Integr Equat Oper Th Vol. 20 (1994)
Generalized Functions M. C. Cs
0378-620X/94/020198-3351.50+0.20/0 (c) 1994 Birkh~iuser Verlag, Basel
Factorization for a Class of n x n Matrix - Partial Indices and Explicit Formulas A. F. dos Santos
The generalized factorization of a class of continuous non-rational n x n matrix-functions is studied. The partial indices are determined and, in the case of existence of a canonical factorization, explicit formulas for the factors are obtained.
1
Introduction In this paper we determine the partial indices and obtain explicit formulas for the factors
in a generalized factorization of non-rational matrix-functions of the so-called Jones class (el. [2], [9]) i.e., matrix functions of the form n
G = ~ a i r j-1 j=l
where aj E L~(P~) and R satisfies the equation R n = p'~I,~. Here In denotes the identity matrix in C ~xn and p~ is a scalar rational function, invertible in the algebra of continuous functions in ~ .
In the present paper we study the case where p~ is the quotient of two
first-degree polynomials (see (2.5) in the following section). It may be interesting to note that these are Toeplitz matrices when we take for R the companion matrix
R =
0
1
0
0
0
1
..
0
0
0
0
..
1
0
..
0
~
0
Cfimara and dos Santos
199
We assume in this study that the eigenvalues of G, bj(j = 1 , . . . , n), admit a bounded factorization. Matrix functions of Jones class appear related to problems in elasticity and diffraction theory. They have been studied in [2], [9], [17] and, for n = 2, in several other papers (el. [5], [12], [141, [151, [19], [22]). To the authors' knowledge, the question of existence of canonical factorization for n > 2, which is an important objective of the present paper, was not considered in any of the above references. Moreover, except for n = 2, the calculation of the partial indices in the noncanonical case was not studied. The method used here can be considered simple and, contrarily to some of the papers on this subject that we mentioned above, does not require the resolution of boundary-value problems on Riemann surfaces. A somewhat unexpected but nice result of the theory expounded in the following sections is that the partial indices can be obtained without complete knowledge of the factorization. This means that the structure of the kernel and image of the Wiener-Hopf operator associated with G can be determined without actually knowing a generalized inverse of the operator. This fact can be of value if, for example, the corresponding Wiener-Hopf equation is to be solved numerically. Another result of interest is the existence of a unilateral inverse for the Wiener-Hopf operator associated with G whenever it is Fredholm. The paper is organized as follows. In section 2 we give the notation and some known results that are needed in the subsequent sections.
In section 3 we give necessary and
sufficient conditions for the factorization to be canonical and determine the partial indices (if the factorization is not canonical).
The existence of a commutative factorization is
established in section 4 for the case where it is canonical and finally in section 5 we present explicit formulas for the factors.
2
Preliminaries For F a piecewise Lyapunov curve system, (cf. [16]), and 1 _< p < +oc, Lp(F) denotes
the Banach space of all complex-valued Lebesgue measurable functions f on I" for which Ifl p is integrable, with the norm defined by
Ilfl,p=(~lf(t)lPldtl) lip
200
C~mara and dos Santos
We define as usual the singular integral operator Sr on Lp(r), 1 < p < +c~ by Srf(t)
1 f r uf ( U = ~ri - )t du ' t er
'
where the integral is understood in the sense of Cauchy's principal value. It is well known that this is a bounded linear operator on Lp(F), for 1 < p < +e~. If r is a closed curve
Lp(r).
system or a straight line, S~. = I r where Ir denotes the identity operator on
For F = [R we define two complementary projection operators in Lp([R), p > 1, P+=I(I+S~) and we denote by
L~(P~) and
, P-=I(I-S~)
(2.1)
Lg (P~) the images of P+ and P - , respectively.
If f + E L+(R), it admits an analytic extension to the open upper half-plane C +, defined by 1 s f+(z) = 7~i
f(~)z d~
( z E C +)
(2.2)
and analogously, if f - C L~-(IR) it admits an analytic extension to the open lower half-plane r
defined by f-(z)-
1 f~ :(~)z d~ 2~ri -
(z C C - ) .
(2.3)
We also define L1~([R) as the space of all functions that admit an analytic extension to C • r such that [r
+ iy)l is integrable for each y E IR• and there is a constant M E R +
such that s162
, for all y e n
+
m
We denote by n+(f{) and n2~(P~)the spaces of all essentially bounded functions f r Loo(P~) that admit a bounded analytic extension to C + and C - , respectively. If F is a line parallel to the real or imaginary axis, we can also define two complementary hounded projections on
L~(r),
p > 1, by replacing S~ in (2.1) by St.
All the results
presented here for F = N correspond to analogous ones when F is a line of the latter type. In particular, when r is the imaginary axis Fi, we define pl =
+ sF,) , F :
89
sF,)
and denote by L~(ri) and L;(Fi), respectively, the ranges of pl and pT in L~(r~). The functions in L~(F~) and L;(I'i) admit analytic extensions to the open left half-plane e l = {z : Re z < 0} and the open right half-plane e ~ = {z : Re z > 0}, respectively (similar to (2.3) and (2.2)). A result from the theory of Lp spaces that will be used in the sequel is
201
C~tmara a n d d o s S a n t o s
the following (cf. [18]): if f E Lp(~) and g 6 Lq(PJ,p,q > 0, then f g 6 L,(~) where s is such that
1 p
I-
1 1 q s
Some other spaces will be needed as well. We define B~([R) (resp. B~-([R)) as the space of all functions f such that
r+f 6 L+(~)
(resp. r _ f 6 L~-(R))
with
r~(()--
i
, (ea.
We have (cf. [23]).
re(a) n ~(a)
= {0}
We denote by 7~(~) the set of all rational functions which are bounded in ~, as well as their inverses. C(~) denotes the space of all functions continuous in IR and, for ~ El0, 1], C~(R) denotes the space of those functions in C(~) which satisfy a HSlder condition with exponent ~ (in ~). To fix the notation concerning vector-valued and matrix-valued functions, we denote by
Xn(n r N) the linear space of n-dimensional vectors with components from a linear space X and by X nx~ the linear space of all square matrices of order n with entries from X. Let ~(A) denote the group of invertible elements in an algebra A. By a generalizedfactorization of G 6 G(Loo([R)) nx~ relative to Lp(~), p > 1, we mean a factorization of the form G(~) = G_(~) diag
~
G+(~)
(2.4)
with a 6 C + and #1 -*2 _< -.. _< #~ (#1, . - . , # n E Z), where the factors Ga: satisfy the following conditions (for p' = p/(p - 1)): (i) r+G+ E (L+(IR)) ~x~, r+G+ 1 6 (L+(Pj) ~x~ (ii) v_G_ E (L~-([R)) ~x~, r - G : ~ C (L~-,(~)) ~x~ (iii) G+~P+G--II is an operator defined on a dense subset of (Lp(PJ) ~ and has a bounded extension to (L,(POF. This factorization is said to be canonical if kl . . . . .
G C 6(L~o(R)) ~•
k~ = 0. It is well known that
admits a generalized factorization relative to Lp(P~) lit" the operator Tc: (L+(IR)) ~ + (L+(P~)) ~ , Tc~o + = P+Cqo +
Cfimara and dos Santos
202
is Fredholm. The factorization is canonical iff Ta is invertible (cf. [4], [16]). It is also well-known t h a t if G E ~ ( C ( ~ ) ) ~x~ then it admits a generalized factorization relative to any Lp(IR), p Ell, +oo[, and this factorization does not depend on the value of p. In that case we refer to it simply as a generalized factorization of G. W h e n G_ and G+ satisfy +
G+ E 6(Loo(R))
nXn
, G_ E ~(L~(IR)) - -
nXn
we say t h a t (2.4) is a bounded factorization of G. A m a t r i x function G E ~(Loo([R)) ~x= which admits a generalized factorization relative to Lp(R), p > 1, is called p - regular or p - non singular. In this case we associate with it a total index/* given by j=l
the integers/*j being called partial indices (see (2.4) for the definition of/*j). Finally, we need in the sequel p E C([R) such that p~(~) -
-i~
, a e p(o) > o
(2.5)
(where 3, 3' E P~ \ {0}, 5 =r 7). We shall denote by the same symbol p the extension to e \ F~ where F~ C Fi is the corresponding branch cut.
3
Partial
Indices
Let In be the identity matrix of order n and let R E R(P~) ~x~ satisfy R ~ = phil, where p~ is the quotient of two first degree polynomials, as defined in section 2 (cf. (2.5). Let d denote the class of all n x n matrix-valued functions of the form
G = alI~ + a2R + . . . + anR n-1 -- ~ aj[~j-1 j=l
where R is as above and al, a 2 , . . . , an E Loo(IR). Let furthermore t'1 = 1,/*2,.-. ,/*n be the n th roots of unity, /*j=exp((j-1) \
2rri
, j=l,2,...,n
n
W i t h these assumptions R ~ = p n I = is equivalent to
(R - H=)(R -/*~H=)--. (R -/*~Hn) = 0
.
(3.1)
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203
a n d it follows t h a t R a d m i t s n distinct eigenvalues Aj=#jp
(j=l,2,...,n)
corresponding to the characteristic e q u a t i o n )`n
=
fin.
We take as a representative of this class of matrix-valued functions the c o m p a n i o n matrix
R =
0
1
0
0
...
0
0
0
1
0
...
0
(3.2) 0
0
0
0
...
1
pn
0
0
0
...
0
i.e., R = [rij]i,j=l ...... where rk,j+l = 5kj for all k = l , 2 , . . . , n - 1
and j=0,1,...,n-1
rn,1 =- pn rn,j = 0
if j = 2 , . . . , n
.
m a t r i x - v a l u e d functions of J for which R is of the form (3.2). In the sequel we a s s u m e t h a t R is of this type. In fact, it can be shown (cf. [13]) t h a t all m a t r i x - v a l u e d functions of the class J, corresponding to the same f u n c t i o n pn, can be related with R by r a t i o n a l invertible matrices, R being defined as in (3.2). I f / ~ satisfies [l n = p~I, then =- Q R Q -1
where Q is any r a t i o n a l n x n m a t r i x of the form Q =
[kn-Jv]~=i for some v C C ~ \ {0}.
col
This is a well-known result in the case where n = 2 (cf. [8]). We can associate with each eigenvalue Aj the eigenvector n--1
Hj = (1,Aj,
)
as can easily be checked. T h u s R can b e p u t in the form (3.3)
R ~- H A H -1
with H =
1
1
..-
1
)'1
A2
"'"
)`n
...
)`~-1
A = diag ()`j)~ t
Camara and dos Santos
204
It is clear from (3.1) and (3.3) that G can be represented in the form G = H B H -~, B being a diagonal matrix-valued function B = diag
(bj)jn__l
(3.4)
where
bj = al q-/~ja2 -1- ... q- .~jn--1 an = al + #jpa2 + ... + #~-lpn-lan
(3.5)
are the eigenvalues of G. We will also assume that G C ~ ( C ( R ) ) ~x~ and all the eigenvalues bj admit bounded factorizations, i.e.,
G = H B _ C B + H -~ with B_ =
9 &ag
( b j _ ) j =nl
9
G(L2~([R)) ~xn, B+
=
(3.6)
diag(bj+)j=l 9 ~ ( L + ( R ) ) ~xn, C n
=
diag (rk~)j~=l where, for fixed a 9 C + and for kj 9 Z, = \~---~a]
forall { 9
.
(3.7)
The class of all such matrix-valued functions G, for which R is of the form (3.2) and p~ is defined by (2.5) will be denoted by Jc. Let us now consider the operator
T o : (L+([R)) ~ -+ (L+(P~)) ~ , Ta~ + = P+Ggo +
(p > 1) .
As a starting point for the analysis of the generalized factorization of G, we study its total index.
P r o p o s i t i o n 3.1. If G E Jc the total index of G is given by #=~
kj j=l
where kj is the index of the eigenvalue bj, for each j = 1 , . . . , n .
Proof: Since G C ~(C(~)=x=), it is clear that det G does not vanish in ~ and # = i n d d e t G = i n d d e t ( H B H -1) = ind 1-I bj = j=l
Theorem
kj . j=l
3.2. Let # be the total index of G C J~. Then Ta is injective iff # > 0 and
surjective iff # <_ O. In particular it is invertible iff # = O.
205
C~mara and dos Santos
Proof." Let us first consider the case where # = ~-~jn__1 kj
~_ O. The equation
T a T + = O,
with qo+ 9 (L+(~)) ~ (p > 1), is equivalent to G~ + = T - ,
with ~+ 9 (L+(~)) ~ , ~ - 9 (L;([R)) ~ ,
(3.9)
or, according to (3.6), CB+H-lc? + = BZIH-1 T-
.
(3.10)
In both sides of (3.10) we have vector-valued functions in (Lp(IR))~ of the form H-1~2 + ( H - I ~ + or H - I ~ - ) . Since
1 H-1
=
p-1
1
1_ n
p--n+l
. . .
/9 - 2
~21f1-1 #~-2p-2 ...
~t2nTlfl--n+ 1
#~1p-1
#nn+lfl-~+ 1
:
1
~n2p
-2
...
it is clear that each component in H - 1 T + has the form --1
( H - l i P * ) j = tP~l + #j P
= r
-]- t t n + 2 - j f l
-1
+
--1
4-
~22 + # 7 2 p - 2 ~ 2
--2
902 + P n + 2 - j , O
r
4"
--n+l
+ . . . + #j n-1
-]- "'" + # n + 2 - j f l
p
--n+l
--n+l,^+
Y~n =
:E
~n
(3.11)
In (3.11) we take, for convenience, #n+l = #1 = 1. If we denote by ~= the function which is obtained from the right-hand side of (3.5) when we replace p by p-1 and al, a2,.., a~ by -., T~, respectively, it follows from (3.11) that (H-1T=t=)j = fln+2-j =t=
, for j = 1 , 2 , . . . , n
,
or, equivalently, =
(3.12)
Here we once again identify the element corresponding to the subscript n + 1 with that corresponding to the subscript 1. Now, multiplying the left-hand sides of the n equations obtained from (3.10) and proceeding in the same way with the right-hand sides, we get n
%"
n
n
l'I(bj+)"
I-I(Y-l~+)j
j=l
j=l
n
= 1-I(b71) 9 I - I ( H - 1 T - ) j j=l
(3.13)
j=l
where # = ~j~=l kj > 0. It follows from (3.12) that (3.13) is equivalent to n
%"
n
n
n
Yi(bJ+) " I I ( f l ~ - ) = I-I(bj--1) 9 1-i(flJ) j=l
j=l
j=l
j=l
(3.14)
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206
Let/~ be the rational matrix which is obtained when we replace p in R by p-1. Then n
1-'[( / ~ ) = det(H diag ( ~ ) 2 = ~ H -~) = j=a
= det(~U+~+....~ The matrix ~ I
--
~
~7~n--1
).
+ c22~/~+ . . . + ~ 4- R~ n - 1 has the form
9 -
-
~ n - 1
Therefore, the left-hand side of (3.14) is analytic in the upper half-plane C + except (at most) for z = i5, if ~ > 0, while the right-hand side is analytic in C- except (at most) for z = i5, if 5 < 0. In any case, z = i5 is a pole of order less or equal to n - 1. Morever, both
sides of (3.14) represent functions which tend to zero when Izl --, +o~ in the corresponding half-plane. Thus both sides of
(3.14)
must be equal to
c0 + c ~ + . . . + Cn_2~ ~-2 (~ - i @ - 1
(3.15)
for some constants Co, C1,... C~-2. On the other hand G e ~(C(fi) ~x") and hence equation (3.9) can be considered in
any (Zp(~R))n, p > 1. Thus, we take p = n / ( n - 1). Since the lef-hand side of (3.14) is the product of n functions in L-~(~R), it is in L_~zr(R) and the same happens with the right-hand side. It follows that Co = 6'1 . . . . .
Cn-2 = 0 in
(3.15)
and therefore both
sides of (3.14) are identically zero. We will now prove that each ~ : is also identically equal to zero9 Let /?+(z) be defined in C + \ Fc by
~2-(z) = ~,+,(z) + G ' p - ' ( ~ )
~+~(~) + . . . + ,;~+'p-~+'(z)
~+~(z) .
Since we must have, from (3.14),
d=l
some t3+(z) must be equal to zero in a set A C @+ \ F~, with a positive measure. Let it be /~+(z), for instance. Then, by analytic continuation, /3+(z) = 0 in e + \ Fc, which implies that 3 + ( z ) = 0 in e +. It follows that c2+ = c2~ + .....
~o+ = 0.
C~mara and dos Santos
207
We have proved that if # > 0 then Ta is injective. If # < 0 we need only to consider the adjoint operator T~ = Ta*, where G* also belongs to Jc, and prove analogously that T~ is injective.
9
Now we s t u d y the partial indices in the generalized factorization of G. Theorem
3.3. Let G and # satisfy the assumptions of Theorem 3.2 and let q , m be
integers such that #=qn+m
, O
.
Then the partial indices in the generalized factorization of G are given by #j = q + m j where 0 < m j <_ rnj+ 1 _< 1, m j E Z f o r aIl j = 1 , . . . , n
(3.16) and ~]=t m i = m.
Proof." Let # be the total index of G and let # 1 , # 2 , . . . ,#~ be its partial indices (#1 _< #2 _< . . . _< #~). There are (uniquely defined) integers q and m such t h a t # = qn + m, 0 _< m < n. Let Go be defined by G = rqGo 9 We shall prove t h a t the partial indices m r , m 2 , . . . ,m~ of G0 (with mt < rn2 _< . . . <_
ran),
whose total index is m, satisfy the condition 0 _< m j <_ 1 (j = 1 , . . . ,n).
In fact, since m >_ 0 and Go E J~, it follows from Theorem 3.2 that mj >_ 0, for all j = 1 , . . . ,n. On the other hand, r - l G o has a total index - n + m which is less or equal to zero and, therefore, the partial indices in its generalized factorization are also less or equal to zero. It follows t h a t 0 _< mj _< 1 for all j = 1 , . . . , n and thus the partial indices #j of G are given by (3.16) with m j satisfying the conditions of this theorem.
Remark
9
3.4. T h e result that we have just proved shows, in particular, the stability of
the partial indices in a generalized factorization of G (cf. [4]) for G C J~.
4
Commutative
Canonical Factorization
Let us now assume t h a t G C d possesses a canonical generalized factorization. We show in this section t h a t in this case G admits a canonical generalized factorization with factors belonging to the class of matrix functions J. This means, in particular, t h a t these factors commute.
Cfimara and dos Santos
208
In what follows we also assume, for simplicity, that p~ 9 L+([R), i.e., 3' < 0 (see (2.5)). Identical results would be obtained if p~ 9 L~(R). T h e o r e m 4.1. Let G be in J and such that it admits a canonical generalized factorization relative to Lp(R), p > 1. Then there exists a canonical generalized factorization,
G = G_G+ with G_ and G+ in J, i.e., a commutative canonicaIfactorization. P r o o f i The equation ar + = 6- + r+l
with r
E (L+(I~)) '~x~, r
C (L~-(PQ) '~x~ and 1
r+(~) -has a unique solution (r
(4.1)
r
, for ~ E P~ ,
(4.2)
where 6+ = r+G; 1
(4.3)
6 - = r + ( 5 _ -- I~) ,
G_ and G+ being such that G = G_G+ is a canonical generalized factorization of G (relative to Lp(P~)) a n d ~_(i-y)
= In. We shall show that it is possible to renormalize the factorization G_G+ in such a way
that a commutative factorization is obtained. To this end let Ct,+, r the n elements in the last column of r r
1 , . . . , n) denote
and 6 - , respectively, and let
= 6~-j+l,n + , ~j- = r
(J = 1 , . . . , n )
.
(4.4)
Since G is of the form (3.1), we have from (4.1): n
E(ajRj-1)r
= ~ - -1- r+I n
j=l
n
~
n
n
E(ajRJ-1)[6+ - E ( ~ + R J - I ) ] = 6 - - E ( a j Rj-1) E ( 9 ~+Rj-1) + r+Zn j=a j=l j=l j=l n
G[r L
~
- E(
J+ R j-~ )l = r
j=l
- E ( c , RJ ') + r+ o j=l
where we took into account that -~
n
n
j=l
j=l
j=l
E ( a j R j-1 ) E(cpj+ R j-1 ) = E ( c j R J - 1 )
(4.5)
C~mara and dos Santos
209
with
n
+
Cl = al~+l -4- fl a n ~ 2 "4- p ~ a n - l ~ + + . . . c~
= .~+1 + al~+~ + P ~
c,,_l
+ p'~a2~ +
t +... + P'a~ t
+ ' + = a n _ l ~ l + + a,~_2p + + . " + allan-1 + g- " a nWn
c~=a~ol ++
a
+
~-1~2 + . . . +
a
+
+
2~-1+a1~
9
Considering the right-hand side of (4.5), we will now put it in a more convenient form by establishing a relation between the functions cj ( j = 1 , . . . , n) and CT,~(/= 1 , . . . , n). In fact we have, from (3.1) and (3.2) al
a2
...
an_l
as
pnan
al
...
an-2
an-1
G= flnan-1 :
flnan :
an-3 :
a~-2 :
pna3
pna4
al
a2
pna 2
pna 3
pnan
a1
..
(4.6)
and therefore the last column of the matrix function in the right-hand side of (4.1) is related to the last column of the matrix function in the left-hand side of (4.1) as follows. For the first element, alr
+ a2r
+
.+
an-lr
+
+ a.r
+
= r
which is equivalent to (see (4.4)) + ale? + + a 2 ~ _ l +
9 . + a o - , ~ ' ~ + + a=~'~+ = COn
Hence ~ = r Analogously for the second element we have P~a~r
+ a,r
+...
+ an-2r +
+ a ~-1r +
= r
which is equivalent to pnan~+ +
+ a + a i ~ , _ i + . . . + ,,-2c22 + a n - l ~ l+ = r
9
210
C~imara and dos Santos
Hence Cn_l z
r
and so on. Thus we obtain for all j = 2 , . . . , n
CJ=r
cl
r
.
Therefore (4.5) is equivalent to n
a
[r
n
- E(~?RJ-1)]
= r
- E(~TR
j=l
Since G = r
~-1) .
(4.7)
j=l
we can also write, from (4.7), n
G+ [r
n
- ~ ( ~j+ R ~-1 )] = ~ : 1
[r
- ~(9~-RJ-1)] .
j=l
(4.8)
j=l
The left-hand side of t'his equation represents a function which is analytic in the upper half-plane C +. However, the right-hand side of (4.8) does not represent a function analytic in the lower half-plane C - due to the term ~ -1 G_ ~-~(~j-- R j - 1 ) .
(4.9)
j=l
In fact, the matrix function ~ = I ( ~ - R j - l ) is in the same class as G and its form can be obtained from (4.6) replacing a l , a 2 , . . . ,an by 9ol ,r
,--. ,9~ respectively. Therefore we
see that all elements in the upper triangular half of (4.9), including the main diagonal, are analytic in C - due to the form of ~ = ~ ( ~ - R j-~) and to the condition C-~(i~/) = I,. However, those elements in the lower triangular half of (4.9) can have a pole of first order for z = i'~ due to the presence of the factor p~. This pole can be cancelled by adding to both sides of (4.8) an appropriate matrix function, as follows: n G+[r
n
-- E ( ~ J - I ) ]
--
r+A -
~-1[r
_
E(ctpy~j-1)]
j=l
j=l
where A is a constant matrix of the form
A
=
0
0
0
0
0[21
0
0
0
oL31
a3:
0
0
O/nl
f~n2
~n3
. . .
0
_ r+A
(4.10)
211
Chmara and dos Santos
The left-hand side of (4.10) is in (B+) ~x~, while the right-hand side is in (B1) ~• Hence they are both zero, so that r~ G+q~+ - G + ~ ( 9 9 j + R j - 1 ) = r + A j=l
(4.11)
r~
~-1r
_ ~_-a ~--~.(qo~-RJ-1) = r+A .
(4.12)
j=l
In equation (4.11), we can use identity (4.3) which yields G+r and write
= r+I~
n
-
= r+A. j=l
This is equivalent to
rt
G+ ~ ( c 2 + R j - l ) = r+(In - A) j=l i.e.~ n
(I~ - A ) - I G + = r+
--1
(4.13)
Analogously, from (4.12) we obtain n G _ ( I n - A ) = r~_ 1 . ~ - ~ ( ~ y / ~ j - 1 )
_}_ i n
.
(4.14)
j=l
Since the class J is inverse closed, it is clear that the right-hand side of (4.13) belongs to this class and the same is true for the right-hand side of (4.14). Therefore we have G = G_G+ = [G_(I~ - A)] 9 [(In - A)-IG+] where the right-hand side is a canonical generalized factorization of G with commutative factors belonging to the class J.
5
Formulas
for G+ and
9
G_
In this section we obtain explicit formulas for the factors G+ and G_ in a canonical
factorization of G as considered in Theorem 4.1. The main result is given in Theorem 5.6.
212
Chmara and dos Santos
However, before we come to t h a t point we show first t h a t the p r o b l e m of obtaining a generalized factorization of G E J can be reduced to t h a t of o b t a i n i n g a canonical generalized factorization plus the factorization of a r a t i o n a l function of the same class J. As in the preceding section, we assume t h a t p" C L+(IR). Proposition
5.1. Let G E J and let its eigenvalues bj,j = 1 , . . . , n admit a generalized
factorization. Then G = GoQ, where Go C J is such that its eigenvalues admit a canonical generalized factorization and Q e J is a rational invertible matrix in ( C ( ~ ) ) ~x~.
P r o o f : Let (j = 1 , . . . , n )
bj = bj_rk bj+
be a generalized factorization of bj (relative to Lp(ff{),p > 1) where rkj is defined by (3.7). Thus G is of the form (3.6), where we consider in p a r t i c u l a r the r a t i o n a l diagonal m a t r i x C. We have
rk'
(rk,~)
=- 1 ~
+ . . . + p n--1 # jn--1
+ p#j m=l
= 71 -t- p # j " / 2 't- . . . + p
with
}f't~r~ " ' r k m m=l
[ # m1--s r km]\
n-1
#j
n-1
~k
tO n - 1
]J
3`n ,
/
3`s =
\
~
j
for
~ = 1,..,n
m=l
Since rk,,(~) --+ 1 as I~1 --+ + o o , w e see t h a t for all s = 2 , . . . , n 13`s(~)[ =
nps_l(~)
1-s _ (#m rkm(~))[
---+ -n m=l ~ - s
I= o
,
as
I~1 -~ + o o
while 3`1 is a r a t i o n a l function such t h a t 3`1(() ~ 1 as I~[ ~ + c o . Therefore % E C ( ~ ) for all s = 1, 2 , . . . , n and for any (~ > 0 there is a rational function q~ (vanishing at infinity for s = 2 , . . . , n ) such t h a t , for K = max{[[p[]oo, []p2[]oo,..., ][p~[[oo}, fi In particular, we can choose ql = 3'1.
Now,
if we choose a sufficiently small ~, we have
for all j = 1 , . . . , n , ql + P#jq2 + ... + P
I[r~,
#j
q~ E
- (ql -]- P#jq2 + . . . + P~-l#2-1q=)l[oo =
= 11(3`~+ p#J3`~ + . , .+ p~-1#y-13`~) _ (ql + p~jq~ + . . . +
p~-l~;~-Xq~)lloo _<
Cfimara and dos Santos
213
< K (Ib2 - q2[l~ + . . .
+ [bn - q.l[~) <
and kj = ind vk, = ind(ql + pl~jq~ + . . . + p'~-l #2-1qn)
It is obvious, on the other hand, that Q=
H diag (ql -F P#jq2 -}-,
-..
~-
P
n--1
n--1
ttj
"~n
rr--I
q n ) j = l I1
"=
qlln + qzR + . . . + qnR n-1
satisfies the following: (i) Q
E
J
(ii) Q is a rational invertible matrix in (C(~)) ~•
(iii) Q-1 G J (iv) G = GoQ where Go = G Q -1 and all the eigenvalues of Go (which are of the form bj = bj(qa + p#jq2 + . . . + pn-a#}~-aq~)-l) admit a canonical generalized factorization
(relative to Lp([R)).
It follows from Proposition 5.1 that if we can obtain a factorization for Go with factors in the same class J, then G = Go-QGo+
and we can also obtain a factorization for G via rational factorization of Q. In general the rational matrix Q of Proposition 5.1 can be determined by inspection without great difficulty. We present here a procedure to obtain Q in the case where i7 = - i 5 (which allows some simplifications and actually corresponds to most applications). Let pn(~) = (~ + i7)/(~ - i7) and assume that p~ r_(~)
-
1 +i-,/
(~
E
L+([R), i.e., 7 < 0. Let furthermore
E
IR)
(5.1)
so that
For n > 3, the image of pr_ in the complex plane is a closed curve passing through the origin (for ~ = +oo).
Cfimara and dos Santos
214
#jpr_ in the complex pr_ by a rotation of 2rr(j - 1)/n. In fig. T h e image of
plane, for j = 2 , . . . , n, can be o b t a i n e d from that 1 we present the images of # j p r _ for j = 1 , 2 , . . . , n
when n = 3 and n = 5, a s s u m i n g 3' = - 1 :
Fig. 1 It is geometrically obvious that we can in each case choose a point a j such that the
#jpr_ has index - 1 relative to a j (cf. [10]) while all the other #~pr_ with s C { 1 , 2 , . . . , n } \ {j}, have index zero relative to that continuous function aj - #~pr_ has index 1 if s = j a n d index zero
curve corresponding to
curves, corresponding to point. Therefore the ifac{1,g,...,n)\{j}.
Hence, if for s o m e j E { 1 , 2 , . . . , n } 9
n
d,ag (bs)s= 1
we have ind by = kj ~ 0, we can write ,~
=
\rz
diag (b,),~=, diag ( ( a j - # , p , ' _ ) ' ) , = ,
where indb,=
indb,
for s =
{1,2,...,n}\{j}
ind bj = 0 . This procedure can be sucessively used to obtain marl ix ftmcliolls (,'o and Q such as described in Proposition 5.1. For n = 2 the same can be achieved in the following way. Assmuil~g that ind bl = 0, ind b2 = k r 0, then G =
GoQ where
Q=H [ (r+ l pr-)kO ~--i
,-,(~)=~+--
~+/
(7"1 Dr=)~01 tf-l=(rll+r-R)k (~)
,
C~,mara and dos Santos
215
a being a constant such t h a t i n d ( r l + pr_) = 0 i n d ( r l - pr_) = 1 . Now we i n t r o d u c e some n o t a t i o n which will be needed for the next theorems. For f such t h a t r + f C Lp(P~),p > 1, we define &
IIzo f = A~oP+(A~-olf) where Zo is a fixed point in r \ •
(5.2)
and Azo(~) = ~ - zo (~ E R). For any Zo E r \[R, II zo + and
II~-o are c o m p l e m e n t a r y projections. In particular, when we restrict the d o m a i n of II+zo to C~(l~), o E]0, 1[, II+o and Hb0 are c o m p l e m e n t a r y b o u n d e d projections. Let now G E J i G = H B H -1 as in section 3, see (3.4) and (3.5). We define 4- = H diag (IIz0 4- a 1 + #jpII~zoa2 + .. 9 + #jn-1 Pn - l I I zo q- a ~)j=l ..... IIzoG u
Proposition
I
(5.3)
5.2. Let G @ J~. Then all the eigenvalues bj are in C ( ~ ) and have the
same limit as l(l ---* +00" Proof:
Since G E ( C ( ~ ) ) nx~, we must have aj E C ( ~ ) for all j = 1 , . . . , n (see (4.6)).
Thus bj E C(IR) and it has finite limits as ~ --+ 4-00, since it is of the form (3.5). The existence of a b o u n d e d factorization for bi now implies t h a t h i ( - 0 0 ) = bj(+00) (of. [6]). Moreover, H(-00)
= H(+00)L
with
L =
0
1
0
0
0
0
i
0
0
0
0
1
0
0
0
1
...
Therefore G(+00) = G(-00)
r H(+00)B(00)H-l(+oo)
= H(-00)B(00)H-l(-00)
r
H(+00)B(oo)H-1(+oo)
= H(+00)LB(oo)L-IH-~(+00)
r
B ( 0 0 ) -~ i B ( o o ) L -1
216
C~,mara and dos Santos
This means that
bn(oo ) = bl(OO )
bl(oo ) = 52(00 )
b,,_,(oo) = b,,(oo)
.
Proposition 5.2 is useful to obtain explicit formulas for the factorization of G.
We
consider first a simpler case, where G belongs to a decomposable algebra (cf. [4]). Theorem
5.3. Let G E J~ N (C~(R)) ~x~, c~ E]0, 1[ and let bj(j = 1 , . . . ,n), defined as
in section 3, admit a bounded canonical factorization.
Then a canonical factorization of G
is given by G = G_G+ with
G_ = exp(n;~ log C) c + = exp(n
log c )
(5.4)
(5.5)
.
P r o o f : Since the eigenvalues b j , j = 1 , . . . , n, admit a bounded canonical factorization we have
d e t G ( ~ ) = 12Ibj(~) r
for all ~ e ~
j=l
and log G = H (log B ) H -1 where log B = diag (log bj)~=a C (C~(~I)) ~x~. Moreover, we can represent log bj in the form logbj = l~ + p f j l 2 + 9 9 + pn-l f2-11n with
1 I ~ - nPs- 1
since ~ = 1
r
s
(#~-s log b,~
rr~=l
,
for s = 1,
...
,n
(5.6)
= 0 for s r 1.
It follows from proposition 5.2 that all the functions log b,~, m = 1 , . . . , n, have the same limit as I{[ ~ +oc. Hence ls C C~(~)
for all s = 1 , . . . , n
.
(5.7)
It is therefore apparent from (5.6) that, as 1{I + +oz, 11 --* logbl(o~) and I, + 0 for S ~ 2,...
~r/,,
Cfimara and dos Santos
217
We have log G = H diag (ll + pfjl2 + . .. + p
n--a
n--ll
#j
\n
lrT--1
~n)j=an
with ls defined by (5.6) and satisfying condition (5.7). On the other hand we can write log G = II~ log G + Hi+ log G where, according to (5.3), IIi~ log G = H diag (IIey/1 + + #jpII~lx + . . . + #jn-a p n-1 lli.ytn)j=lll rr-t- . . . . . . 1
+ , (II a= , ~l. ,) R n-1
= (n,V,)I+ (ntj~)n+
Let C;(IR.) = C~'(~) gl L+(IR) , C_~(~) = C~'(IR) n L;(IR). + It is clear from (5.7) and from the definition of ll~l~, s = 1,...,n
(see (5.2)) that,
assuming p" G L+([R), as we did in the beginning of this section, II+~ log G E so that G+ = exp(II + log G)
e g( c +~ (~) ) ~•
On the other hand we see from (5.2) that
(tI~lj)(i~) = 0 hence ( [ I ~ l j ) R j-a E (C_~(~)) ~•
(5.s)
for all j = 1 , . . . ,n. Therefore
n:~loga c (<(a))n• and G_ =- e x p ( I I ; log G)
e
6(CZ(~)) "•
It follows that G = G_G+ is a (bounded) canonical factorization.
9
Next we show that for G E J~ (but not necessarily in a decomposable algebra) formulas (5.4) and (5.5) still hold if G has a canonical generalized faetorization. We begin with some auxiliary results. L e m m a 5.4. Let f satisfy the following conditions f
f--1
L~(a)
A~- , ~ L~(~)
(5.9)
218
C~mara and dos Santos
for p > 1, p' = p l ( p - 1 ) ,
n 9
n > 2 and
A•177 Then
log f -~ L~(a) ~:~
for ~11 ,7 > 1 .
P r o o f : Let ~ = f/A+ and (* = A~_/vq~ . Then, since c2 9 L+(~), we have (5.1o)
On the other hand, it follows from (5.9)
that
~,7'%/",~-' 9
/~7n(19-1 9 Lp+,(P0 and thus
s~(a)
.
This in turn is equivalent to
A;('~-2/P)P'(* -p' 9 L+(PQ whence A+(1-21(~'))~r-l/" 9 L+p,(a) .
(5.11)
Let now k = n p and let k' and s be real numbers such that 1
1
~+~r
1
=1
'
1
-8+ - - =g tkp I
1
=
(5.12)
which yields k' = npt(np - 1) and s = n l ( n - 1). Then
A+21U(,-,in = X+(:-21k) G-*l~ = ),+, (A+Cl-2/k)a-~/,~) and the right-hand side of (5.13)is in s
(5.13)
(P0 because A+1 9 L+([R), A+('-~/k)er -1/~ 9 L+v,([R )
according to (5.11) and ~1 = ;, + @ according to (5.12). Thus we have
A;2/k' (*-V~ 9 L +, (JR) . Defining afro) = (* (\ i~~ -+11))
, for ~ 9 1 4 9 1 6 2
and denoting by H;(ft) the Hardy space of all functions analytic in ft such that sup 0<,X
/;
IF(a?~ ~r
< + o o (p > 1)
(5.14)
C~,mara and dos Santos
219
we see that (5.10) is equivalent to
gr e H~(fl) while (5.14) is equivalent to ~--lln
~
Hk,(Ft)
(cf.[ll]). Since ~ and ~--1 n e v e r vanish in f~, we can define log & as an analytic function in
Hq(f~) if
a (of. [1]) and it will be in
sup 0_),<1
f
IRe
log~(Ad~
< -t-oo
(5.15)
r
(cf. [7], [21]). We will now prove that (5.15) holds for any q > 1. Defining, for each 9 [0,1[ [1 = {O 9 [--7r, Tr] : ]5(Aei~
~ 1}
I~ = {0 9 [ - ~ , ~ ] : 0 < I~(~d~ we
< 1)
have
Z IRe log~(:,~~ ir
=
7r
Ilog I~(~r176 q dO =
<_M1 fz [#(Aeie)]PdO+M2~ lh-l/n(~eie)]k'dO < 1
< M~
[
2
[5(Ae~~ 7r
+ M2
f
](T-1/n(Ar
dO
,
71
M1 and M2 being positive constants. To obtain this we used the fact that (log q x)/x p is a bounded function in [1, + ~ [ for all p, q > O. Since # E Hp(fl) and &--l/n E H # ( ~ ) , log ~ is in Hq(fl) and it follows that A+2/q log a 9 L+([R)
(5.16)
.
Therefore, if q >_ 2, (5.16) yields A+~log~r 9 L+(IR).
This implies, in turn, that
.~_1 l o g f 9 L+(~[). If 1 < q < 2 the same conclusion can be reached considering that A+~ log ~r =
A+(~-2/k)(A+2/klog a)
(5.17)
220
C~mara and dos Santos
with k = 2q/(q - 1). The first factor in the right-hand side of (5.17) is a function in L+~ (R) and the second factor is in L+(IR) (according to (5.16)). Therefore the product in k--3
the right-hand side of (5.17) is in L+(N) because k-3
1
~-+k
1
q
L e m m a 5.5. Let G 9 (L~(R)) "~x~ admit a canonical 9eneralized factorization relative to L~(R),p > n, G = G_G+. I f det G admits a bounded canonical factorization, then
det G+ 9 GL+([R)
,
det G_ 9 GL~o(~ ) .
(5.18)
P r o o f : Let det G = d_d+ be a bounded canonical factorization, i.e., d+ 9 g L + ( a )
,
d_ 9 g L L ( a )
.
We have, on the other hand, det G = det G _ . det G+ whence d+ det G+ 1 = d -z det G_ . Now, det G+ 1 is a sum of products of the form gig2 r+gj 9 L+(l{)
9
9
, j= l,...,n
9
(5.19)
gn where each factor satisfies .
Thus r ~ g l g 2 . . . g n 9 L+/~(P~), and it follows that r ; det G+ 1 9 L+p/~(a) .
(5.20)
Analogously we obtain (5.21)
r2 det G_ C L~(R) n
On the other, we see from (5.19) that r ; d + det C ; 1 _
(5.22)
= r ; d -1 det C _ -
j=l
j=l
where K j , j = 1 , . . . , n, are constants defined by K~_j
-
(j
!
1 ) ! { d~[d-'(z) z-
deta_(z)]}~=~
C~mara and dos Santos
221
It follows from (5.20) and (5.21) that the left-hand side of (5.22) is in B + while the righthand side is in B~-. Therefore they are both zero. This yields d+ det G+ 1 = ~
=
d -1
det G_
(5.23)
where ~ is a polynomial. However, this polynomial cannot have any zeros in the upper half-plane r
(due to the left-hand side of (5.23)), or in r
(due to the right-hand side of
(5.23)), or in IR. Hence ~ must he a constant K aad we have det G+ ~ = Kd+ 1
E gL+(IR) EgL~(N) .
det G_ = K d _
T h e o r e m 5.6. Let G E J~ and let the eigenvalues bj(j = 1 , . . . , n ) admit a bounded canonical factorization. Then a canonicaI.factorization of G is G = G_G+ where G_ and
G+ are given by formulas (5.4) and (5.5) respectively. Proof." If G_,G+ E J (see theorem 4.1), then G+: E 3 and G_ = ~o~-I+ ~ - R + . G+ 1 = ~i+I + c2+R +
. + c2~R~-1 +
(5.24)
n-1
(5.25)
where ~ y , ~ + ( j = 1 , . . . , n ) s a t i s f y
r_~; EL;(a)
, r+9+ EL+(~) forall p > l a n d ~ - f ( i T ) = O f o r j > _ 2
,
(5.26)
r+ and r_ being defined by (4.2) and (5.1), respectively. Let us take p = n. We can also write, from (5.24) n H-i G - = H d i a g ( r J)y=l with ej defined by
I@~'~l ~-#JP~92 ~-''''b#j --
n--1
P
n--1
~n , j = l , . . . , n --
Analogously, from (5.25), G+ 1 = H diag (~j)j=IH ~ -1
.
(5.27)
C~mara and dos Santos
222
with rb defined by ,j=
n-1
l
p
n-1
+
, j=
l,...,n
(5.28)
.
Since G G +1 = G_, we have log G + l o g G+ 1 = log G_
(5.29)
log G = H diag (log b j ) ~j = l H -1
(5.30)
where
log G_ = H diag (log r
~
-1
(5.31)
log G + I = H diag (log rlj)~:l H - 1
(5.32)
.... 1 log G_ = H diag (l~ + #dpl~ + . . . -• #jn-1 p n-1 .q~. .)j=11-1
(5.33)
From (5.31) we obtain
with l~- defined by 1
Iy -- npJ_ 1
[log ~bl + #;0-1) log ~b2 + .
..
+ ~n (j-1) lOg
r
(5.34)
for j = 1 , . . . ,n. Analogously, from (5.30) we obtain rr--1 log G = H diag (ll + #jpl2 + . . . + #jn - - 1 p n - - 1 7l ~~n )j=ln
(where la, I2,. 9 9 In are defined in terms of log bl, log b e , . . . , log b~ by expressions similar to (5.34)) and from (5.32) we obtain log G+ a
~_
'ln 14---1 H d i a g (l + + p j p l + + . . . . .A- #jn - - 1 Pn - - l j +o,~jj=l--
with ll, + 12 + , . . . , l+ defined in terms of log rh, log r/2,..., log r/~ in a similar way. Thus, (5.29) yields n--1
diag(ll + #jple + . , . + #j
p
n--17 ".n
n--1
L,~)j=I+ diag(t + + # j p l + + . . . + #j n--1
= diag (1l + # j p l 2 + . . . + # j
p
p
n--l~-~',n
~,,~)j=a =
n--ll--',n
q~)j=l 9
(5.35) On the other hand we also have (see (5.2)) n--1
diag (11 + #jpl2 + . . . + #j
p
n--ll
~n
~n)d=l =
-n = diag (II~lt + #jpl]~12 + . . . -t- #dn - - 1 p n--1 II~.rl~)j=~+
"r~--I n--1
+ diag (II+/1 + #dpII+l~ + . . . + #d
P
+
n
rIi"/ln)j=l "
(5.36)
223
C~mara and dos Santos
It follows from (5.35) and (5.36) that p o - 1 (l,+ +
diag ((l + + II+l,) + I~jp(l + + II+/2) + . . . + #j :
diag ((li- - II~h)
+
.jp(I;
--
II7~/2 )
-~-
. . .
n--1
-~ #j
p
n--l/l--
[l n
,j=l l
-- n~
=
n n))j=l
The last identity now implies that , for all j = 1,. .., n .
1+ + IIi.~I i + = 12 -- H ; l j
(5.37)
According to Lemma 5.7 (next to this proof), r_l; e n;(~:)
, r+l + e L + ( N )
, for all j = 1 , . . . , n
.
Therefore the left-hand side of (5.37) satisfies
while the right-hand side satisfies r_(l; - II~li) 9 n;(a)
.
Therefore, both sides of (5.37) are constant (cf. [23]). However, this constant must be zero, since it must be equal to zero for z = i'~. In fact we have, on one hand, (5.8). On the other hand, we can assume withofit lack of generality that ~ - ( i ~ ) = 1 and, since the values ~ - ( i 7 ) , . . . ,qo~(i~f) must all be equal to zero (see (5.26)), it follows from (5.27) that r
= 1 and thus tT(i"/) = 0. So, 12(i"/) - II~/j(iT) = 0. Now, since both sides of (5.37) are zero, we have = --~Ii~l j
l}- = II~lj whence we obtain from (5.33) I ,~)j=l ,n H-1 log G_ = H diag (II~/1 + #jpH~12 + . . . -~- #jn - 1 p n-~ H -i-~
According to (5.3) this means that log G_ = I I ~ l o g G which yields (5.4). The analogous formula (5.5) for G+ is obtained in a similar way.
224
C~mara and dos Santos
L e m m a 5 . 7 . Let G satisfy the assumptions of Theorem 5.6 and let l~,l + be defined, for j = 1 , . . . , n , as in the proof of that theorem. Then
Proof." The proof involves two steps; in the first step we show that r_ly (j = 1,... n) is a function analytically extendible to the whole lower half plane C-; in the second step we prove that this function actually belongs to L~([R) which is a deeper result. (i) For convenience of exposition we shall consider instead of p a new function/5 satisfying /5~ = p~,/5(0) = p(0), analytic in e \ I'c where the branch cuts ['c are shown in Fig. 2.
Im
z
re i8 m
Re z
I'c
Fc Fig. 2 Since the two functions coincide on a strip containing the real axis the functions Iy, l+ are given by the same expressions with p replaced by/3. Let ~s and 05 be the functions pi, qj with p replaced by/5 (cf. (5.27), (5.28)). It should be noted that this does not affect ~ and r/which were defined on IR but affects their analytic extensions as the branch cut changes from Pc to f'c- We have, for 17 (el. (5.34)),
n/sJ -1
log fl
(~)~
( ~ = ;k-')
D
77Z~1
As we cross the branch cut l)c, /5 changes by a factor #2; hence ~ - changes to @j-+l and ~
introducing the factor •tt2 in front of log into the exponent of ( ~ + 1 ) ~m changes vm into urn+l, i.e., the product in the argument of the logarithm remains invariant. This shows that the function l~- is continuous across the branch cut. The theorem of analytic continuation now yields the analiticity of lj in the whole open half plane C-. This argument can be repeated for l+. (ii) Next we show that r_l7 ~ L;(A). To this end we need several auxiliary results which we proceed to prove.
CAmara and dos Santos
225
To begin with we show that there exists a constant M > 0 such that
f+_~ l~_(x § iy)r;(x + i y ) F d x
<_ M for
all
y ~]-~,0[ .
(5.38)
This is obviously true for j = 1, since (cf. (5.27) and (5.34)) l{ = l ( l o g ~1 + l o g ~2 + . . . + l o g On) = = 1 l o g (~b1~2...~bn) = 11og (det G_) n
n
and, from (5.18), det G_ 9 G L ~ ( ~ ) . So we prove (5.38) for j = 2, considering that the proof is entirely similar for any other value of j. We have l~- = l ( l o g
r
+ ~t21 log r + . . . + #n I log ~2n)
Q
The identity G G +1 = G_ leads to diag (bj)j~=l . diag (~J)j~l = diag (r
-
(5.39)
where yj and Cj(j = 1 , . . . ,n) are defined by (5.27) and (5.28). Considering that each bj admits a bounded canonical factorization, we can also write, from (5.39), bj+rlj
=
by~,j
for all j = 1 , . . . , n
.
It follows (cf. [3], [23]) that we can define functions M j , j = 1 , . . . , n, analytic in the whole cut plane C \ F~, such that My =
bj+rlj -1
in
r
\ r~
b~_r
in
e-kr~
(5.40)
Let us show that (of. section 2 for the definition of L~(FI)) (5.41) r~-lM-lt
j
9 Lt/(Fi)
(5.42)
where, for a given a 9 C + n C l, 1 ~(t) -
t + a ' (t 9 r~)
.
As regards (5.40), we see that M j is analytic in C ~ for all j - 1 , . . . , n, the same being true for its inverse, because M2M3...M,~ M{-1 = M I M 2 M 3 . . . M n
..
'"
M~
'
1 =
M1M2...M,~-I M1M2...M,~-IMn
(5.43)
226
C;imara and dos Santos
where
1-I(bj+)detG+ 1, M~M2M3... Mn =
in
e lAe +
j=l
~ 1-I (b;_~) det G_ ,
(5.44) in
e ' C-1e -
j=l
does not vanish in C ~. On the other hand, since r_p~- 9 Lp-(PQ, see (5.26) r+~ + 9 L+(IR), it follows from a result of M. Riesz (cf. [20]) that L +~ Ir+~-(x0 + iy)lPdy _< JV1 ;
[r+~+(()l pd~
for all xo < 0, where N1 and N2 are constants depending only on the value of p. It follows from here that lT M (x + iy)l p dy <_ N for all x < 0 ,
where N is a constant (independent of the value of x) and therefore we have r, Mj 9
L~(ri),
for all j = 1 , . . . , n . From (5.43) and (5.44) we now obtain, as in the proof of Lemma 5.5
~?-'M 7' 9 U_~_, (rd:_ = U~,~ (rD = L',(r{) . As a consequence of (5.41) and (5.42) we have, according to Lemma 5.4, r~ log Mj 9 Ltp(Fi) , for p > 1 Resorting once again to the above mentioned result of M. Riesz (cf. [20]), we conclude that rl log Mj satisfies
i?
lrtlogMj(x + iyo)lPdx <_ co
_< N,/~, I,', log MJ(~)lPd(
It follows that, in particular for ~ < y0 < 0, _+51 r_ log ~j(z + iyo)l'dx < N~
where N2 is a constant. Thus it is also true that there exists N3 > 0 such that
i+( l, ,r(x + iy)l p dx < Na
(5.45)
227
C~mara and dos Santos
in the strip 7 < I m z < 0. A similar result is true for
fj~
lr+lf(x + iy)] p dx
in the strip 0 < I m z < 5. (iii) Let F be the line y = 7, oriented as usual (from left to right).It is clear t h a t
r_~j e L~-(F),
for all j = 1 , . . . , n
.
(5.46)
Furthermore, ~b1~2~3... ~ Since r - r
det G_
, r - ~ n E L~-(F) for all p > 1, if we take p = n we obtain -
<_-~ r
Cn e L2(F)
n
_
with s - - n-1
p,
and thus r2-1r -' C L ~ ( F ) .
(5.47)
It follows from (5.46) and (5.47) that, according to Lemma 5.4, r_ l o g ~ l E L~(F) . Analogously, we can prove t h a t r _ l o g ~ j E L ; ( F ) for all j = 1 , 2 , . . . , n
.
Therefore, see (5.34),
r_lj ~ L;(r)
for all j = 1 , . . . , n
.
Now, it follows from the boundedness of (5.45) in 7 < I m z < 0 and (5.48), t h a t L~-([R), in view of the analiticity of The relation
6
r+l+ E L+(R)
r_l-j
(5.48)
r_l-~ E
in C - as shown in (i).
can be obtained in a similar way, for all j = 1 , . . . ,n.
,,
Example As an example we give the canonical factorization of a 2 x 2 matrix function of the class
considered here. In this case it is possible to present the factors G• given by (5.4) and (5.5),
C~mara and dos Santos
228
in such a way that they involve only algebraic functions and some known transcendental functions which are not more complicated than those appearing in the original matrix symbol itself. It should be noted that the generalized factorization of 2 x 2 matrix functions of this type was studied in [14], but a complete study, leading to explicit formulas, was carried out only for a family of functions which does not include the symbol considered below (although the method given in [14] could be extended to include the present example). Let
+ 2i +i
p2(~)
(1 ~ + i~ , t(~) = th \ ~ - ~ - - ~ 1
,(ER,
The eigenvalues of G are given by dl = 1 - t and thus
in
(did2) =
dI ln~(~) 9
,
d2 = l + t
In (1 - t 2) ~ -'}-i
(~ za)
Using this in (5.4) and (5.5) we obtain
G+ = [ ch (pb+) P-lsh(pb+) 1 p sh (pb+) ch (pb+) with
pb+(~)-
2
v'g (~ - i)
G- =sech (1-~+i~ [ psh(pb-) p-ash(pb-)
with
~4U4-~ 4~ + 2i
C~imara and dos Santos
229
REFERENCES [1] Ahlfors, L. : Complex analysis; Mc-Graw Hill, 1979, 3rd ed. [2] Cs M. C., Lebre, A. B. and Speck, F.-O.: Generalized factorization for a class of Jones form matrix functions; accepted for publication in Proc. R. Soc. Edinburgh. 1992. [3] Carleman, J.: L'int~grale de Fourier et questions qui s'y rattachent, Uppsala, Almquist & Wiksells, 1944. [4] Clancey K. and Gohberg, I.: Factorization of matrix functions and singular integral operators; Birkh~user, Basel, 1981. [5] Daniele, V. G.: On the solution of two coupled Wiener-Hopf equations; SIAM J. Appl. Math., 44 (1984), 667-680. [6] Duduchava, R.: Integral equations in convolution with discontinuous pre-symbols, Leipzig, 1979. [7] Duren, P.: Theory of Hp spaces, Academic Press, 1970. [8] Hurd, R. A.: The explicit factorization of Wiener-Hopf matrices, Preprint 1040, Fachbereich Mathematik, Technische Hochschule Darmstadt, 1987. [9] Jones, D. S.: Commutative Wiener-Hopf factorization of a matrix, Proc. R. Soc. London A 393 (1984), 185-192. [10] JSrgens, K.: Linear integral operators, Pitman, 1982. [11] Koosis, P.: Introduction to Hp spaces, London Mathematical Society Lecture Notes Series 40, Cambridge University Press, 1980. [12] Lebre, A. B.: Factorization in the Wiener algebra of a class of 2 x 2 matrix functions, Int. Eq. and Op. Th., 12 (1989), 408-423. [13] Lebre, A. B.: A note on the reduction of Jones-form n x n matrix functions to a special form, Manuscript. [14] Lebre, A. B., dos Santos, A. F.: Generalized factorization for a class of non-rational 2 x 2 matrix functions; Int. Eq. and Op. Th., 13 (1990) 671-700.
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[15] Meister, E. Penzel, F.: On the reduction of the factorization of matrix functions of Daniele-Khrapkov type to a scalar boundary value probelm on a Riemann surface, Preprint 1351, 1991, Fachbereich Mathematik, Technische Hochschule, Darmstadt. [16] Mikhlin, S. G. and PrSssdorf, S.: Singular integral operators, Springer; Berlin 1986 (in German, 1980). [17] Moiseev, N. G.: On factorization of matrix-valued functions of special form, Soviet Math. Dokl. vol. 39 (1989), n ~ 2, 264-267. [18] Okikiolu, G. O.: Aspects of the theory of bounded integral operators in LP-spaces, Academic Press, 1971. [19] PrSssdorf, S., Speck, F.-O.: A factorization procedure for 2 x 2 matrix functions on the circle with two rationally independent entries; Proc. R. Soc. Edinburgh, 115 A (1990) 119-138. [20] Riesz, M.: Sur les fonctions conjugu~es, Math. Z., vol. 27 (1927), 218-244. [21] Rudin, W.: Real and complex analysis, 2na ed., Mc Graw-Hill. [22] Teixeira, F.: Generalized factorization for a class of symbols in [PC(~)]2•
appl.
Anal., 36 (1990), 95-117. [23] Widom, H.: Singular integral equations in L~, Trans. Amer. Math. Soc., 97 (1960), 131-160.
Departamento de Matems Instituto Superior TScnico Av. Rovisco Pais 1096 Lisboa Codex Portugal
A M S 47B35,45EI0 Submitted; June 16, 1993 Revised; April 18, 1994
or
Centro de An~lise e Processamento de Sinais Complexo I do I.N.I.C. Av. Rovisco Pais 1000 Lisboa Portugal