-8
- Hjalmar Rosengren and Michael Schlosser, Elliptic determinant evaluations and the Macdonald identities for affine root systems, Compos. Math. 142 (2006), no. 4, 937–961. MR 2249536, DOI 10.1112/S0010437X0600203X
- Walter Rudin, Positive definite sequences and absolutely monotonic functions, Duke Math. J. 26 (1959), 617–622. MR 109204
- I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108. MR 5922, DOI 10.1215/S0012-7094-42-00908-6
- J. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1–28 (German). MR 1580823, DOI 10.1515/crll.1911.140.1
- Harkrishan Vasudeva, Positive definite matrices and absolutely monotonic functions, Indian J. Pure Appl. Math. 10 (1979), no. 7, 854–858. MR 537245
Additional Information
- Apoorva Khare
- Affiliation: Indian Institute of Science, Bangalore – 560012, India; and Analysis and Probability Research Group, Bangalore – 560012, India
- MR Author ID: 750359
- ORCID: 0000-0002-1577-9171
- Email: khare@iisc.ac.in
- Received by editor(s): March 25, 2021
- Received by editor(s) in revised form: September 7, 2021, September 17, 2021, and September 21, 2021
- Published electronically: December 22, 2021
- Additional Notes: This work was partially supported by Ramanujan Fellowship grant SB/S2/RJN-121/2017, MATRICS grant MTR/2017/000295, and SwarnaJayanti Fellowship grants SB/SJF/2019-20/14 and DST/SJF/MS/2019/3 from SERB and DST (Govt. of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), and by a Young Investigator Award from the Infosys Foundation.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 2217-2236
- MSC (2020): Primary 15B48; Secondary 05E05, 15A24, 15A45, 26C05, 26D10
- DOI: https://doi.org/10.1090/tran/8563
- MathSciNet review: 4378092
Dedicated: To Roger A. Horn and the memory of Charles Loewner, with admiration