数理解析研究所講究録
1535 巻 2007 年 11-14

11SYMMETRIC OPERATOR WORD EQUATIONSYONGDO LIM

ABSTRACT. Symmetric matrix word equations have recently been the topic of aotive investigation because of their relationship to thetrace, an open conjecture arising from statistical physics. In this paper weconsider infinite dimensional (operator) word equations and show the uniquenessof positive definite solutions of some word equations via the non-positive curvature
property of Thompson’s part metric on the space of positive definite operators in aMbert space.
$\mathrm{B}\infty \mathrm{s}\mathrm{i}*\mathrm{M}\mathrm{o}\mathrm{u}\mathrm{s}\epsilon \mathrm{a}\cdot \mathrm{V}i\mathrm{u}\dot{\mathrm{m}}$$\mathrm{c}\mathrm{o}\dot{\mathrm{r}}\infty \mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}$For a nonempty alphabet
words of the form$\mathrm{A}$, we consider the concatenation monoid generalized$W=A_{1}^{\mathrm{p}\iota}A_{2}^{\mathrm{P}2}\cdots A_{k}^{p_{k}}$,where eachnumber, subject to the standardA and each exponent is a, theexponential laws for adjacent powers with a common base. In particular
identity, for each . The reversalof the word $W$ is the word written in reverse
order, and the word is symmetric (or “palindromic”) if it is equal to its reversal. Asymmetric word equation for $\mathrm{A}=\{X, A, B\}$ is an equation of the form $S(X, A)=B$ ,where $S(X, A)$ is a symmetric word in $X$ and ; we further assume that the exponents
of $X$ are all positive, and other exponents are nonnegative.
$A_{j}\in$$\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{J}$$p_{j}$$A_{j}^{0}=I$$W^{*}$$j$$A$

Definition 1. A symmetric word equation $S(X, A)=B$ is called (uniquely) solvable
if there exists (uniquely) a positive definite solution $X$ of $S(X, A)=B$ for every pairof positive definite operators and on a Hilbert space.
$A$

$B$

Example 2. The Riccati
ation $XAX=B$ is uniquely solvable. It hasa uniqu positive definite solution, the geometric mean:andof$A^{-1}\neq B=A^{-1/2}(A^{1/2}BA^{1/2})^{1/2}A^{-1/2}$ . See $([8]-[10])$for more general setting.
$mat\dot{m}eq\mathrm{u}$$e$$A^{-1}\neq B$$A^{-1}$$B$

Date: January 26, 2007.
Key words and phrases. Operator word equation, positive definite operator, nonlinear operator
equation, Thompson’s part metric, contraction.

12YONGDO LIM

Example 3. The symmetric word equation $X(AX)^{m}=B$ appears in [4] which has
the unique solution $X=A^{-1}\#_{\frac{1}{m+1}}B$ , where $A\neq_{t}B=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}$ denotesthe -power mean of and .
$A$

$t$

$B$

In [6], Hillar and Johnson proved that every symmetric word equation is solvablefor finite-dimensional case and Armstrong and Hillar [1] have recently showed via theBrowner mapping degree that the symmetric word equation of degree 6$\cdot\cdot$(1)$XBX^{2}B^{3}X^{2}BX=A$

has multiple positive definite solutions for somepositive definite matrices andsolving the conjecture negatively, but it has a unique positive definite solution in2
real positive definite letters remaining the conjecture open for 2
positivedefinite matrices.
$A$$3\cross 3$$B$$\mathrm{x}2$$\mathrm{x}2$

Theorem 4 (Armstrong and Hillar, [1]). The symmetric word equation (1) is notuniquely solvable for 3
positive definite letters, but is uniquely solvable foroeal positive definite letters.
$2\cross 2$$\mathrm{x}3$

In [14], an explicit form of the unique solution of (1) in 2
letters is given by$\mathrm{x}2$real positive definite$X=(sB^{-1}+B^{-3})\#(tI-A)^{-1}$where$t>\mathrm{t}\mathrm{r}(A)$and $s>0$ are uniquely determined by the simultaneous equations$\frac{s+\mathrm{t}\mathrm{r}(B^{2})}{t-\mathrm{t}\mathrm{r}(A)}$$=$$\frac{t}{s}=\mathrm{t}\mathrm{r}([(sB^{-1}+B^{-3})\neq(tI-A)^{-1}]^{2}B)$.

For a Hilbert space $E$ , let $B(E)$ denote the set of bounded linear operators,$B(E)$ the symmetric operators, andthe set of positive definite operatorson . We define a closed positive order on $S(E)$ by $A\leq B$ if $B-A$ is positive
semidefinite. The Thompson (or part) metric on given by$S(E)\subseteq$$\Omega\subseteq S(E)$$E$$\Omega$$d(A, B)= \max\{\log M(A/B),\log M(B/A)\}$$M(A/B):= \inf\{\lambda>0 :A\leq\lambda B\}$

. A. C. Thompson [17] (cf. $[15]-[16]$ ) has shownthat is a complete metric space with respect to this metric and the corresponding
metric topology on agrees with the relative norm topology. It is easy to see that$\Omega$$\Omega$$d(A, B)=d(A^{-1}, B^{-1})=d(M^{*}AM, M^{*}BM)$13OPERATOR WORD EQUATIONS

for any invertible operator $M$ . The L\"owner-Heinz inequality$0\leq A\leq B\Rightarrow A^{t}\leq B^{t},$$t\in[0,1]$

is equivalent to the following nonpositive curvature property of Thompson’s part
metric (cf. [2], [3], [12], [13]).

Theorem 5. For $A,$ $B,$$C\in\Omega$,$d(A\# tB, A\#{}_{t} C)\leq td(B, C),$ $t\in[0,1]$.

Theorem 6. Symmetric word equations through degree 5 is uniquely solvable.

Proof. Let $S(X, A)=B$ be a symmetric word equation with degree $n\leq 5$ . By theRiccati Lemma (Example 2), we assume that $n\geq 3$ . If $n=3$ , then the only nontrivial equation is XAXAX $=B$ , which has the unique positive definite solution
XAXAX $=X(AX)^{2}=B$ if and only if $X=A^{-1}\# 1/\epsilon B$(Example 3). If $n=4$ , then up to equivalence it is enough to consider the equations$XAXA^{m}XAX=B$ and $XAX^{2}AX=B$ . By the Riccati Lemma,$X=A^{-1}\# 1/3B\mathrm{h}\mathrm{o}\mathrm{m}$$XAXA^{m}XAX=B\Leftrightarrow XAX=B\neq A^{-m}\Leftrightarrow X=(B\neq A^{-m})\# A^{-1}$$XAX^{2}AX=B$$\Leftrightarrow$,$(XAX)^{2}=B\Leftrightarrow XAX=B^{1/2}\Leftrightarrow X=A^{-1}\# B^{1/2}$.

Let $n=5$ . In this case, non-trivial equations (up to equivalence) are $XAX^{3}AX=B$and $XAXA^{m}XA^{m}XAX=B$ . Let$f(X)=(B\neq X^{-1})\# A^{-1},$$g(X)=[B\#(A^{m}XA^{m})^{-1}]\# A^{-1}$.

By the invariance properties and non-positive curvature property, and are strict
contractions for the Thompson metric and hence by completeness of the metric haveunique fixed points, respectively. Then the proof follows $\mathrm{h}\mathrm{o}\mathrm{m}(XAX)X(XAX)=B$if and only if $XAX=B\neq X^{-1}$ if and only if $X=(B\# X^{-1})\# A^{-1}$ if and only if $X=$$f(X)$ , and $(XAX)(A^{m}XA^{m})(XAX)=B$ if and only if $XAX=B\neq(A^{m}XA^{m})^{-1}$ if
and only if $X=[B\neq(A^{m}XA^{m})^{-1}]\neq A^{-1}$ if and only if $X=g(X)$ .
$f$

$g$

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14
YONGDO LIM[2] G. Corach, H. Porta, and L. Recht, Convexity of the geodesic distance on spaces of positive
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[9] J. D. Lawson and Y. Lim, Means on dyadic symmetric sets and polar decompositions, Abh.
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[11] J. D. Lawson and Y. Lim, Solving symmetric matrix word equations via symmetric spacemachnery, Lin. Alg. Appl., 414 $(20\infty)$ , 560-569.
[12] J. D. Lawson and Y. Lim, Symmetric spaces with convex metrics, to appear Fomm Math.
[13] J. D. Lawson and Y. Lim, Metric convexity of symmetric cones, submitted.
[14] Y. Lim, On the symmetric matrix word equation $XBX^{2}B^{\theta}X^{2}BX=A$ , submitted.
[15] R. D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps, Memoirs of Amer.
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[16] R. D. Nussbaum, Finsler structures for the part metric and Hilbert’s projective metric andapplications to ordinary differential equations, Differential and Integral Equations 7 (1994),

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$\mathrm{D}\mathrm{E}\mathrm{P}\mathrm{A}\mathrm{R}\Gamma \mathrm{M}\mathrm{E}\mathrm{N}\mathrm{T}$OF MATHEMATICS, KYUNGPOOK NATIONAL

REA-mail address: ylimGknu.ac. kr$E$, TABGU 702-701, Ko-$\mathrm{U}\mathrm{N}\mathrm{I}\mathrm{V}\mathrm{E}\mathrm{R}\mathrm{S}\iota\tau \mathrm{v}$