## Operator monotone functions and Löwner functions of several variables.(English)Zbl 1268.47025

Ann. Math. (2) 176, No. 3, 1783-1826 (2012); corrigendum ibid. 180, No. 1, 403-405 (2014).
For positive integers $$d$$ and $$n$$, let CSAM$$_n^d$$ be the set of $$d$$-tuples of commuting self-adjoint $$n\times n$$ matrices and let CSA$$^d$$ denote the set of $$d$$-tuples of commuting self-adjoint operators acting on an infinite-dimensional separable Hilbert space. Let $$E$$ be open in $$\mathbb{R}^d$$ and $$f$$ be a real-valued $$C^1$$-function on $$E$$. Then $$f$$ is said to be locally $$M_n$$-monotone on $$E$$ if, whenever $$S$$ is in CSAM$$_n^d$$ with $$\sigma(S)$$ consisting of $$n$$ distinct points in $$E$$ and $$S(t)$$ is a $$C^1$$-curve in CSAM$$_n^d$$ with $$S(0)=S$$ and $$\frac{d}{dt}S(t)|_{t=0}\geq 0$$, then $$\frac{d}{dt}f(S(t))|_{t=0}$$ exists and is $$\geq 0$$. If in the above definition, we replace all occurrences of CSAM$$_n^d$$ by CSA$$^d$$, then the function $$f$$ is called locally operator monotone.
In [Math. Z. 38, 177–216 (1934; Zbl 0008.11301, JFM 49.0714.01)], K. Löwner completely characterized functions of one variable ($$d=1$$) that are matrix or operator monotone. In the paper under review, the authors generalize Löwner’s results to higher dimensions. Let $$E$$ and $$f$$ be as above. They show that $$f$$ is locally $$M_n$$-monotone if and only if $$f$$ belongs to the Löwner class $$\mathcal{L}^{d}_{n}(E)$$. For operator monotone functions, they establish the equivalence of the following statements: (a) $$f$$ is $$M_n$$-monotone for all $$n\geq 1$$; (b) $$f$$ is operator monotone; (c) $$f$$ belongs to the Löwner class $$\mathcal{L}(E)$$.
There are also notions of global matrix and operator monotone functions. However, the results for these functions are less complete. The authors successfully characterize rational functions of two variables that are operator monotone on rectangles in $$\mathbb{R}^2$$, but the problem remains open for rational functions of more than two variables, or for non-rational functions.
The paper ends with a list of open questions.
Reviewer: Trieu Le (Toledo)

### MSC:

 47A63 Linear operator inequalities 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A60 Functional calculus for linear operators 47A13 Several-variable operator theory (spectral, Fredholm, etc.)

### Citations:

Zbl 0008.11301; JFM 49.0714.01
Full Text:

### References:

  M. Abate, ”The Julia-Wolff-Carathéodory theorem in polydisks,” J. Anal. Math., vol. 74, pp. 275-306, 1998. · Zbl 0912.32005  J. Agler, ”On the representation of certain holomorphic functions defined on a polydisc,” in Topics in Operator Theory: Ernst D. Hellinger Memorial Volume, Basel: Birkhäuser, 1990, vol. 48, pp. 47-66. · Zbl 0733.32002  J. Agler and J. E. McCarthy, Pick Interpolation and Hilbert Function Spaces, Providence, RI: Amer. Math. Soc., 2002. · Zbl 1010.47001  J. Agler, J. E. McCarthy, and N. Young, ”A Carathéodory theorem for the bidisk via Hilbert space methods,” Math. Ann., vol. 352, iss. 3, pp. 581-624, 2012. · Zbl 1250.32005  General Topology. I, Arkhangel’skiuii, A. and Pontryagin, L., Eds., New York: Springer-Verlag, 1990, vol. 17. · Zbl 0778.00007  J. A. Ball and V. Bolotnikov, ”A tangential interpolation problem on the distinguished boundary of the polydisk for the Schur-Agler class,” J. Math. Anal. Appl., vol. 273, iss. 2, pp. 328-348, 2002. · Zbl 1015.47004  J. A. Ball and V. Bolotnikov, ”Canonical de Branges-Rovnyak model transfer-function realization for multivariable Schur-class functions,” in Hilbert Spaces of Analytic Functions, Providence, RI: Amer. Math. Soc., 2010, vol. 51, pp. 1-39. · Zbl 1210.47041  J. A. Ball, C. Sadosky, and V. Vinnikov, ”Scattering systems with several evolutions and multidimensional input/state/output systems,” Integral Equations Operator Theory, vol. 52, iss. 3, pp. 323-393, 2005. · Zbl 1092.47006  J. A. Ball and T. T. Trent, ”Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables,” J. Funct. Anal., vol. 157, iss. 1, pp. 1-61, 1998. · Zbl 0914.47020  K. Bickel, ”Differentiating matrix functions,” Operators and Matrices, vol. 7, pp. 71-90, 2013. · Zbl 1268.26019  J. D. Chandler Jr., ”Extensions of monotone operator functions,” Proc. Amer. Math. Soc., vol. 54, pp. 221-224, 1976. · Zbl 0308.47028  B. J. Cole and J. Wermer, ”Ando’s theorem and sums of squares,” Indiana Univ. Math. J., vol. 48, iss. 3, pp. 767-791, 1999. · Zbl 0945.47010  M. J. Crabb and A. M. Davie, ”Von Neumann’s inequality for Hilbert space operators,” Bull. London Math. Soc., vol. 7, pp. 49-50, 1975. · Zbl 0301.47007  W. F. Donoghue Jr., Monotone Matrix Functions and Analytic Continuation, New York: Springer-Verlag, 1974, vol. 207. · Zbl 0278.30004  G. E. Dullerud and F. Paganini, A Course in Robust Control Theory, New York: Springer-Verlag, 2000, vol. 36. · Zbl 0939.93001  F. Hansen, ”Operator monotone functions of several variables,” Math. Inequal. Appl., vol. 6, iss. 1, pp. 1-17, 2003. · Zbl 1035.47005  G. Herglotz, ”Über Potenzreihen mit positivem, reelem Teil in Einheitskreis,” Leipz. Bericht, vol. 63, pp. 501-511, 1911. · JFM 42.0438.02  R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge: Cambridge Univ. Press, 1991. · Zbl 0729.15001  F. Jafari, ”Angular derivatives in polydiscs,” Indian J. Math., vol. 35, iss. 3, pp. 197-212, 1993. · Zbl 0820.32001  D. S. Kaliuzhnyi-Verbovetskyi, ”On the Bessmertnyi class of homogeneous positive holomorphic functions of several variables,” in Current Trends in Operator Theory and its Applications, Basel: Birkhäuser, 2004, vol. 149, pp. 255-289. · Zbl 1077.47012  G. Knese, ”Polynomials with no zeros on the bidisk,” Anal. PDE, vol. 3, iss. 2, pp. 109-149, 2010. · Zbl 1226.42019  G. Knese, ”Rational inner functions in the Schur-Agler class of the polydisk,” Publ. Mat., vol. 55, iss. 2, pp. 343-357, 2011. · Zbl 1219.47028  A. Korányi, ”On some classes of analytic functions of several variables,” Trans. Amer. Math. Soc., vol. 101, pp. 520-554, 1961. · Zbl 0111.11501  P. D. Lax, Functional Analysis, New York: Wiley-Interscience [John Wiley & Sons], 2002. · Zbl 1009.47001  K. Löwner, ”Über monotone Matrixfunktionen,” Math. Z., vol. 38, iss. 1, pp. 177-216, 1934. · Zbl 0008.11301  R. Nevanlinna, ”Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentproblem,” Ann. Acad. Sci. Fenn. Ser. A, vol. 18, 1922. · JFM 48.1226.02  R. Nevanlinna, ”Remarques sur le lemme de Schwarz,” C.R. Acad. Sci. Paris, vol. 188, pp. 1027-1029, 1929. · JFM 55.0768.01  V. V. Peller, ”Hankel operators in the theory of perturbations of unitary and selfadjoint operators,” Funktsional. Anal. i Prilozhen., vol. 19, iss. 2, pp. 37-51, 96, 1985. · Zbl 0587.47016  V. V. Peller, ”The behavior of functions of operators under perturbations,” in A Glimpse at Hilbert Space Operators, Basel: Birkhäuser, 2010, vol. 207, pp. 287-324. · Zbl 1258.47023  M. Riesz, ”Sur certaines inégalités dans la théorie des fonctions avec quelques remarques sur les geometries non-euclidiennes,” Fysiogr. Sällsk. Lund Forh., vol. 1, pp. 18-38, 1931. · Zbl 0003.25907  M. Rosenblum and J. Rovnyak, ”Restrictions of analytic functions. I,” Proc. Amer. Math. Soc., vol. 48, pp. 113-119, 1975. · Zbl 0304.30025  W. Rudin, Function Theory in Polydiscs, New York: W. A. Benjamin, 1969. · Zbl 0177.34101  W. Rudin, Lectures on the Edge-of-the-Wedge Theorem, Providence, RI: Amer. Math. Soc., 1971. · Zbl 0214.09001  W. Rudin, Real and Complex Analysis, Third ed., New York: McGraw-Hill Book Co., 1987. · Zbl 0925.00005  D. Sarason, ”Nevanlinna-Pick interpolation with boundary data,” Integral Equations Operator Theory, vol. 30, iss. 2, pp. 231-250, 1998. · Zbl 0906.30031  M. Singh and H. L. Vasudeva, ”Monotone matrix functions of two variables,” Linear Algebra Appl., vol. 328, iss. 1-3, pp. 131-152, 2001. · Zbl 0990.15010  M. H. Stone, Linear Transformations in Hilbert Space and their Applications to Analysis, New York: Amer. Math. Soc., 1932, vol. 15. · Zbl 0005.40003  T. N. Varopoulos, ”On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory,” J. Funct. Anal., vol. 16, pp. 83-100, 1974. · Zbl 0288.47006
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