On some functional equation in the theory of geometric objects

(1)

Seria I: PRACE MATEMATYCZNE X II (1969) ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X II (1969)

Z. K areńska (Kraków)

On some functional equation in the theory of geometric objects

In this paper we find all solutions of the functional equation (

⁰

.

¹

) g(oo-y) = F{ x ) - g( y) + g{x),

where x and у are non-singular

2

x

2

real matrices, g is an unknown function whose values are 3 x 1 real matrices, and F is a given function of the form

x\x 2 х Х1 х 12 x\2

(0.2) 1

я И = ~ j XlxX21 ffil\X22~\- xx2x2X <^i2x22 2 x 2X x 22 ffi22

where Л is the determinant of x and x lx, x l2, x 21, x 22 are the entries of x.

The general linear group G L (

2

,B ) will shortly he denoted by G2. W e do not make any assumptions concerning the regularity of the matrix func

tion g.

Equation (

0

.

1

) appears in the theory of geometric objects when we want to find the solutions of the system of functional equations

F ( x - y ) = F { x ) - F ( y ) , g{x-y) = F{ x) - g{ y) + g{x)

([2], p. 152) in order to determine the geometric objects of type [3, 2 ,1]

with linear non-homogeneous transformation rule.

The method used below is analogous to that used in [5] and the function (0.2) is similar to that appearing in [5 ]; the present case, however, is more interesting and more complicated, and we get some non-meas- urable solutions whereas all solutions in [5] were measurable.

The main result of the paper is

T heorem

0

.

1

. Any solution of (0.1) defined on G2 is of the form g{x) = [ F { x ) - E ] - q + g(x),*

(0.3)

(2)

220 Z. K a r e ń s k a

where F is given by (

⁰

.

²

), E is the unit 3 x 3 matrix, q is a 3 x 1 matrix whose entries &2, &3 are real parameters,

/у» /V»

^ 1 1 ^ 1 2

* 1

9 (я) = — ł <p(xu) ę>(®22) (p(x21) <p{x12)

Л &11 X22 + 21 *^12

<P(®2l) ^(^

2 2

)

# 2 1 ® 2 2

«md

^{9 9}

is any derivation of R.

B y a derivation of we mean any function <p: R R satisfying the conditions

(0.4) . <p{£+y) = <pW + <p(y),

(0.5)

9

?(£??) = w ( £ ) +

for all i, у in R (cf. [9], p. 120 and [3]).

Let

9

0 be such a derivation. It is well known that (p(0) =

9

?(

1

) = 0,

<p(— I) = — <p{i) and

(

⁰

.

⁶

) q>{f) = p | p- V ( D

for £ in R and p = 1, 2, ... I f у Ф 0, then <p{lly) = — <p{y)ly2 and

(0.7) J l \ = w W z W ' i ) .

\yf 7)2

Furthermore

^{9 9}

vanishes on the algebraic closure Q of the field Q of rationale (in R ) and has a dense set of periods. I t is well known that any measurable derivation of R is trivial (i.e., identically 0), but there are non-trivial derivations of R. W e thus get the following corollary.

C

o r o l l a r y 0

.

¹

. Any measurable solution of (0.1) is of the form g(x)

= [ F ( x ) ~ E] - q, where F , E and q are as in Theorem 0.1.

§1 . Some properties of solutions of (0.1). A straightforward veri

fication shows that the function F defined by (0.2) satisfies the equation F ( x ' y ) = F ( x ) - F ( y ) (cf. [4]) and any function of the form (0.3) satisfies (0.1). W e have thus to show the converse statement. Let us notice that the matrix equation (

⁰

.

¹

) is equivalent to the system of three equations

Yi (a-y) = ~ [ ^hyi ( y) + 2x11x 12y2(y) + x2 12yz(y)] + yl (x),

1

УЛХ' У) = -j[oOu002iyi{y) + {co11^22 + co12xn )y 2{y)-\-x12x 22y3{y)'] + yi (x), 1

1

Уз(х ‘ У) =-^-[ooh ri(y) + 2x2lx 22y2{y) + xl2y3(y)] + y3{x),

(3)

where

y x( x ) , y 2( x ) ,

Уз(ж) are the entries of the matrix

g ( x ) .

Let

g ( x )

denote the matrix

\\yik{x)\\,

i, к — 1 ,2 where

У11И = У1О»),

^{y X2( x )}

= у ²¹ (ж) = ya(®)> У22И = y ³ (a?).

The above system is equivalent to the matrix equation

1 T „

(1.1) 0(®-y) = - j x - g { y ) - x + g ( x ) ,

where ж2 denotes the transposed matrix and A = deta;. I f g is a solution of (0.1), then g is a solution of (1.1) and conversely. I f g satisfies (1.1), then g(e) = ||0||, where [|0|| is the 2 x 2 zero matrix and e is the unit 2 x 2 matrix. Moreover, if г = fie, where fieB and fi Ф 0, then g(z) = ||0||.

Indeed, since г commutes with any 2 x 2 matrix, we get g( x-z) — g( z-x) and

1 „ T „ «

— x- g{ z) - x + g { x ) = g{x)Ą-g{z), and hence

(1.2) A~ 1 x- g{ z) - xT = g(z).

Substituting x — j, where j denotes the matrix (bik) with bxx — b 22 — 0,

&i

2

= b 21 = 1, we get —j - g { z ) - j = g{z). Hence yX 2 (z) = 0 and y 22 (z)

— ~Уи( я) - Consequently,

9() =* У u («) 0 0 — yxl(z)

Now, substituting x = n in (1.2), where n is the matrix (eik) with cn = — 1, c22 = 1, c 12 = c 21 = 0, we get — n- g( z ) - n = g(z); therefore Уи{у) = 0. This concludes the proof of the identity g(z) = ||0||.

Substituting x = у = j in (1.1) we get

II0II = g(e) = g ( j - j ) = - j ’ § U) - j + g{j)- Hence

(i- 3) 5 (j) =

§ 2. The functions r and h. We are going to find the general form of a solution g( x) of (1.1). W e begin with the following identities used by Kucharzewski and Kuczma ([6], p. 3), which guide us in our consid

erations :

(

2

^.

1

⁾

x xx 0 1 0 ---- 0 A 1 0

\T

0 1 x 2X 1 0 1 Ж12 ^

®11

(4)

222 Z. K a r e ń s k a

provided that x xx Ф 0, and 0

x X2

= (3D’j ) - j =

x X2

0 (2.2)

x

=

x 2i x 22 X 22 X 21

provided that

x xx

= 0 and det

x

Ф 0. Consequently, in order to find the general solution

g ( x )

of (1.1) it is enough to know g( j ) and the functions

r ( e) = 9 Q Ф 0, Щ ) = 9

The identity g(e) = ||0|| proved above implies that r( 1) and h( 0) are equal to the zero matrix. Applying (2.1), (1.1) we get, when x xx ф 0, (2.3) g{®) = — - j x - g ( j ) ’ ®

0 x xi A

A

0 ---- Хц

X11 X21

3D 11 3C2X 1 0 xxl •r\ — )■ 1 A 0 1 X11 1

^{x 21}

\ / 3D a ^X ²¹ + — 1

Хц 0

’9(3)'

Хц X2x +

X11 3D 21 f 0 1

+

x xx 0

0 1 Ji(x21) Хц 0

0 1 ф г ( х

₁₁₎

Thus, we have to find r(g), h(r)) and g{j). An easy computation yields

(2.4) га?] ) = 1 0

1 'Г(т)У I 0

0 - 0 1

+ r(£)

provided that ф 0. Since the matrix g( x) is symmetric, r(| ) is also symmetric, i.e., r 12{£) = r 21(£), where rik(£), i, к = 1 , 2 denotes the entries of r(|). Hence

(2.5) = Г 12 {£) + Г 12 {

г

}),

if irj ф 0 . Applying (2.4) to the identity r(irj) = r(r)tj) and comparing the corresponding terms of matrices we get

- - l W 2(|)

П I

( l — i ) r n ( v ) = (v — i ) r n U ) ,

(5)

for all £,r) different from 0. I f g = 2, then

r lx{£) = A J ! — 1) and r22(£) = — 1), where Kx = ru (2) and 13 =

(2.6) r(|) =

— 2r22(2). Thus, if f # 0, Ях( | - 1 ) rla( f )

^12 ( £) ^3 1 j Now, since g is a solution of (1.1), we obtain*

(2.7) &(£ + »?) = l о I i •*to)-

i I

o 1 f ()•

Arguing as above and taking into account the symmetry of the matrix Ji(rj) we infer that £hxl(T)) — ф х1{£) and

12^

и

(^) + 2 ^ 12(^) = У 2 Ьп (%) + 2 ф Х2(£)-

In particular, if rj — 1, then tjhxl( 1) = hxx(£) and 12йп (1) + 2|Л12(1)

= ^ u ( l ) + 2Л12(£). Thus,

(

2

.

8

)

W ) =

2

t

l £ ^l^2+ ^ 2 l 'rl^ 2+ T2l ^22 (£)

where r x = \hxx{ 1), r2 = hx 2 { l ) — p u (l), and £ is any real number.

Substituting rj = — I in (2.7) and applying (2.8) we get h22(£) + /i22( — £)

= 2r2 £2. In virtue of the identity

1 0| 1 0 1 0 1 I 0

0 1 1

1 ^Sy 1 ^Г) 1 0 1 and equation (1.1), we get

1 7

i о

0 1 • й (£>?)•

I о

0 1 ■r(£) I1 0

I V 1

•r(|) 1 ?7

0 1 + h(rj).

Substituting rj = 1, applying (2.6) and (2.8) and comparing the corresponding terms of matrices we infer that

r ^ - l J ^ O ,

(£ — i ) [ T'i ( £ + i ) + T2— Л-

i

] = о

and A22(£) = £Л22(1) + Лг£(£ — l ) + 2 fria( f ) for all f # 0. Consequently, Ti = 0, r 2 = Ax, and

(2.9) * ..(* ) = 2£[r12(£ )-< 9 ] + r 2£2,

(6)

224 Z. K a r e ń s k a

where 0 — | [т2— й22(1)]. Combining the above results with (2.6) and (2.8) we get

(

2

.

10

)

if £ yŁ 0, and ( 2 ^. 11 ⁾

r ( i ) =

t

2( £ - 1 ) ria(£)

W )

* w (f) Mr-)

0

t

2£

t

2£

^<M

for all £. I f £ ^ 0, then the relation (2.9) between r12 and /г22 may be written as r12(£) = (2£)~1ft22( ! ) — -|

t

2£ + <9- On the other hand, substi

tuting (2.11) to (2.7) and comparing the corresponding terms we get

” ^22 ( £) "I" ^22 (??) "H 2

t

2 £>7 .

Let £Г(£) = &22(£) —

t

2£2. An immediate computation shows that i f ( £ + ?

⁷

) = И (| ) + Я (^ ) for all £ and rj. Let

⁹

?(£) = IT (£) + 2(9£. Then

9 9

(

1

) =

2

£r12(£) if £ Ф

0

. Consequently, by (2.5),

9 9

is a derivation of B, (2.12) fl2( £ ) = A l A and * „ ( £ ) = 99(£)-2<9£ +

t

2£2.

2£

W e thus get a description of functions in (2.10) and (2.11).

§ 3. Computation of g{ j ) . The symbols j and n have the same meaning as in § 1. Let gu, g12, g21, g 22 be the entries of the matrix g ( j ); by (1.3), g 12 = g21. In virtue of the identity n - j - n - j = j - n - j - n , (1.1) yields

g (j - n -j - n) = g { n - j - n - j ) = g ( { n - j - n ) - j ) = g { - j - j ) = g { - e ) = ||0||.

On the other hand, combining the identity r2 0 g ( n) = /( — 1)

with (1.3) we get

g ( j - n - j - n ) = g( { j - n) - { j - n) ) =

0 Я3

2 [^ u + (T 2+ ^3)] ⁰

0 2

[

0

n -f-(т

2

+ Я 3)]

Consequently, gn = g 22 = — (r

²

+ Я3). Further, by (1.1),

— 2<7i2-f~ Я3 + 20 2gr12 2(9 r 2

^ 9 l 2 2© T2

/ 1 1 0 Д /

¹ ⁰

Д

= 5 •j H •1

\ 1

^- ¹ ¹

I \

¹ ¹

/_

l i

- 1 0

v

¹ ⁰

t

2 l y

⁰ ¹

)

¹ ¹

^g

^{1 2} ^{r 2} ^{^ 2}

^{^3}

(7)

Hence g x2 = 20. Gathering the above equations we infer that

9(3) =

— (

t

24- Я3) ^{2 0} 2 0 - (т24- Я3) J

§ 4 . The concluding p art o f the proof. Using (2.3) and the above formulae for r(£), /i(f), g (j) and the corresponding matrix terms, and applying (0.4) and (0.5) we get the following relations, when xxx Ф 0

(4.1) ' yxx{x) = yx{x) = | i [ r а®?! — 20жи я?12 + Я8я?;2] — raJ +

9?(

ж

12) 'ii

(4.2) yia(a?) = y2(®) [т 2 ХххХ 21 — в {хххх 22 + х Х 2 х2Х) + % 2 Хх 2 Х22\ + в\ф 1

2 Л

4>(хп) (Р(Х 22 ) +

<р{х2х) <р{хХ2)

$9.1 $Л9

(4.3) У2г(ж) = Уъ(х ) — \У 2 Х 2 \ х г\х 22 ~\~ h$X22\ ^з| ~Ь д>(х2Х) <р(%22) +

'21

I t remains to determine the form the function g( x) for matrices . with x xx = 0.

Let

' О X 10

e(x 2 (X 12X21 b)

X =

0 x X2 00 2X x 22

and x = x-j . Then g{x) = g{x-e) = g(oc-j-j) = g [ ( x - j ) - j ] = g( x- j ) . Com

bining (1.1) and the above results we get (if x xx = 0) yxx{x) — yi(x) — ^3Ж12 U

1 1

Уi

2

(x ) = Уг(ж) ^ ~pp[ 0 xX 2 x 2 X-j- Л 3 х Х 2 х22] -j- 0 -b

У гг^ ) = Уз(^г) = [г 2ж21 2 0 ж 21 ж 22+ Я3 ж 22] ^з| рр

9»(®2l) ^ (^

^{1 2}

)

'2 1 X i

9? (#2l) (P(X 22 )

X 21 X 22

This means that the formulae (4.1)-(4.3) are also true for x xx = 0.

W e thus get (0.3); the entries of the matrix q turn out to be equal

to T „ - 0 , A 3.

(8)

326

Z. K a r e ń s k a R e fe r e n c e s

[1] J. A c z e l,

Y orlesungen iiber Funktionalgleiclmngen und ihre Anwendungen,

Basel und Stuttgart 1961.

[2] J. A c z e l und S. G-ołąb,

Funktionalgleichungen der Theorie der geometri- schen Objekte,

Warszawa 1960.

[3] W . V. D. H o d g e and D. P e d o e ,

Methods of algebraic geometry,

Vol. I, Cambridge 1947.

[4] Z. K a r e ń s k a ,

Ogólne rozwiązanie równania funkcyjnego F { x - y ) — F ( x )

x

x F ( y ) , F macierz

3 x 3 ,

х , у

<=Grl(2,

B )

(to appear).

[5] —

O pew nym równaniu funkcyjnym mającym zastosowanie w teorii obiektów geometrycznych,

Zeszyty Naukowe Politechniki Krakowskiej 9 (1965), pp. 3-25.

[6] M. K u c h a r z e w s k i and M. K u c z m a ,

O n the functional equation F (A • Б )

= F ( A ) - F ( B ) ,

Ann. Polon. Math. 13 (1963), pp. 1-17.

[7] —

O n a system of functional equations occurring in the theory of geometric objects,

ibidem 14 (1963), pp. 59-66.

[8] M. K u c z m a ,

O n linear differential geometric objects of the first class with one component,

Publ. Math. Debrecen 6 (1959), pp. 72-78.

[9] O. Z a r is k i and P. S am u el,

Commutative algebra,

On some functional equation in the theory of geometric objects

Z. K areńska (Kraków)