数理解析研究所講究録1039 巻 1998 年 135-140

135

COMMUTATION RELATIONS AND RELATED TOPICSMASATOSHI ENOMOTO*ABSTRACT.

We shall describe some aspects of commutation relations.

1. Introduction.
$\mathrm{H}.\mathrm{U}\mathrm{m}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{k}\mathrm{i}[12]$\’etablished the foundation of the noncommutative probability

theory. The concept of covariance has not been focused in this setting. Recently
M.Fujii,T.Furuta, R.Nakamoto andextensively studied noncommutative covariance with some applications for several inequaliti\’e. Related results
are also studied in $[4,11]$ .
.Fujii introduced the covarianc\’e and the variances for operators as follows.
Let $H$ be a separable Hilbert space and $x\in H,$ $||x||--1$ . The covariance of (notnec\’esarily bounded) operators and in a state is defined by$\mathrm{S}.\mathrm{T}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{i}[3]$$\mathrm{J}.\mathrm{I}$

$A$

$B$

$x$

$c_{ov_{x}}(A, B)=(A^{*}Bx, x)-(A^{*}x, x)(B_{X}, x)$ ,and the variance of $T$ in astate$x$

is defined by$Var_{x}(A)=||Ax||2-|(Ax, x)|^{2}$.

The following covariance-variance inequality is established in [3]. The inequality
is a fundamental result in noncommutative probability.
Theorem 1.1(The covariance-variance inequality). The square of the absolute of the covariance of $oper.at_{ors}$
and is not . reater than the product of thevariance of and :
$A$

$A$

$B$

$g$

$B$

$|Cov_{x}(A, B)|2\leq Var_{x}(A)\cdot Var(xB)$.

We shall point out that the covari\‘ance variance inequality is exactly the generalized Schr\"odinger inequality. We briefly sketch the history of the uncertainty
relation $[1,5]$ . The Heisenberg relation$\Delta x\cdot\triangle p\sim\hslash$.
1991 Mathematics Subject Classification.
Key wods and phrases. covariance of operators, variance of operators, Heisenberg uncertainty$47\mathrm{A}99$

principle, noncommmutative probability.

$-$

Typeset by$A_{\mathcal{M}}\theta \mathrm{I}\mathrm{E}\mathrm{X}$136
is expr\’esing the uncertainty principle in quantum physics. This relation was discussed by Heisenberg [6] in 1927. The derivation of the uncertainty principle fromthe mathematical formalism of wave mechanics was given by Kennard [7] as follows:$\frac{\hslash}{2}\leq\Delta x\cdot\Delta p$.

Afterward Robertson [9] proved the generd uncertainty principle, valid for an arbitrary pair of the selfadjoint operators and $B$ . The Robertson inequality reads:$A$$\frac{\hslash}{2}|<C>|\leq\Delta A\cdot\backslash \Delta B$,

where the operator $C$ is defined as$C=(AB-BA)/i\hslash$ ,the brackets
by the formula$<>\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}$the average values and the standard deviations are defined$(\Delta A)^{2}=<(A-<A>)^{2}>$.

An improved inversion of the Robertson inequality is due to Schr\"odinger [10].
Schr\"odinger generalized Robertson’s derivation by introducing $D=A+\alpha B+i\beta B$ ,where and are real numbers, and deducing from $0\leq(Dx, DX)$ the inequality
expr\’esing the nonpositivity of the discriminant ; by an appropriate choice of andhe then obtained the following Schr\"odinger inequality:$\alpha$$\beta$$\alpha$$\beta$$( \frac{1}{2}<AB+BA>-<A><B>)^{2}+\frac{\hslash^{2}}{4}|<C>|^{2}\leq(\Delta A)^{2}\cdot(\Delta B)^{2}$.

In this paper we shall point out that the generalized Schr\"odinger inequality isexactly the covariance variance inequality. We also look at the Hilbert-modulecase. Finally we also look at the -Fock space case and describe its relation to-Kantrovich inequality.
$C^{*}$$\mathrm{q}$$\mathrm{q}$2. Schr\"odinger inequality.
In this section we shall point out that the absolute value of the covariance of two
selfadjoint operators is represented by the term of the Schr\"odinger inequality.
Theorem. Let and be (not necessarily bounded) selfadjoint operators
on a Hilbert space H. Let $D(AB)$ and $D(BA)$ be the domain of AB and $BA$
respectively. Let $x\in D(AB)\cap D(BA)$ and $||x||=1$ . Let $\{A, B\}$ be the Jordanproduct $AB+BA$ . Let $[A, B]$ be the commutator AB–BA. Then$A$

$2.1\sim$$B$$|Cov_{x}(A, B)|2= \{\frac{1}{2}(\{A, B\}X, x)-(A_{X}, x)(Bx, X)\}2+\{\frac{1}{2}([A, B]x, x)\}^{2}$

Proof.
$ReCov_{x}(A, B)= \frac{1}{2}(\{A, B\}X, x)-(A_{X}, x)(Bx, x)$$Imc_{ov_{x}}(A, B)= \frac{1}{2i}([A, B]X, X)$,.

Therefore using by$|Cov_{x}(A, B)|2=|Recov_{x}(A, B)|2+|ImCov_{x}(A, B)|^{2}$ ,this theorem holds.

.

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3The case of Hilbert

Let $X$ be aHilbert$C^{*}$$C^{*}$-module.

-modul\’e over a$C^{*}$-algebra$A$.

$\mathcal{L}_{A}(X_{A})=\{T|T:Xarrow X, \exists\tau^{*}, \tau(X\cdot a)=(Tx)a\}$.

The inner product $(x|y)$ has the following properties ;$(x|y.\cdot a)|=(x|y)a$for$x,$ $y\in$(x\cdot a|y)=a^{*}(X|y)$Then we shall define the covariance $Cov_{x}(S,T)$
follows ;..

X,$ $a\in A$,.
for

$S,T\in \mathcal{L}_{A}(X_{A}),X.\in X$

as$\cdot$$Cov_{x}(S, T)=(Sx-x(X|s_{X})A|TX-x(X|\tau_{X})A)$$Var_{x}(S)=Cov_{x}(S, S)$.

Then
$Cov_{x}(S, T)=(Sx|TX)-(Sx|x(X|Tx)_{A})-(X(x|s_{X)}A|Tx)+(x(X|SX)A|x(X|Tx)A)$$=(T^{*}Sx|X).-(Sx|X)(X|\tau_{X})A-X|sX)*(x|X)(X|T_{X)_{A}}$(We assume the following condition here.
$(x|x)=1.)$$=(T^{*}Sx|X)-(Sx|X)(X|\tau_{X)-}(Sx|x)(X|\tau X)+(Sx|X)(X|Tx)$$=(T^{*}sx|X)-(Sx|X)(X|Tx)$ .

Then we have
$Cov_{x}(S, T)=(T^{*}Sx|X)-(Sx|X)(x|Tx)$ .

On the other hand, $Cov_{x}(S, T)$ is linear for and conjugate linear for. Therefore. Thus $Cov_{x}(S,T)$ is an inner product for$S$$S,$ $T\in \mathcal{L}_{A}(X_{A})$$T,\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$\mathcal{L}_{A}(X_{A})$$-||Cov_{x}(s, T)||2\leq||Var_{x}(s)||\cdot||Var_{x}(T)||$.

Remark 3.1. The condition $(x|x)=1$ is different from the condition $||x||=1$ .
part and the imaginary part of $Cov_{x}(S, T)$ are as follows;The$rea\dot{l}$$\Re covx(S, T)=\frac{1}{2}((s^{*}\tau+T^{*}S)_{X}|X)-\Re((\tau X|x)(x|s_{x}))$,$\Im Covx(S, T)=\frac{1}{2i}\{((s*\tau-\tau*S)x|X)-(Tx|X)(x|sX)+(Sx|X)(x|TX)\}$.

Remark 3.2. ’We have$\frac{1}{4}||(S^{*}\tau-\tau^{*}S)x|X)-(Tx|X)(x|sX)+(Sx|X)(X|\tau X)||^{2}\leq||Var_{x}(s)||\cdot||Var_{x}(T)||$

his is an uncertainty relation for Hilbert$\dot{T}$$C^{*}$-module.

.

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4 Parametrization of variance-covariance inequality Rom the point ofq-statistics.

$\cdot$There is a recent topic of -parametrization between Boson statistics and Fermionlstatistics. In this section we shall consider the -parametrization of variancecovariance inequality. Then we remark a -parametrization of Kantrovich inequality
from the point of [FFNT]. First we shall review the fact of -parametrization between Boson statistics and Fermion statistics [cf.2]. Fix a number $q\in[-1,1]$ . Takea separable complex Hilbert space . Let
be the linear space of thevector of the form
andvacuumvector). We put the sesquilinear form
ondefind by$\mathrm{q}$$\mathrm{q}$$\mathrm{q}$$\mathrm{q}$$\mathcal{H}$$f_{1}\otimes\cdots\otimes f_{n}\in \mathcal{H}^{\otimes n}$$\mathcal{F}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}(\mathcal{H})$$(n\in \mathrm{N}_{0})$$(\cdot, \cdot)_{q}$$\mathcal{H}^{\otimes 0}\simeq \mathbb{C}\Omega(\Omega=\mathrm{t}\mathrm{h}\mathrm{e}$$\mathcal{F}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}(\mathcal{H})$$<f_{1^{\otimes\cdots f_{n’ g}q^{i(\pi}}}1\otimes\cdot\cdot.\cdot g_{n}>m^{\Sigma(f_{1,g\pi})\cdots(f\pi(}q^{=\delta}n)(1)h,gh))$and put$i(\pi)=\#\{(i,j)|1\leq i<j\leq n, \pi(i)>\pi(j)\}$

Then we shall define the -Fock space$\mathrm{q}$Definition 4.1. For $A,$$x,$$||x||=1$ by$B\in B(\mathcal{H})$$\mathcal{F}_{q}(\mathcal{H})$by.

$\overline{\mathcal{F}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}(\mathcal{H})}^{(},\cdot)_{q}$. using this ,in astatefor$q$, we define the -cova$q$$r\dot{\eta}ance$$A,$ $B$$x\in \mathcal{H},$$Cov_{x}^{q}(A, B)=(Ax\otimes x, Bx\otimes x)_{q}$.

Under this setting ,Proposition 4.2. For $q\in[-1,1]$ , we have$(Ax\otimes x, Bx\otimes x)_{q}=(Ax, x)+q(AX, X.)\cdot(x, Bx)$

Corollary$4.3(\mathrm{s}_{\mathrm{e}}\mathrm{o})$.

. For $q=-1$ ,$(Ax\otimes x, Bx\otimes x)_{-1}=(Ax\wedge x, Bx\wedge x)=c_{ov_{x}}(A, B)$.

Corollary 4.4. For $q=1$ ,$.(Ax\otimes x, Bx\otimes x)_{1}=(Ax\vee x, B_{X}\vee x)=(Ax, BX)+(A_{X}, x)(X,$ $B_{X)}$

5 The bound of -variances and q-covariances.
$\mathrm{q}$In this section we shall consider the bound of -variances and -covarianc\’e. Let$||x||=1,$ $q\in[-1,1]$ and $0<m\leq A\leq M$ .
Then we have$\mathrm{q}$$\mathrm{q}$$x\in \mathcal{H},$$Var_{x}^{q}(A)=(M+q(AX, X))((AX, X)-m)-((M-A)(A-m)_{X}, X)+m(1+q)(A_{X},x)$ .
$Var_{x}^{q}(A)\leq(M+q(A_{X}, x))((Ax, X)-m)+m(1+q)(Ax, x)$ .

We put$f(t)=(M+qt)(t-m)+m(1+q)t$ .

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Proposition 5.1. For$- \frac{M+m}{2M}<q\leq 1$,$Var_{x}^{q}(A)\leq M^{2}(1+q)$

Proposition 5.2. For.

$- \frac{M+m}{2m}\leq-1\leq q\leq-\frac{M+m}{2M}$$Var_{x}^{q}(A) \leq-\frac{(M+m)^{2}}{4q}$,–Mm.

Using this ,we shall show the bound of q-covariances.
For $0<m_{1}\leq A\leq M_{1},0<m_{2}\leq B\leq M_{2}$ , we have the following.
Proposition 5.3. For $0\leq q\leq 1$ ,$|Cov_{x}^{q}(A, B)|\leq M_{1}M_{2}(1+q)$

Proposition 5.4. Let(1) For(2) For.

$-1 \leq q\leq 0and-\frac{M+m}{2M_{1}}=-Marrow+m2M_{2}^{\cdot}$$- \frac{M_{1}+m_{1}}{2M_{1}}<q$$q \leq-\frac{M+m}{2M_{1}}$,$|Cov_{x}q(AB):|\leq M_{1}M_{2}(1+q)$.

,.

$|Cov_{x}^{q}(A, B)|\leq\sqrt{\{-\frac{(M_{1}+m_{1})^{2}}{4q}-M_{1}m_{1}\}\{-\frac{(M_{2}+m_{2})^{2}}{4q}-M2m_{2}\}}$

Proposition 5.5. Let(1) $Let$$-1\leq q\leq 0$$.- \frac{M_{2}+m_{2}}{2M_{2}}<-\frac{M_{1}+m_{1}}{2M_{1}}$$F_{or-^{M}\mathrm{m}+m}<q2M_{1}$$(\alpha)$and$- \frac{M+m}{2M_{1}}\neq-Marrow+m2M_{2}^{\cdot}$,’$|Cov_{x}^{q}(A, B)|\leq M_{1}M_{2}(1+q)$

For$(\beta)$$- \frac{M_{2}+m_{2}}{2M_{2}}<q\leq-\frac{M_{1}+m_{1}}{2M_{1}}$.

,$|Cov_{x}^{q}(A, B)|\leq\sqrt{\{-\frac{(M_{1}+m_{1})^{2}}{4q}-M_{1}m_{1}\}M_{2}^{2}(1+q)}$For$(\gamma)$$q\leq-Marrow+m2M_{2}$.

’$|Cov_{x}^{q}(A, B)| \leq\sqrt{\{-}\frac{(M_{1}+m_{1})^{2}}{4q}-M_{1}m_{1}\}\{-\frac{(M_{2}+m_{2})^{2}}{4q}-M2m_{2}\}$

(2)

For
$(\alpha).$$-M\mathrm{m}_{1}+m2M<-Marrow+m2M_{2}$For.

$- \frac{M_{2}+m_{2}}{2M_{2}}<q$’,$|Cov_{x}^{q}(A, B)|\leq M_{1}M_{2}(1+q)$$(\beta)$For.

$- \frac{M_{1}+m_{1}}{2M_{1}}<q\leq-\frac{M_{2}+m_{2}}{2M_{2}}$.

,.

$|Cov_{x}^{q}(A, B)|\leq\sqrt{M_{1}^{2}(1+q)\{-\frac{(M_{2}+m_{2})^{2}}{4q}-M2m_{2}\}}$$(\gamma)$For$q \leq-\frac{M_{1}+m_{1}}{2M_{1}}$,.

$|Cov_{x}^{q}(A, B)|\leq\sqrt{\{-\frac{(M_{1}+m_{1})^{2}}{4q}-M_{1}m_{1}\}\{-\frac{(M_{2}+m_{2})^{2}}{4q}-M2m_{2}\}}$

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Remark( -Kantrovich inequality).
.
(1) For $q=0$,$0<q\leq
1$,(2) For$\mathrm{q}$$1 \leq\frac{M}{m}$$(A_{X}, x)(A^{-1_{X,x}}) \leq\frac{M+Mq-m}{mq}$For-l $\leq q<0$ ,

(3)

and

$- \frac{m+M}{2M}<q$,$(A_{X}, x)(A^{-1_{X,x}}) \leq\frac{M+Mq-m}{-mq}$

For-l $\leq q<0$ and$q \leq-\frac{m+M}{2M}$.

.

,$(Ax, x)(A^{-1_{X,x}})\leq-$$-q1(1+\sqrt{\{-\frac{(M+m)^{2}}{4q}-Mm\}\{-\frac{(\frac{1}{m}+\frac{1}{M})^{2}}{4q}-\frac{1}{Mm}\}})$.

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