Book Review
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Book Information:
Author:
Barry Simon
Title:
Loewner’s theorem on monotone matrix functions
Additional book information:
Grundlehren der mathematischen Wissenschaften, Vol. 354,
Springer Nature Switzerland,
2019,
xi+459 pp.,
ISBN 978-3-030-22421-9, 978-3-030-22422-6 (ebook)
Jim Agler and John E. McCarthy, Aspects of non-commutative function theory, Concr. Oper. 3 (2016), no. 1, 15–24. MR 3491879, DOI 10.1515/conop-2016-0003
Jim Agler, John E. McCarthy, and N. J. Young, Operator monotone functions and Löwner functions of several variables, Ann. of Math. (2) 176 (2012), no. 3, 1783–1826. MR 2979860, DOI 10.4007/annals.2012.176.3.7
T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90. MR 155193
John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
M. J. Crabb and A. M. Davie, von Neumann’s inequality for Hilbert space operators, Bull. London Math. Soc. 7 (1975), 49–50. MR 365179, DOI 10.1112/blms/7.1.49
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
Carl H. Fitzgerald, On analytic continuation to a schlicht function, Proc. Amer. Math. Soc. 18 (1967), 788–792. MR 219712, DOI 10.1090/S0002-9939-1967-0219712-9
Otte Heinävaara, Local characterizations for the matrix monotonicity and convexity of fixed order, Proc. Amer. Math. Soc. 146 (2018), no. 9, 3791–3799. MR 3825834, DOI 10.1090/proc/13674
G. Herglotz. Über Potenzreihen mit positivem, reellem Teil in Einheitskreis, Leipz. Bericht, 63 (1911), 501–511.
Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of free noncommutative function theory, Mathematical Surveys and Monographs, vol. 199, American Mathematical Society, Providence, RI, 2014. MR 3244229, DOI 10.1090/surv/199
Fritz Kraus, Über konvexe Matrixfunktionen, Math. Z. 41 (1936), no. 1, 18–42 (German). MR 1545602, DOI 10.1007/BF01180403
Peter D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1892228
Charles Loewner, On schlicht-monotonic functions of higher order, J. Math. Anal. Appl. 14 (1966), 320–325. MR 190354, DOI 10.1016/0022-247X(66)90033-3
Karl Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216 (German). MR 1545446, DOI 10.1007/BF01170633
R. Nevanlinna. Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentproblem, Ann. Acad. Sci. Fenn. Ser. A, 18 (1922).
J. E. Pascoe and R. Tully-Doyle, Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables, J. Funct. Anal. 273 (2017), no. 1, 283–328. MR 3646301, DOI 10.1016/j.jfa.2017.04.001
Georg Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1915), no. 1, 7–23 (German). MR 1511844, DOI 10.1007/BF01456817
Barry Simon, Operator theory, A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI, 2015. MR 3364494, DOI 10.1090/simon/004
Joseph L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973), 1–34. MR 315446, DOI 10.1090/S0002-9904-1973-13077-0
N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83–100. MR 0355642, DOI 10.1016/0022-1236(74)90071-8
Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124
References
- Jim Agler and John E. McCarthy, Aspects of noncommutative function theory, Concr. Oper. 3 (2016), no. 1, 15–24. MR 3491879, DOI 10.1515/conop-2016-0003
- Jim Agler, John E. McCarthy, and N. J. Young, Operator monotone functions and Löwner functions of several variables, Ann. of Math. (2) 176 (2012), no. 3, 1783–1826. MR 2979860, DOI 10.4007/annals.2012.176.3.7
- T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90. MR 155193
- John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
- M. J. Crabb and A. M. Davie, von Neumann’s inequality for Hilbert space operators, Bull. London Math. Soc. 7 (1975), 49–50. MR 365179, DOI 10.1112/blms/7.1.49
- 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, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 207. MR 0486556
- Carl H. Fitzgerald, On analytic continuation to a schlicht function, Proc. Amer. Math. Soc. 18 (1967), 788–792. MR 219712, DOI 10.2307/2035112
- Otte Heinävaara, Local characterizations for the matrix monotonicity and convexity of fixed order, Proc. Amer. Math. Soc. 146 (2018), no. 9, 3791–3799. MR 3825834, DOI 10.1090/proc/13674
- G. Herglotz. Über Potenzreihen mit positivem, reellem Teil in Einheitskreis, Leipz. Bericht, 63 (1911), 501–511.
- Dmitry S. Kaliuzhnyi-Verbovetskyi and Victor Vinnikov, Foundations of free noncommutative function theory, Mathematical Surveys and Monographs, vol. 199, American Mathematical Society, Providence, RI, 2014. MR 3244229, DOI 10.1090/surv/199
- Fritz Kraus, Über konvexe Matrixfunktionen, Math. Z. 41 (1936), no. 1, 18–42 (German). MR 1545602, DOI 10.1007/BF01180403
- Peter D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1892228
- Charles Loewner, On schlicht-monotonic functions of higher order, J. Math. Anal. Appl. 14 (1966), 320–325. MR 190354, DOI 10.1016/0022-247X(66)90033-3
- Karl Löwner, Über monotone Matrixfunktionen, Math. Z. 38 (1934), no. 1, 177–216 (German). MR 1545446, DOI 10.1007/BF01170633
- R. Nevanlinna. Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentproblem, Ann. Acad. Sci. Fenn. Ser. A, 18 (1922).
- J. E. Pascoe and R. Tully-Doyle, Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables, J. Funct. Anal. 273 (2017), no. 1, 283–328. MR 3646301, DOI 10.1016/j.jfa.2017.04.001
- Georg Pick, Über die Beschränkungen analytischer Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden, Math. Ann. 77 (1915), no. 1, 7–23 (German). MR 1511844, DOI 10.1007/BF01456817
- Barry Simon, Operator theory, A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI, 2015. MR 3364494, DOI 10.1090/simon/004
- Joseph L. Taylor, Functions of several noncommuting variables, Bull. Amer. Math. Soc. 79 (1973), 1–34. MR 315446, DOI 10.1090/S0002-9904-1973-13077-0
- N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83–100. MR 0355642, DOI 10.1016/0022-1236(74)90071-8
- Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124
Review Information:
Reviewer:
John E. McCarthy
Affiliation:
Department of Mathematics and Statistics, Washington University in St. Louis, St. Louis, Missouri
Email:
mccarthy@wustl.edu
Journal:
Bull. Amer. Math. Soc.
57 (2020), 679-684
DOI:
https://doi.org/10.1090/bull/1688
Published electronically:
December 16, 2019
Review copyright:
© Copyright 2019
American Mathematical Society
