数理解析研究所講究録
第 1839 巻 2013 年 40-4240

Semi-operator monotonicity for operator
monotone functions

山形大学理学部数理科学科 佐野 隆志 (Takashi SANO)
Department of Mathematical Sciences, Faculty of Science,
Yamagata University

We review results on operator monotone functions. For details, we refer[11].

Loewner and Kwong matrices
Let $f(t)$ be a continuously differentiable function from the interval, we define the $n\cross n$ matrix
ininto itself. For distinct$(0, \infty)$$t_{1},$$L_{f(t)}(t_{1}, \ldots,t_{n})$$\ldots,$$t_{n}$$(0, \infty)$as$L_{f(t)}(t_{1}, \ldots, t_{n}):=[\frac{f(t_{i})-f(t_{j})}{t_{i}-t_{j}}],$

where the diagonal entries are understood as the first derivatives $f’(t_{i})$ .
This matrix is called a Loewner matrix. Similarly we define the $n\cross n$
asmatrix$K_{f(t)}(t_{1}, \ldots, t_{n})$$K_{f(t)}(t_{1}, \ldots, t_{n}):=[\frac{f(t_{i})+f(t_{j})}{t_{i}+t_{j}}],$

which we call an Kwong matrix. (In [2, 8] it is called an anti-Loewnermatrix.) See [3, 4, 5] on Loewner and Kwong matrices.
asandmatrix
We also define the$n\cross n$$L_{f(t)}^{(m)}(t_{1}, \ldots, t_{n})$$K_{f(t)}^{(m)}(t_{1}, \ldots , t_{n})$$L_{f(t)}^{(m)}(t_{1}, \ldots, t_{n}) ;= [\frac{\{f(t_{i})\}^{m}-\{f(t_{j})\}^{m}}{t_{i}^{m}-t_{j}^{m}}],$$K_{f(t)}^{(m)}(t_{1}, \ldots, t_{n}) ;= [\frac{\{f(t_{i})\}^{m}+\{f(t_{j})\}^{m}}{t_{i}^{m}+t_{j}^{m}}]$for a positive integer $m.$
It is well-known that $f(t)$ is operator monotone if and only if for all andare positive semidefinite;, the Loewner matrices
see [10]. If $f(t)$ is operator monotone, the Kwong matrices
are positive semidefinite; see [9]. The latter is recently characterized by$n$$t_{1},$$\ldots,$$t_{n}$$L_{f(t)}(t_{1}, \ldots, t_{n})$$K_{f(t)}(t_{1}, \ldots, t_{n})$41
Audenaert [2]. On the other hand, it is known that if $f(t)$ is operator
monotone, so is the function $t\mapsto\{f(t^{1/m})\}^{m}$ for any positive integer $m.$
See [1, 7]. Hence, combining them, we see that if is operator monotone, then the Loewner matrices
and the Kwong
matricesare positive semidefinite; therefore, so are$f$$L_{\{f(t^{1/m})\}^{m}}(t_{1}, \ldots, t_{n})$$K_{\{f(t^{1/m})\}^{m}}(t_{1}, \ldots, t_{n})$and.
We have an alternative proof for operator monotonicity of the function $t\mapsto\{f(t^{1/m})\}^{m}$ by Theorem 1.6 that if
is operator monotone,then
are positive semidefinite for all andin. We also have in Theorem 1.5 that if is operator monotone, then$K_{f(t)}^{(m)}(t_{1}, \ldots, t_{n})$$L_{f(t)}^{(m)}(t_{1}, \ldots, t_{n})$$f$$L_{f(t)}^{(m)}$$(t_{1}, \ldots , t_{n})$$n$$(0, \infty)$$t_{1},$$\ldots,$$t_{n}$$f$

are positive semidefinite for all andin.
We recall several facts:
Theorem 1.1 (L\"owner [10]) Let be afunction on. Then is
operator monotone if and only if
are positive semidefinite
for all positive integers andin.
$K_{f(t)}^{(m)}(t_{1}, \ldots, t_{n})$$n$$t_{1},$$C^{1}$$f$$\ldots,$$t_{n}$$(0, \infty)$$(0, \infty)$$f$$L_{f(t)}(t_{1}, \ldots, t_{n})$$n$$t_{1},$$\ldots,$

Theorem 1.2 (Kwong [9]) Let
is operator monotone, thenall positive integers and$f$$f$$(0, \infty)$$t_{n}$

be a positive$K_{f(t)}(t_{1}, \ldots, t_{n})$$n$$t_{1},$$\ldots,$

Theorem 1.3 (Audenaert [2])For all positive integers anditive semidefinite if and only if$n$$t_{n}$$(0, \infty)$in $(0, \infty)$ .

Let$t_{1},$

function on. If
are positive semidefinite for$C^{1}$$\ldots,$$f$

be a positive
function on $(0, \infty)$ .
in $(0, \infty)K_{f(t)}(t_{1}, \ldots, t_{n})$ are posis operator monotone.
$C^{1}$$t_{n}$$f(\sqrt{t})\sqrt{t}$Theorem 1.4 (Ando [1], Fujii-Fujii [7]) Let be an operator monotonefunction from
into itself. Then so is the function $t\mapsto\{f(t^{1/m})\}^{m}$ for
any positive integer $m.$$f$$(0, \infty)$

We have the following theorems in [11]:Theorem 1.5 Let
be an operator monotone function from
into itself. Then for any positive integer $m,$
are positive semidefinite for all positive integers
andin: orare positive semidefinite for all positive integers
andin.
$f$$(0, \infty)$$K_{f(t)}^{(m)}(t_{1}, \ldots, t_{n})$$n$$t_{1},$$\ldots,$$t_{n}$$(0, \infty)$$K_{\{f(t^{1/m})\}^{m}}(t_{1}, \ldots, t_{n})$$t_{1},$$\ldots,$$t_{n}$$n$$(0, \infty)$

Theorem 1.6 Let
be an operator monotone function from
into itself. Then for any positive integer $m,$
are positive semidefinite for all positive integers
andin: orare positive semidefinite for all positive integers
andin.
$f$$(0, \infty)$$L_{f(t)}^{(m)}(t_{1}, \ldots, t_{n})$$n$$L_{\{f(t^{1/m})\}^{m}}(t_{1}, \ldots, t_{n})$$t_{1},$$\ldots,$$t_{n}$$(0, \infty)$$t_{1},$$\ldots,$$t_{n}$$(0, \infty)$

$n$

42
$\star\vee’\doteqdot Xffi$[1] T. Ando, Concavity of certain maps on positive definite matrices andapplications to Hadamard products, Linear Algebra Appl., 26 (1979),

203-241.
[2] K. M. R. Audenaert, characterisation of anti-L\"owner functions, Proc.
Amer. Math. Soc., 139 (2011), 4217-4223.
$A$[3] R. Bhatia, Positive Definite Matrices, Princeton University Press(2007).
[4] R. Bhatia and T. Sano, Loewner matrices and opemtor conve city, Math.
Ann., 344 (2009), 703-716.
[5] R. Bhatia and T. Sano, Positivity and conditional positivity of Loewner
matrices, Positivity 14 (2010), 421-430.
[6] W. F. Donoghue, Monotone Matrix Functions and Analytic Continuation, Springer (1974).

[7] J. I. Fujii and M. Fujii, An analogue to Hansen’s theory of generalized
L\"owner’s functions, Math. Japonica, 35 (1990), 327-330.
[8] C. Hidaka and T. Sano, Conditional negativity of anti-Loewner matrices,
Linear and Multilinear Algebra, 60 (2012), 1265-1270.
[9] M. K. Kwong, Some results on matrix monotone functions, Linear Algebra Appl., 118 (1989), 129-153.

[10] K. L\"owner,\"Uber monotone Matrixfunctionen, Math.

$Z$., 38 (1934),

177-216.
[11] S. Tachibana and T. Sano, On Loewner and Kwong matrices, to appear
in Sci. Math. Japonicae.