Applied Mathematics, 2010, 1, 283-287
doi:10.4236/am.2010.14036 Published Online October 2010 (http://www.SciRP.org/journal/am)
Analyticity of Semigroups Generated by Singular
Differential Matrix Operators
Ould Ahmed Mahmoud Sid Ahmed1, Adel Saddi2
1Department of Mat hem at ic s, C oll e ge of Scie nce, Al j ouf University, Aljouf, Sa u di Arabia
2Department of Mathematics, College of Educa tion for Girls in Sarat Ebeidah, King Khalid University, Abha, Saudi
Arabia
Received March 9, 2010; August 3, 2010; August 6, 2010
Abstract
In this paper we prove the analyticity of the semigroups generated by some singular differential matrix op-
erators of the form 2
2
()
=,
dBxd
x
dx dx  in the Banach space ([0,],( )),
d
CMC

 with suitable boundary con-
ditions. To illustrate the work an example is discussed.
Keywords: Dissipative Operators, Positive Operators, Spectrum, Analytic Semigroups, Evolution Equations
1. Introduction
As we will see in the sequel the problem of characte-
rizing operator matrices generating strongly continuous
semigroups or analytic semigroups is quite difficult. Th e
main problem consists in finding appropriate assumptions
on the matrix entries allowing general results but still
including the concrete examples we have in mind.
The evolution of a physical system in time is usually
described in a Banach space by an initial value problem
of the form
0
() ()=0, 0
(0) =.
dU tUt t
dt
UU





 (1.1)
Problem of type (1.1) is well posed in a Banach space
X
if and only if the operator (,())D generates a
0
C-semigroup >0 ()
tt
Ton
X
. Here the solution ()Ut
is given by 0 ()= t Ut TU for the initial data 0()UD

.
For operator semigroups we refer to [1-5] and to [6] for
the theory of op erator matrices. The harmonic analysis
for a class of differential operators with matrix
coefficients was treated in [7,8]. In this work we are
interested in a generalization of the analyticity and the
positivity of the semigroup generated by a matrix
singular differential operator (,())D. A similar
study was realized in [9] for a class of differential
operators with matrix coefficients and interface. For the
scalar case we refer to [10].
This paper is organized as follows. In the second
section we introduce some notations and give prelimina-
ries results. In the third section we investigate some
properties of the operator (,())D in particular we
prove that it is closed, densely defined, dissipative and
satisfies the positive minimum principle. Some spectral
properties of the operator (,())D are obtained in
this section. In section fourth we established the
analyticity of the semigroup generated by an operator in
the form (,())D. In this case the problem (1.1) has a
unique solution for all 0 UX. In section five we
present a concrete example of application of the results
obtained.
2. Notations and Preliminary Results
Let ()
d
M
C be the space of dd matrices with
complex coefficients and d
I
the identity matrix. The
norm of a matrix


1,
=()
ij d
ijd
A
aMC

 wil l b e defi ned
by

1,
|| ||=||
max ij
ijd
A
a
 (2.1)
The reason of this special choice will be justified in
Lemma 2.1 and Lemma 2.2.
A matrix


1,
=()
ij d
ijd
A
aMC

 is called nonnega-
tive (resp positive) if all entries ij a are real numbers
and nonnegative (resp positive). In this case we write
0A, (.>0)respA and we call

1,
||=| |
ij ijd
Aa



O. A. M. S. AHMED ET AL. 284
the absolute value of
A
.
For

1,
=()
ij d
ijd
A
aMC

 and

1,
=ij ijd Aa


()
d
M
C, we will write
0.ABifAB
and
>>0.ABifAB

A matrix-valued functions is said to be continuous,
differentiable or integrable if all its elements are continuous,
differentiable or integ r able fu n ctions. If th e matrix ()
A
x
is integrable over [,]ab, then
()().
bb
aa
A
xdxAx dx

For two matrix functions

1,
()=()
ij ijd
Axa x

 and

1,
()=()
ij ijd
Bxb x , we shall write the negligibility
0
()=(()),
A
xoBx xx if,

0
()= ()
ij ij
axobxxx
for all 1, .ij d Similarly we define the notions of
domination ()=(())
A
xOBx and the equivalence
()~ ()
A
xBx (see for more details [11]).
Let
X
be the space ([0,],( ))
d
CMC

 of continuous,
matrix-valued functions on [0, ] . On the space
X
we define the norm ||.|| ,
X by 0
||||= sup ||()||,
Xx
f
fx
 for
all .
f
X
Note that the normed space (,||.||)
X
X is a Banach
space.
We denote by (]0,],( )),=1,2,
kd
CMCk
  the
space of all ktimes continuously differentiable matrix-
valued functio ns U defined on ]0, [ such that
()
lim( )
p
xUx  exists and finite forall0pk.
Consider a singular second order differential operator
(,())D with matrix coefficients defined on
X
by
2
2
()
=,
dBxd
UU U
x
dx dx 
where Uand Bare matrix-valued functions, with the
domain
2
0
()= {([0,],())(]0,],
()),()= 0}.
lim
d
dx
DUCMC
MC Ux


 

We assume that B is a real valued continuous and
bounded matrix-valued function on [0,[ and if
(0) =d BI we add the assumption
|() (0)|
D
RxBx B (2.2)
in a neighborhood of 0, for nonnegative constants
diagonal matrices R and D. Here
D
x
is the diagonal
matrix with diagonal entries =,=1,2,
di
i
x
id  ,
where , =1,2,,
i
di d are the diagonal entries of D.
We will now recall some results needed in the sequel.
Lemma 2.1 Let ()
d
A
MC

. The following properties
hold
1) 0A if and only if 0AB  for all 0.B
2) ||||||,
A
BAB

hence | |||||
A
BAB if 0A.
Lemma 2.2 Let ,()
d
A
BMC

. The following proper-
ties hold
1) ||||
A
B

implies .
A
B
2) ||= .
A
A
3) |||||| ||,
A
EdA

where


1,
=,=1,1,.
ij ij
ijd
Ee eijd
 
Proposition 2.1 Let
A
be a nonnegative matrix with
spectral radius ()rA.
1) The resolvent (,)RA

is positive whenever
>().rA

2) If ||>()rA

, then |(,)|(| |,).RAR A

 
Proof.
1) We use the Neumann series representation for the
resolvent
11 11
1
0
(,)=() =()
=||>().
d
n
n
n
RAAI A
AforrA
 








If 0A then 0,
n
A for all n, hence for
>()rA

, we have
1
=0
(,)= 0
lim k
n
k
nk
A
RA 

 

since the finite sums are positive and convergence holds
in every entry.
2) For ||>(),rA

we have
1
=0
1
=0
|(,)|=
lim
=(||,).
lim ||
k
n
k
nk
k
n
k
nk
A
RA
ARA










Theorem 2.1 ([12], Perrons Theorem)
If
A
is a nonnegative matrix, then ()rA is an
eigenvalue of
A
with positive eigenvector.
Definitio n 2 .1
1) An operator (, ())SDS on a Banach Lattice
(,)X is called positive if
0()={(),0}.SuforalluD SvD Sv


2) A semigroup >0
()
tt
T on
X
is positive if and only
if t T is positive for all >0t.
Remark 2.1 An operator (, ())SDS defined on a
()K

space(
K
compact) satisfies the positive minimum
principle if for every (),uDSu

positive and
x
K

,()=0ux , then ()()0.Sux 
Theorem 2.2 ([1])

O. A. M. S. AHMED ET AL. 285
Let (, ())SDS be the generator of a semigroup
>0
()
tt
T on ()CK , then the semigroup is positive if and
only if (, ())SDS satisfies the positive minimum
principle.
3. The Diagonal Case
In this section we assume that the matrix function B is
diagonal.
3.1. Characterization of the Operator
(,())D
The proofs of Proposition 3.1 and Proposition 3.2 can be
deduced from the scalar case given in Proposition 2.2
and Prop osi tion 2.3 from [10].
Proposition 3.1 Let >(0)
d
IB and
2(]0,],( ))
d
UXC MC

 . Then ()UD if and
only if 2([0, ],())
d
UMC   for some >0

and
(0) =(0)= 0UU
.
Proposition 3.2 Let (0) =d BI

and
2(]0,],( ))
d
UXC MC

 . Then ()UD if and
only if 1([0,]),( ))
d
UC MC   for some >0

,
(0) = 0U and
0
0
1(() ())=0.
lim x
xUxUtdt
x

 


Moreover, if ()UD then ()= (log)Ux oxE

and

1()=Uxo logxE
x as 0,x where E is the
constant matrix introduced in Lemma 2.2.
Proposition 3.3 The operator (,())D is a
densely defined, closed, dissipative and satisfies the
positive minimum princi ple.
Proof. Put
1
2
() 00
0()
()= 0
00()
d
bx
bx
Bx
bx










and for =1,2, ,id let () =i
i
bx
uu u
x
 
 
with
20
()=
{([0, ],)(]0, ],),()=0}.
lim
i
xi
D
uC CCux


 


Then

1,
=()
ij ijd
UuD

if and only if
()
ij i
uD for all ,=1,2,.ij d Hence from ([10].
Lemma 2.4), the operator (,())D is a densely
defined and closed.
Let us show that (,())D is dissipative:
||||||||,for>0and()
XX
UUU UD




Let


1,
=()
ij ijd
UuD

 and >0

. According
to [10], for all 1,ijd

 we have


|||| ||||
ijiji ij
uuu 



then


||||||||,
XX
UUU  
and hence the dissipativity holds.
In order to prove that (, ())D satisfies the positive
min i mu m p r in ci p le , a ss u me t h at

1,
=()
ij ijd
UuD


and positive such that 0 ()=0Ux for some 0[0, [.x


If 0>0xthen


0
ij
uand 0 ()=0,
ij
ux for all i, j = 1, 2,
,d. Hence 0 '() =0
ij
ux and 0 '( )0
ij
ux
 for all i, j =
1, 2,,d. That means 0
()0Ux. If 0=0x then
0
()=0Ux.
3.2. Spectral Analysis of the Operator
(,())D
The purpose of next th eorem is to deduce under re asonable
hypothesis on the coefficients of B a precise description
of the spectrum of the operator (,())D.
Theorem 3.1 If (0)
d
I
B

, then the spectrum of
(,( ))D is contained in ],0].
Proof. Let


1,
,= ()
ij ijd
CU uD 


and 1, =( ).
iji j d
Vv X


We have()=()=
for all,=1,2,,
iij ij
UVu v
ij d






1
=
=1
=(), =1,2,,
().
ijiij
id
i
i
uvfor allijd
and









This is sufficient to note that the spectrum of
(,( ))D verifies =
=1
()( )
id
i
i
   and then the
result hold by ([10], Lemma 2.6).
Theorem 3.2 The operator (,())D with
(0)
d
I
B

, generates an analytic semigroup of angle
.
2

Moreover, the semigroup is positive and contractive.
Proof. For 0< <

 , put
*
={;|()|}zC argz






. It is clear from
theorem 3.1 that ()

 

 and then for

 

and =1,2, ,id, the operator ()
i

 is invertible
and verifies 1
|| ()||||
i
ii
C



 for some positive cons-
tant i C (see [10]). So ()

 is invertible and veri-

O. A. M. S. AHMED ET AL. 286
fies 11
max
|| ()||||
i
id
X
C


. Hence the operator
(,())D with (0)
d
I
B, generates an analytic
semigroup of angle .
2

For the positivity using the rela-
tion between the operators , =1,2,,
iid and 
and from the fact that each operator (,( )),
ii
D
=1,2, ,id, generates an analytic positive and contrac-
tive semigroup, we deduce the result.
4. The Non Diagonal Case
In this section we consider the operator
()
='' '
Bx
UU U
x
 
Assume that 1
()=() ()()BxPxDxPx
, where D is
diagonal matrix function and P is nonsingular matrix
function. If P is constant, put 1 =VPU
, then
=,= =
' ''' ''
UPVU PVandUPV
so we obtain
1()
='' '
Dx
PPVV V
x





Hence, from Theorem 3.2 we can easily verify the
following
Proposition 4.1 The operator (,())D with
(0)
d
ID , generates an an alytic semi g rou p of a ngle
.
2

Remark that the condition (0)
d
ID

 is equivalent
to the fact that the spectrum ((0))B

of the constant
matrix (0)B verifies ((0))],1].B


We turn now to the general case in which we proceed
with a perturbation argument. For this we recall the
following definition.
Definiti on 4.1 ([2].Definition 2.1).
Let :()
A
DA XX be a linear operator on the
Banach space
X
. Any operator :()BDB X X is
called
A
-bounded if )()DAD B and if there exist
constants ,ab in 
 such that
||||||||||||( ).BU aAU bUforallUDA  (5.1)
The
A
-bound of B is
0=inf{ 0:0(5.1)}aathere existsbsuch thatholds
Proposition 4.2 ([2].Theorem 2.10).
Let the operator (, ())
A
DA generates an analytic
semigroup on a Banach space
X
. Then there exists a
constant >0

such that (,())
A
BDA generates an
analytic semigroup for every
A
-bounded operator B
having
A
-bound 0<.a

Introduce now the operators 00 (,())D and
(,( ))D given by
2
02
(0)( )(0)
==
dBd BxBd
and
x
dxx dx dx


with,
0
20
0
()=( )={([0,],
())(]0, ],()),()=0}.
lim
dd
x
DD UC
MC CMCUx





Then we have 0
=  and if we choose
0
()=()DD, we obtain the principal result of the
paper.
Theorem 4.1 Assume that (0)B is diagonalizable
and ((0))],1]B

 or (0) =d BI and (2.2)
holds. Then the operator (,())D, generates an analy-
tic semigroup of angle .
2

Proof. Let ()UD

 and observe that
2
2
2
2
0
()
=
(0)( )(0)
=
=,
dBxd
UU U
xdx
dx
dBdBxBd
UU U
x
dxxdx dxUU






Let >0,

there exists >0

such that for all
0< <x

we have ||()(0) ||<.Bx B


The formula
1
0
()
=()
Ux Uxtdt
x


 implies that
||()||||||,0<<.
X
RU xUx




On the other hand from the Taylor expansion to order
2 at >x

and for all ()UD there exists a
constant >0C

such that
0
||()||||||||||
(0)
|||||||||||| .
XX
XXX
RU xUCU
B
UUCU
x








 
Since (0)B is diagonalizable and ((0))],1[B


or B satisfies the condition (2.2) for (0) =,
d
BI

the
map
(0) , Bd
UU
x
dx

from ()D into ([0,],( ))
d
CMC

 is continuous
(see [10]. Remark 2.5), so we deduce that the operator
 is -bounded with -bound is equal to zero.
Hence, the desired result follows by applying Theorem
3.2 and Proposition 4.2.

O. A. M. S. AHMED ET AL. 287
5. Application and Example
Assume now that the operator (,( ))D satisfies the
assumptions of Theorem 4.1, it generates so an analytic
semigroup, and consider the evolution equation problem
0
()()=(), 0
(0) =
dU tUtftt
dt
UU





 (5.2)
Corollary 5.1 If =0f then the problem (5.2) has a
unique solution for all 0 UX

. This solution is of
infinitely continuously differentiable on ]0, [.
For general case we have by Pazy [3] and R.autry
[13].
Corollary 5.2 If 1, =( )
iji jd
ff
 , and for all
1
,,(]0,[, )
ij
ijfLTC and for every ]0, [tT there is a
,ij
t

and a continuous function ,:[0,[ [0,[
ij
t
  
such that
,
,
0
()
()()(||)< .
ij
ij
t
ij t
ij ijt
d
ftfsts and



 

Then the problem (5.2) has a classical solution.
EXAMP LE
Let the Banach space =([0,],( ))
d
X
CMC and
0.

 Put ()=
x
x

 and define the linear transforma-
tion P

on
X
in itself by =.PUUo


It is clear that
P

is invertible and 11 ()= .PP



Consider now the operator (, ())D defined on
X
by
22 12
=()Ux Ux BxU
 
  

with
2
0
()= {(]0,],()),
,( )()=0},
lim
d
x
DUXC MC
UX oPUx


 



and BX is a diagonal matrix real valued function
satisfying (0)<(12).
d
BI

 A simple calculus gives
21
=()PP



 (5.3)
where
1
11()
=.
Bx
UU U
x






 




Put
2
0
()={([0, ], ())(]0, ],
()), ()=0}.
lim
d
dx
DUC MCC
MC Ux




From Theorem 3.2 the operator (,())D generates
an analytic semigrou p of angle 2

moreover, the semi-
group is positive and contractiv e. Hence the relation (5.3)
implies that the operator (, ())D generates an analytic
semigroup of angle 2

and the semigroup is contractive
if 1.


6. Acknowledgements
The Authors wish to thank Professor A. Rhandi for many
helpful discussions and comments on the manuscript.
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