On monotone matrix functions of two variables
Author:
Harkrishan Vasudeva
Journal:
Trans. Amer. Math. Soc. 176 (1973), 305318
MSC:
Primary 47A60; Secondary 26A48
MathSciNet review:
0313855
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Abstract: The theory of monotone matrix functions has been developed by K. Loewner; he first gives some necessary and sufficient conditions for a function to be a monotone matrix function of order n, and then, as a result of further deep investigations including questions of interpolation he arrives at the following criterion: A realvalued function defined in (a, b) is monotone of arbitrary high order n if and only if it is analytic in (a, b), can be analytically continued onto the entire upper halfplane, and has there a nonnegative imaginary part. The problem of monotone operator functions of two real variables has recently been considered by A. Koranyi. He has generalized Loewner's theorem on monotone matrix functions of arbitrary high order n to two variables. We seek a theory of monotone matrix functions of two variables analogous to that developed by Loewner and show that a complete analogue to Loewner's theory exists in two dimensions.
 [1]
Julius
Bendat and Seymour
Sherman, Monotone and convex operator
functions, Trans. Amer. Math. Soc. 79 (1955), 58–71. MR 0082655
(18,588b), http://dx.doi.org/10.1090/S00029947195500826554
 [2]
E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series. Vol. 1, Dover, New York, 1958. MR 19, 1166.
 [3]
A.
Korányi, On a theorem of Löwner and its connections
with resolvents of selfadjoint transformations, Acta Sci. Math. Szeged
17 (1956), 63–70. MR 0082656
(18,588c)
 [4]
Adam
Korányi, On some classes of analytic functions
of several variables, Trans. Amer. Math.
Soc. 101 (1961),
520–554. MR 0136765
(25 #226), http://dx.doi.org/10.1090/S00029947196101367656
 [5]
Karl
Löwner, Über monotone Matrixfunktionen, Math. Z.
38 (1934), no. 1, 177–216 (German). MR
1545446, http://dx.doi.org/10.1007/BF01170633
 [6]
, Advanced matrix theory, Mimeographed Notes, Stanford University, Stanford, Calif., 1957.
 [7]
Béla
Sz.Nagy, Remarks to the preceding paper of A. Korányi,
Acta Sci. Math. Szeged 17 (1956), 71–75. MR 0082657
(18,588d)
 [1]
 J. S. Bendat and S. Sherman, Monotone and convex operator functions, Trans. Amer. Math. Soc. 79 (1955), 5871. MR 18, 588. MR 0082655 (18:588b)
 [2]
 E. W. Hobson, The theory of functions of a real variable and the theory of Fourier's series. Vol. 1, Dover, New York, 1958. MR 19, 1166.
 [3]
 A. Korányi, On a theorem of Löewner and its connections with resolvents of selfadjoint transformations, Acta Sci. Math. Szeged 17 (1956), 6370. MR 18, 588. MR 0082656 (18:588c)
 [4]
 , On some classes of analytic functions of several variables, Trans. Amer. Math. Soc. 101 (1961), 520554. MR 25 #226. MR 0136765 (25:226)
 [5]
 K. Loewner, Über monotone Matrixfunctionen, Math. Z. 38 (1934), 177216. MR 1545446
 [6]
 , Advanced matrix theory, Mimeographed Notes, Stanford University, Stanford, Calif., 1957.
 [7]
 Béla Sz.Nagy, Remarks to the preceding paper of A. Korányi, Acta Sci. Math. Szeged 17 (1956), 7175. MR 18, 588. MR 0082657 (18:588d)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303138554
PII:
S 00029947(1973)03138554
Article copyright:
© Copyright 1973
American Mathematical Society
