8_502_khalil.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 5, 2010, 881-898

ISSN 1307-5543 – www.ejpam.com

Tensor Product Semi-Groups

Roshdi Khalil1,∗, R. Al-Mirbati2, and D. Drissi3

1 Department of Mathematics, University of Jordan, Amman-Jordan
2 Gulf University for Science and Technology, Kuwait
3 Department of Mathematics, Kuwait University, Kuwait

Abstract. Let X and Y be Banach spaces and L(X , Y ) be the space of all bounded linear operators from

X to Y .If X = Y we write L(X ) forL(X , Y ). Let X ⊗ Y be the tensor product of X and Y , and X
α⊗ Y be

the completion of X ⊗Y with respect to a uniform cross norm α. In this paper, we present an extension

of the Hille-Yosida Theorem to tensor product semigroups.

2000 Mathematics Subject Classifications: Primary 47D03 , Secondary 47D99

Key Words and Phrases: Semigroups of operators, Tensor product

1. Introduction

One parameter semigroups of operators have been a useful tool in the study of the so-

called abstract Cauchy problem. Such a problem states as follows: Let A be a linear operator

on a Banach space X , find a continuously differentiable function T (., x) from [0,∞) into the

domain of A such that T satisfies the differential equation d

d t
T (t, x) = AT (t, x), (t ≥ 0),

T (0, x) = x , for all x ∈ Dom(A). So much work has been done on one parameter semigroups

of operators as well as its relation to the abstract Cauchy problem. For more on such topics

we refer to [3, 4, 9].

We begin recalling some standard definitions. Let X be a Banach space and L(X ) be the

space of bounded linear operators on X . By a one parameter semigroup of operators on X we

mean a map T : [0,∞)→ L(X ) such that

(i) T (0) = I , the identity operator on X .

(ii) T (s+ t) = T (s) T (t) for all s, t ≥ 0, the semigroup property.

∗Corresponding author.

Email address: roshdi�ju.edu.jo (R. Khalil)

http://www.ejpam.com 881 c© 2010 EJPAM All rights reserved.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 882

The linear operator A whose domain D(A) is given by

D(A) =

�
x ∈ X : lim

t→0+

T (t)x − x

t
exists

�

such that

Ax = lim
t→0+

T (t)x − x

t
=

d+

d t
(T (t)x) |t=0 for x ∈D(A)

is called the infinitesimal generator of the semigroup (T (t))t≥0. The generator A is always a

closed, densely defined operator. It is well known that, when A is a densely defined linear op-

erator with non empty resolvent set, then the abstract Cauchy problem has a unique solution,

for all x in the domain of A, if and only if A generates a strongly continuous semigroup. (Pazy,

[9, 10], Goldstein, [3]).

There are many important results on one parameter semigroups of operators. We mention

two of such results:

I. Characterization of the infinitesimal generator of a semigroup.

II. “Hille-Yosida Theorem”: The norm of the resolvent operator Rλ (A) of the infinitesimal

generator of a C0 semigroup tends to zero at infinity. More precisely,


Rλ(A)



≤ M

λ−ω for

large λ, which is known as the Hille-Yosida Inequality.

In this paper, we introduce what we call a tensor product semigroup. We show that every

tensor product semigroup is a two parameter semigroup. We study the relation between a

tensor product semigroup and its components. As not every two parameter semigroup on

X
α⊗ Y defines a T.P.S., we present a condition under which a two parameter semigroup be a

T.P.S. We show that the operator A1⊗ I + I ⊗ A2, is the infinitesimal generator of a C0 T.P.S.,

where A1,A2 generate the semigroup components of the T.P.S. . Equality of A1 ⊗ I + I ⊗ A2

and A1 ⊗ I + I ⊗ A2 is proved as well.

Throughout this paper, X
∨⊗Y (X

∧⊗Y ) denote the completion of the injective (the projective)

tensor products of X and Y . If P and Q are elements in L(X ) and L(Y ) respectively, then P⊗Q

denotes the tensor product operator on X ⊗ Y . Further, we write X
α⊗ Y to denote either one

of the tensor products (the projective or the injective). For more details on tensor product

spaces and tensor product of operators, we refer the reader to [8].

2. Tensor Product Semigroups

Definition 1. Let X , Y be Banach spaces, and (T (s))s≥0, (S(t))t≥0 be one parameter families of

operators in L (X ) , L (Y ) respectively. The family (T (s)⊗ S(t))s,t≥0 is called a tensor product

semigroup, (abbreviated T.P.S.) on the Banach space X
α⊗ Y if

1. T (0)⊗ S(0) = IX⊗Y ,

2. T (s1 + s2)⊗ S(t1 + t2) =
�

T (s1)⊗ S(t1)
��

T (s2)⊗ S(t2)
�

,



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 883

This is equivalent to

T (0)
α⊗ S(0) = I

X
α⊗Y

and T (s1 + s2)
α⊗ S(t1 + t2) =

�
T (s1)

α⊗ S(t1)

��
T (s2)

α⊗ S(t2)

�
.

Thus the family

�
T (s)

α⊗ S(t)

�

s,t≥0
is a T.P.S. defined on the complete space X

α⊗ Y . For

short, we will write (T (s)⊗ S(t))s,t≥0 for

�
T (s)

α⊗ S(t)

�

s,t≥0
, and I for each of IX⊗Y , and

I
X
α⊗Y

. It should be remarked that if we know T (s)⊗ S(t) on X ⊗ Y , then we know T (s)
α⊗ S(t)

on X
α⊗ Y .

One can define a T.P.S. (T (s)⊗ S(t))s,t≥0 , to be uniformly continuous on X
α⊗ Y if

Lim
(s,t)→(0+,0+)

‖T (s)⊗ S(t)− I ⊗ I‖ = 0, and to be strongly continuous on X
α⊗ Y (C0) if

Lim
(s,t)→(0+,0+)




T (s)
α⊗ S(t)z − z




 = 0

for all z ∈ X
α⊗ Y .

One can easily see that the limit in (2) can be replaced by

Lim
(s,t)→(0+,0+)



(T (s)⊗ S(t))
�

x ⊗ y
�− x ⊗ y



 = 0,

for all x ∈ X , y ∈ Y . The proof is a consequence of the following

Lemma 1. Let X , Y be Banach spaces and α be a uniform crossnorm on X ⊗ Y . If

�Ai ⊗ Bi

�
z − (A⊗ B) z



→ 0 as i→∞, for all z ∈ X ⊗ Y and none of the sequences
�
Ai

�
,
�
Bi

�

has a subsequence that converges to zero pointwise, then
�
Ai ⊗ Bi

�
i is uniformly bonded. More-

over, each of
�
Ai

�
i ,
�
Bi

�
i is uniformly bounded.

Proof. For a fixed 0 6= x0 ∈ X one can see that each vector in the space
�

x0

� ⊗ Y

is of the form x0 ⊗ y for some y ∈ Y . Therefore,
�

x0

� ⊗ Y is a Banach space. Since

�Ai ⊗ Bi

�
z − (A⊗ B) z



 i→∞,→ 0 for every z ∈ �x0

�⊗Y, then
�
Ai ⊗ Bi

�
is a pointwise bounded

sequence of bounded operators on the Banach space
�

x0

�⊗Y . Which implies by the Uniform

Boundedness Principle that

��
Ai ⊗ Bi

�
|[x0]⊗Y

�

i
is uniformly bounded. That is




�
Ai ⊗ Bi

�
|[x0]⊗Y




≤ c for all i. But




�
Ai ⊗ Bi

�
|[x0]⊗Y




 = sup
y∈Y

‖x0⊗y‖=1



�Ai ⊗ Bi

��
x0 ⊗ y

�


= sup
y∈Y

‖y‖= 1

‖x0‖



�Ai ⊗ Bi

��
x0 ⊗ y

�




R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 884

= sup
y∈Y

‖y‖=1



Ai x0 ⊗ Bi y


 = sup

y∈Y

‖y‖=1



Ai x0





Bi y


 .

Thus sup
i


 sup

y∈Y

‖y‖=1



Ai x0





Bi y




 ≤ c. In other words



Ai x0





Bi y


 ≤ c for all i for all y ∈ Y .

Under the assumption that Ai x0 does not converge to zero, we obtain that
�
Bi

�
is uniformly

bounded on Y . Repeating the same approach and choosing 0 6= y0 ∈ Y, one can show that�
Ai

�
is uniformly bounded on X . The lemma is then completely proved.

One can easily prove the following result.

Lemma 2. Let X , Y be Banach spaces and (T (s))s≥0, (S(t))t≥0, be one parameter families of

operators in L (X ) , L (Y ) respectively. Then the following are equivalent:

a. T (s) is a one parameter semigroup on X .

b. T (s)⊗ I is a one parameter semigroup on X
α⊗ Y .

c. I ⊗ T (s) is a one parameter semigroup on Y
α⊗ X .

The following Lemma is essential for Theorem 1. Its proof is different from the proof in

[7].

Lemma 3. Let X , Y be Banach spaces, α any crossnorm on X ⊗ Y . Let a, c ∈ X , b, d ∈ Y be

nonzero vectors. If a⊗ b = c ⊗ d, then there exists a nonzero scalar β such a = β c, b = 1

β
d.

Proof. Let x∗ ∈ X ∗. Then x∗ (a) b = x∗ (c) d . In particular, this holds for an x∗ satisfying

that x∗ (c) = ‖c‖. That is,
x∗(a)
‖c‖ b = d . It is clear that x∗ (a) is not zero. Choose

x∗(a)
‖c‖ = β .

Then
�
a− β c

�⊗ b = 0. Thus, x∗
�
a− β c

�
b = 0 for all x∗ ∈ X ∗. Choosing x∗ ∈ X ∗, such that

x∗
�
a− β c

�
=


a− β c



 completes the proof.

Theorem 1. Let X , Y be Banach spaces, (T (s))s≥0, (S(t))t≥0 one parameter families of operators

in L (X ) , L (Y ) respectively. Then the family T (s)⊗ S(t) is a T.P.S. on X
α⊗ Y if and only if there

is a unique 0 6= β ∈ R, and unique one parameter semigroups
�bT (s)

�
s≥0

,
�bS(t)

�
t≥0

on X , Y

respectively, such that

βT (s) = bT (s) and
1

β
S(t) = bS(t) for all s, t ≥ 0.

Proof. If β = 1 then (T (s))s≥0, (S(t))t≥0 define one parameter semigroups. Therefore,

from Lemma 3, each of T (s)⊗ I and I ⊗ S(t) is a one parameter semigroup on X
α⊗ Y . Conse-

quently,

(T (s)⊗ I) (I ⊗ S(t)) = T (s)⊗ S(t) = (I ⊗ S(t)) (T (s)⊗ I)



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 885

is a T.P.S. on X
α⊗ Y .

If β 6= 1, then T (s),S(t) are not semigroups of operators since T (0) = 1

β
I 6= I even though,

T (s)⊗ S(t) is a T.P.S.

To show necessity, let T (s)⊗ S(t) be a T.P.S. on X
α⊗ Y . then T (0)⊗ S(0) = I ⊗ I , and by

Lemma 3, there exists 0 6= γ ∈R such that T (0) = γI , and S(0) = 1

γ
I . Define the families bT (s)

and bS(t) from R
+2

into L

�
X
α⊗ Y

�
so that bT (s) = 1

γ
T (s) and bS(t) = γS(t), s, t ≥ 0. Clearly,

bT (s)⊗ bS(t) is the T.P.S. T (s)⊗ S(t). Moreover, bT (s), bS(t) are one parameter semigroups on

X , Y respectively. Indeed, bT (0) = 1

γ
T (0) = I and bS(0) = γS(0) = I . To show the semigroup

property for bT (s), let s1, s2 ∈R+
2

and let x ∈ X . Then for any 0 6= y ∈ Y we have



bT (s1 + s2)x − bT (s1)bT (s2)x



=
1

y





�bT (s1 + s2)x − bT (s1)bT (s2)x

�
⊗ y





=
1

y





��bT (s1 + s2)⊗ I

�
−
�bT (s1)bT(s2)⊗ I

���
x ⊗ y

�



=
1

y





�bT (s1 + s2)⊗ bS(0+ 0)

�
−
�bT (s1)bT(s2)⊗ bS(0)bS(0)

��
x ⊗ y

�



=




��bT (s1 + s2)⊗ bS(0+ 0)

�
−
�bT (s1)⊗ bS(0)

��bT (s2)⊗ bS(0)
���

x ⊗ y
�





y



=
1

y




��T (s1 + s2)⊗ S(0+ 0)

�− �T (s1)⊗ S(0)
��

T (s2)⊗ S(0)
���

x ⊗ y
�

 .

Therefore bT (s1 + s2) = bT (s1)bT (s2). Similarly,
�bS(t)

�
t≥0

satisfies the semigroup property.

Hence bT (s) and bS(t) are one parameter semigroups on X , Y respectively.

The proof of Theorem 1 shows that if (T (s))s≥0, (S(t))t≥0, are one parameter semigroups

on X , Y respectively, then the family (T (s)⊗ S(t))s,t≥0 is a T.P.S. on X
α⊗ Y .

As for the continuity of tensor product semigroups it is not difficult to see

Lemma 4. Let X , Y be Banach spaces, (T (s))s≥0, (S(t))t≥0, one parameter families of operators

in L (X ) , L (Y ) respectively. If T (s)⊗S(t) is a T.P.S. and bT (s), bS(t) are as in Theorem 1, then the

following are equivalent

a. T (s)⊗ S(t) is uniformly (strongly) continuous.

b. bT (s)⊗ I and I ⊗ bS(t) are uniformly (strongly) continuous.

c. bT (s) and bS(t) are uniformly (strongly) continuous.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 886

Now, if (T (s))s≥0, (S(t))t≥0, are one parameter families of operators in L (X ), L (Y ) respectively

and T (s)⊗ S(t) is a T.P.S., then:

If T (s)⊗ S(t) is uniformly (strongly) continuous, then the map

F(s, t) : R+
2

= [0,∞)× [0,∞)→ L

�
X
α⊗ Y

�
defined by F (s, t) → T (s)⊗ S(t) is continuous

in the uniform (strong) operator topology. Further, F(s, t) is uniformly (strongly) continuous if

and only if it is separately uniformly (strongly) continuous.

The proof of the following proposition is straight forward, and will be omitted.

Proposition 1. Let L(s, t) be a 2-parameter semigroup on the Banach space X
α⊗ Y , such that

L(s, 0)
�

x ⊗ y
�
=

�
f (s)x

�⊗ y for all x ∈ X , y ∈ Y,

L(0, t)
�

x ⊗ y
�
= x ⊗ �g(t)y

�
for all x ∈ X , y ∈ Y,

where f , g are any functions on X , Y respectively. Then

1.
�

f (s)
�

s≥0, and
�

g(t)
�

t≥0 are one parameter semigroups on X , Y respectively.

2. L(s, t) is uniformly (strongly) continuous if and only if each of the one parameter semi-

groups L(s, 0) and L(0, t) is uniformly (strongly) continuous.

It follows from the definition of a two-parameter semigroup [7], we observe that a T.P.S.

(T (s)⊗ S(t))s,t≥0 defines a two-parameter semigroup (L(s, t))s,t≥0 = L(s, t) = T (s) ⊗ S(t).

Note that L(s, t) = L(s, 0)L(0, t), where L(s, 0) = T (s)⊗ S(0) = T (s)⊗ I and

L(0, t) = T (0)⊗ S(t) = I ⊗ S(t).

3. The Infinitesimal Generator of a T.P.S.

Let (T (s)⊗ S(t))s,t≥0 be a C0 T.P.S. on X
α⊗ Y and A1, A2 be the infinitesimal genera-

tors of the one parameter C0 semigroups
�bT (s)

�
s≥0

,
�bS(t)

�
t≥0

on X , Y respectively, where�bT (s)
�

s≥0
,
�bS(t)

�
t≥0

are as in Theorem 1.

Remark 1. Let us recall the followings.

1. Let X be a normed space and A be a linear operator, A : D (T ) ⊆ X → X . A subspace

Z of the domain D (A) is called a core for A if Z is dense in D (A) for the graph norm

‖A‖A := ‖x‖+ ‖Ax‖. [2]

2. A function G : R+
2 → X

α⊗ Y , is said to be differentiable at (0,0) if there exists a linear

transformation L : R+
2 → X

α⊗ Y such that

lim
(s,t)→(0+,0+)

‖G (s, t)− G (0,0)−L ((s, t)− (0,0))‖
‖(s, t)‖ = 0.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 887

In other words,

G (s, t)− G (0,0) = L (s, t) + R(s, t),

where

lim
(s,t)→(0+,0+)

‖R(s, t)‖
‖(s, t)‖ = 0.

3. The transformation L above, if it exists, is unique, and it is called the derivative of G at

(0,0).

4. For a fixed z ∈ X
α⊗ Y , if G(s, t)z = (T (·)⊗ S (··)) z, then (2) becomes

�
T (s)

α⊗ S (t)

�
z − z = L (s, t) z + R(s, t)z,

and (2) comes to

lim
(s,t)→(0+,0+)

‖R(s, t)z‖
‖(s, t)‖ = 0,

for all z where (2) holds.

5. If it is shown that for any z satisfying (4), one has the same L (·, ··) in (4), then one

can consider the derivative as the linear transformation from R
+2 → L

�
X
α⊗ Y

�
, where

L
�

X
α⊗ Y

�
is the space of linear (not necessarily bounded) operators on X

α⊗ Y , in the

following sense:

For all z such that (4) holds, there is a linear transformation bL : R+
2 → L

�
X
α⊗ Y

�
,

where (s, t) 7→ bL (s, t) such that bL (s, t) z = L (s, t) z.

6. In case of item 5 holds, if moreover, L is of the form
�
L1,L2

�
then for any (s, t) ∈R+2

the

domain of bL (s, t) is D
�
sL1 + tL2

�
, the domain of sL1 + tL2 which is D

�
L1

�∩D�L2

�
.

Definition 2. Let (T (s)⊗ S (t))s,t≥0 be a T.P.S. on X
α⊗ Y . The infinitesimal generator A of

(T (s)⊗ S (t))s,t≥0 is defined as follows

D (A) =

§
z ∈ X

α⊗ Y :

�
T (s)

α⊗ S (t)

�
z is differentiable at (0,0)

ª
,

Az = D

�
T (s)

α⊗ S (t)

�
z|(s,t)=(0,0)

for z ∈D (A) ,

where D (A) is the domain of A, and D

�
T (s)

α⊗ S (t)

�
z|(s,t)=(0,0)

is the derivative of T (s)
α⊗ S (t) z

as a function of two variables at (s, t) = (0,0).

Lemma 5.
�
A1 ⊗ I

��
x ⊗ y

�
= ∂

∂ s

�
(T (s)⊗ S (t))

�
x ⊗ y

��
|(s,t)=(0,0)

, and
�

I ⊗ A2

��
x ⊗ y

�
=

∂

∂ t

�
(T (s)⊗ S (t))

�
x ⊗ y

��
|(s,t)=(0,0)

for all x ∈ D(A1), y ∈ D(A2), where A1 and A2 are the

infinitesimal generators of the coordinate semigroups respectively.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 888

Proof. Let x ∈D(A1), y ∈D(A2). Then

∂

∂ s

�
T (s)⊗ S (t)

�
x ⊗ y

��
|(s,t)=(0,0)

= Lim
h→0+

(T (h)⊗ S (0))
�

x ⊗ y
�− (T (0)⊗ S (0))

�
x ⊗ y

�

h

= Lim
h→0+

��
T (h) − I

h
x

�
⊗ y

�

=

�
Lim
h→0+

�
T (h)− I

h
x

��
⊗ y =

�
A1 x

�⊗ y.

Similarly for I ⊗ A2.

Now, Let

�
T (s)

α⊗ I

�

s≥0
, (T (s))s≥0 be one parameter C0 semigroups on the Banach spaces

X
α⊗ Y , X with infinitesimal generators A,A1 respectively. Then, one can easily see:

a. D(A1)⊗ Y is a subspace of D (A).

b. D(A1)⊗ Y is dense in X
α⊗ Y .

c. D(A1)⊗ Y is invariant under T (s)
α⊗ I .

d. D(A1)⊗ Y is a core for A.

Lemma 6. Suppose that (T (s))s≥0 , (S (t))t≥0 are one parameter C0 semigroups on the Banach

spaces X , Y with infinitesimal generators A1, A2 respectively. Then A1 ⊗ I and I ⊗ A2 are the

infinitesimal generators of the one parameter C0 semigroups

�
T (s)

α⊗ I

�

s≥0
,

�
I
α⊗ S (t)

�

t≥0

respectively on X
α⊗ Y .

Proof. First let z = x ⊗ y, for some x ⊗ y ∈ D(A1)⊗ Y . If A is the infinitesimal generator

of

�
T (s)

α⊗ I

�

s≥0
, then Az =

�
A1⊗ I

�
z. This means that A|D(A1)⊗Y

= A1 ⊗ I . In other words,

A is an extension of A1 ⊗ I from the subspace D(A1)⊗ Y to the domain D (A) of A. Being the

infinitesimal generator of a one parameter C0 semigroup, A is closed [9]. Thus A is a closed

extension of A1 ⊗ I . But A1 ⊗ I is closable [6]. Since the closure of an operator is its smallest

closed extension, then A⊃ A1 ⊗ I ⊃ A1 ⊗ I . On the other hand, by [6], A1⊗ I is the maximal

extension of A1 ⊗ I . Therefore A ⊂ A1 ⊗ I . Hence A = A1 ⊗ I . Similarly, one can show that

I ⊗ A2 generates

�
I
α⊗ S (t)

�

t≥0
.

Theorem 2. The infinitesimal generator of a C0 T.P.S. (T (s)⊗ S (t))s,t≥0 is the linear transfor-

mation L : R
+2 → L

�
X
α⊗ Y

�
, (a, b) 7−→

��
A1 ⊗ I , I ⊗ A2

�
(a, b)

�
=
�

aA1⊗ I + bI ⊗ A2

�
,

where A1, A2 are the infinitesimal generators of the one parameter C0 semigroups�bT (s)
�

s≥0
,
�bS (t)

�
t≥0

respectively.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 889

Proof. First, we should notice that L(a, b)
�

x ⊗ y
�
=
�

aA1 ⊗ I , bI ⊗ A2

��
x ⊗ y

�
for all

x ∈ D�A1

�
, y ∈D�A2

�
. Now, Let (T (s)⊗ S (t))s,t≥0 be a C0 T.P.S., A its infinitesimal genera-

tor and let z ∈ D (A). That is, z ∈ X
α⊗ Y such that

�
T (s)

α⊗ S (t)

�
z is differentiable at (0,0).

Thus D

��
T (s)

α⊗ S (t)

�
z

�

|(s,t)=(0,0)

exists. In other words, there exist z1, z2 in X
α⊗ Y such that

lim
(s,t)→(0+,0+)

‖{(T(s)⊗S(t))−T(0)⊗S(0)}z−sz1−tz2‖
‖(s,t)‖ = 0. In particular, choose (s, t) to be (s, 0) where

s→ 0+. Then lim
s→0+

‖{(T(s)⊗S(0))−T(0)⊗S(0)}z−sz1‖
s

= 0. Therefore,

lim
s→0+




¦�bT (s)⊗ I

�
− I ⊗ I

©
z − sz1




s

= lim
s→0+







¦�bT (s)⊗ I
�
− I ⊗ I

©

s
z − z1






= 0.

From Lemma 6, z1 =
�

A1 ⊗ I
�

z, where A1 generates the C0 semigroup
�bT (s)

�
s≥0

. Sim-

ilarly, one can show that z2 =
�

I ⊗ A2

�
z. Since z was arbitrarily chosen in D(A), this

shows that D(A) is a subspace of X
α⊗ Y . Further, D(A) ⊆ D

�
A1⊗ I

�
∩ D

�
I ⊗ A2

�
. Now

let z ∈D
�

A1⊗ I
�
∩D

�
I ⊗ A2

�
and s, t > 0. Set

J(s, t) = (T (s)⊗ S (t))− (T (0)⊗ S (0))−
�

A1 ⊗ I , I ⊗ A2

��s

t

�
.

Then

‖J (s, t) z‖ =







�
T (s)

α⊗ S (t)

�
(z)− (z)

−
�

sA1 ⊗ I
�
(z)−

�
t I ⊗ A2

�
(z)








≤









�
bT (s)

α⊗ I

��
I
α⊗ bS (t)

�
(z)

−
�
bT (s)

α⊗ I

�
(z)−

�
t I ⊗ A2

�
(z)









+





�
bT (s)

α⊗ I

�
(z)− (z)−

�
sA1 ⊗ I

�
(z)






≤ t









�
bT (s)

α⊗ I

� �
I
α⊗bS(t)

�
(z)−

�
I
α⊗I

�
(z)

t

!

−
�

I ⊗ A2

�
(z)









+s










bT (s)

α⊗ I − I
α⊗ I

s


 (z)−

�
A1 ⊗ I

�
(z)







.

Divide both sides by ‖(s, t)‖ =
p

s2 + t2 to get

‖J (s, t) z‖
‖(s, t)‖ ≤ ψs,t





�
bT (s)

α⊗ I

���
1

t

�
I
α⊗ bS (t)−

�
I
α⊗ I

��
−
�

I ⊗ A2

��
(z)

�






R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 890

+φs,t





�

1

s

�
bT (s)

α⊗ I − I
α⊗ I

�
−
�

A1 ⊗ I
��
(z)





 ,

where ψs,t =
tp

s2+t2
,φs,t =

sp
s2+t2

. But ψs,t ≤ 1, φs,t ≤ 1 for all s, t > 0. Therefore,

‖J (s, t) z‖
‖(s, t)‖ ≤





�
bT (s)

α⊗ I

���
1

t

�
I
α⊗ bS (t)−

�
I
α⊗ I

��
−
�

I ⊗ A2

��
(z)

�




+





�

1

s

�
bT (s)

α⊗ I − I
α⊗ I

�
−
�

A1⊗ I
��
(z)





 ,

As (s, t) → �
0+, 0+

�
, the second norm in the right hand side converges to zero, whereas

the first norm converges to zero by Lemma 6, the strong continuity of

�
bT (s)

α⊗ I

�

s≥0
, and

the uniform boundedness principle. Therefore,
‖J(s,t)z‖
‖(s,t)‖ → 0 as (s, t)→ (0+, 0+). Now, define

L : R+
2 →L (X α⊗Y ) by (L (s, t)) z =

��
A1 ⊗ I , I ⊗ A2

��s

t

��
z for every z ∈D

��
A1 ⊗ I

��s

t

��∩
D

��
I ⊗ A2

��s

t

��
=D

��
A1 ⊗ I

��
∩D

��
I ⊗ A2

��
. Then

D (T (s)⊗ S (t)) |
(s,t)=(0,0)

= (A1⊗ I , I ⊗ A2), as a linear transformation from R
+2

to L (X α⊗ Y )

is the derivative of the C0 T.P.S.(T (s)⊗ S (t))s,t≥0 at (0,0). Hence the linear transformation

L=
�

A1 ⊗ I , I ⊗ A2

�
is the infinitesimal generator of the C0 T.P.S. (T (s)⊗ S (t))s,t≥0 .

Remark 2. One can show that for any nonzero (a, b) ∈ R
+2

, D
�
A1

� ⊗D
�
A2

�
is a core for�

A1 ⊗ I , I ⊗ A2

��a

b

�
.

1. In general, if A, B are closable, or even, closed linear operators on the Banach space X , then

A+B need not be closed. But, Theorem 1.1 in [5] ensures that aA1⊗ I+ b, I⊗A2, a, b 6= 0

is closable. Moreover, its closure is aA1 ⊗ I + bI ⊗A2.

2. Since the restriction of L(a, b) to X ⊗ Y is defined by

L(a, b)
�

x ⊗ y
�
=
�
aA1 ⊗ I + bI ⊗ A2

��
x ⊗ y

�
=
�
A1⊗ I + I ⊗A2

��a

b

��
x ⊗ y

�
,

for all x ∈ D
�
A1

�
, y ∈ D

�
A2

�
, and since X ⊗ Y is dense in X

α⊗ Y , it is enough to study

T (s)⊗ S (t) instead of its extension T (s)
α⊗ S (t), and

�
A1 ⊗ I + I ⊗ A2

��a

b

�
instead of its

closure
�

A1 ⊗ I , I ⊗ A2

��a

b

�
.

From now on, the infinitesimal generator of (T (s)⊗ S (t))s,t≥0 will be denoted by�
A1 ⊗ I , I ⊗ A2

�
.

Lemma 7. If (T (s)⊗ I)s≥0 is a C0 semigroup on X
α⊗Y with infinitesimal generator A1 ⊗ I where

A1 is a linear operator on X , then (T (s))s≥0 is a C0 semigroup on X with infinitesimal generator

A1.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 891

The proof follows from general functional analysis arguments and will be omitted.

Lemma 8. If (T (s)⊗ S (t))s,t≥o is a C0 T.P.S. on X
α⊗ Y , then for every (a, b) ∈ R+2

, the family

(T (as)⊗ S (bs))s≥o is a one parameter C0 semigroup on the Banach space X
α⊗ Y .

Proof. Let Q(h) = T (ah)⊗ S (bh). Then Q(0) = I , where I is the identity on X ⊗ Y , and

Q(h1+ h2) =
�

T
�
ah1

�⊗ S
�

bh1

���
T
�
ah2

�⊗ S
�

bh2

��

= Q(h1)Q(h2).

Put bh= t, ah= s. Since

h→ 0+ if and only if s = ah→ 0+ if and only if t = bh→ 0+,

then the function Q(h) = T (s)⊗S (t) converges to I as h→ 0+ in the strong operator topology.

Lemma 9. Let 0 6= (a, b) ∈ R
+2

. Then the infinitesimal generator of the one parameter C0

semigroup (T (as)⊗ S (bs))s≥o is the linear operator

aA1⊗ I + b, I ⊗ A2.

Proof. The generator of the one parameter semigroup (T (as)⊗ S (bs))s≥o is given by

d+

ds
(T (as)⊗ S (bs))|s=0

=
d+

ds
(T (as)⊗ I) (I ⊗ S (bs))|s=0

=
d+

ds

�bT (as)⊗ I
��

I ⊗ bS (bs)
�
|s=0

.

Being the derivative of a function of one variable at s = 0, the derivative is

�
d+

ds

�bT (as)⊗ I
�
|s=0

��
I ⊗ bS (0)

�
+
�bT (0)⊗ I

��d+

ds

�
I ⊗ bS (bs)

�
|s=0

�

= a

�
d+

d (as)

�bT (as)⊗ I
�
|s=0

�
(I ⊗ I) + (I ⊗ I) b

�
d+

d (as)

�
I ⊗ bS (bs)

�
|s=0

�
(1)

and this is just aA1⊗ I + b, I ⊗ A2.

Corollary 1. The linear operator aA1 ⊗ I + b, I ⊗ A2 is closed and densely defined.

Corollary 2. For every 0 6= (a, b) ∈ R
+2

the linear operator aA1 ⊗ I + b, I ⊗ A2 is closable,

densely defined and its closure is aA1 ⊗ I + b, I ⊗ A2.

Proof. Being closable is proved in [6]. Since

D
�
aA1 ⊗ I + b, I ⊗ A2

�
=

�
D
�
A1

�⊗ Y
�∩ �X ⊗D

�
,A2

��

= D
�
A1

�⊗D
�
,A2

�
,



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 892

and since A1,A2 are densely defined in X , Y respectively, one can show that the subspace

D
�
A1

�⊗D
�
,A2

�
is dense in X ⊗ Y , which is in turn dense in X

α⊗ Y . Thus aA1⊗ I + b, I ⊗A2

is densely defined on X
α⊗ Y . So

�
aA1 ⊗ I + b, I ⊗ A2

��
x ⊗ y

�
=
�
aA1 ⊗ I + b, I ⊗A2

��
x ⊗ y

�
,

for all x ⊗ y ∈D�A1

�⊗D
�
,A2

�
. That is

�
aA1 ⊗ I + b, I ⊗ A2

�
|(X⊗Y )∩D(aA1⊗I+b,I⊗A2)

= aA1 ⊗ I + b, I ⊗ A2.

Therefore, B = aA1 ⊗ I + b, I ⊗ A2 is an extension of A= aA1⊗ I+ b, I ⊗A2 from the subspace

D
�
A1

�⊗D�,A2

�
to D (B). From Corollary 1, B is a closed extension of A. Since A is closable,

and the closure is the smallest closed extension, A ⊂ B. On the other hand, A is closable,

and the closure of a closable operator is its maximal extension. Thus B ⊂ A. Hence A = B

completes the proof of the corollary.

Corollary 3. Let (T (s)⊗ S(t))s,t≥0 be a C0 T.P.S. on X
α⊗ Y , with infinitesimal generator�

A1 ⊗ I , I ⊗ A2

�
and 0 6= (a, b) ∈R+2

. Then the infinitesimal generator of the one parameter C0

semigroup (T (as)⊗ S (bs))s≥o is the linear operator aA1 ⊗ I+b, I ⊗ A2 = a
�
A1 ⊗ I

�
+ b

�
I ⊗ A2

�
.

As a consequence of Corollary 3, we obtain Nagel’s result [1, Proposition, Sec. 3.7].

Corollary 4. The infinitesimal generator of the one parameter C0 T.P.S. (T (t) ⊗ S (t))t≥0, is

(A1⊗ I) + (I ⊗ A2) defined on the core D
�
A1

�⊗D
�
A2

�
of the generator.

Proof. From Corollary 3 the operator

a
�
A1 ⊗ I

�
+ b,

�
I ⊗ A2

�
= a

�
A1⊗ I

�
+ b

�
I ⊗ A2

�

generates (T (at)⊗ S(bt))t≥0. As a particular case take (a, b) = (1,1). Then

�
A1⊗ I

�
+
�

I ⊗ A2

�

is the infinitesimal generator of the one parameter C0 semigroup (T (t) ⊗ S(t))t≥0. But�
A1 ⊗ I

�
+
�

I ⊗ A2

�
is defined on D

�
A1

�⊗D
�
A2

�
, which is a core for the infinitesimal gen-

erator
�
A1 ⊗ I

�
+
�

I ⊗ A2

�

Definition 3. Let (T (s))s≥0 and (S(t))t≥0 be one parameter C0 semigroups on the Banach spaces

X and Y respectively. For u= (a, b) ∈R+2

, the almost directional derivative a.Du of

T (s)⊗ S(t) at (0,0) is defined by

D

�
a.Du

�
T (s)

α⊗ S(t)

�

|(s,t)=(0,0)

�
=



z ∈ X

α⊗ Y : lim
h→0+

T (ah)
α⊗ S (bh)z − z

h
exists







R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 893

and �
a.Du

�
T (s)

α⊗ S(t)

�
|

(s,t)=(0,0)

�
z = lim

h→0+

(T (ah)
α⊗ S(, bh))z − z

h

It follows from the definition that the almost directional derivative a.Du

�
T (s)

α⊗ S(t)

�

|(s,t)=(0,0)

is the infinitesimal generator of the one parameter C0 semigroup (T (at)⊗S(, bt))t≥0. Further,

for u= (a, b) ∈R+2

, a.Du(T (s)
α⊗S(t)) |

(s,t)=(0,0)

=
�

aA1 ⊗ I + bI ⊗ A2

�
= a

�
A1 ⊗ I

�
+ b

�
I ⊗ A2

�
.

Also, since ∇ (T (s)⊗ S(t)) = ∂

∂ s
T (s)⊗ S(t)i + ∂

∂ t
T (s)⊗ S(t) j, then for u = (a, b) ∈R+2

a.Du(T (s)
α⊗ S(t)) |

(s,t)=(0,0)

=∇T (s)⊗ S(t) |
(s,t)=(0,0)

.u.

Theorem 3. Let (T (t))t≥0 and (S(t))t≥0 be one parameter C0 semigroups on Banach spaces X

and Y with infinitesimal generators A1 and A2 respectively. Then

D (T (s)⊗ S(t))

�
a

b

��
x ⊗ y

�
=
�
A1 ⊗ I , I ⊗ A2

��a

b

�
(T (s)⊗ S(t))

�
x ⊗ y

�
(2)

for all (a, b) ∈R+2

, all x ∈D�A1

�
and y ∈D�A2

�
.

Proof. Let (a, b) ∈R+2

, x ∈D�A1

�
, and y ∈D�A2

�
. Then D(T (s)⊗S(t)) as a function of

two variables is given by

(D(T (s)⊗ S(t)))

�
a

b

��
x ⊗ y

�

=

�
∂

∂ s
(T (s)⊗ S(t)) ,

∂

∂ t
(T (s)⊗ S(t))

��
a

b

��
x ⊗ y

�

=

�
a
∂

∂ s
(T (s)⊗ S(t)) + b

∂

∂ t
(T (s)⊗ S(t))

��
x ⊗ y

�

=

�
a

d (T (s)⊗ I)

ds
(I ⊗ S(t)) + b

d (I ⊗ S(t))

d t
(T (s)⊗ I)

��
x ⊗ y

�
.

Then, by Lemma 2-c, Theorem 2 and Lemma 6 we have

(D(T (s)⊗ S(t)))

�
a

b

��
x ⊗ y

�

= a
��

A1⊗ I
�
(T (s)⊗ I)

��
x ⊗ S(t)y

�

+b
��

, I ⊗ A2

�
(I ⊗ S(t))

��
T (s)x ⊗ y

�

= a
�
A1⊗ I

��
T (s)x ⊗ S(t)y

�
+ b

�
, I ⊗ A2

��
T (s)x ⊗ S(t)y

�

=
�
A1 ⊗ I , I ⊗ A2

��a

b

�
(T (s)⊗ S(t))

�
x ⊗ y

�
.

which is (2).

As in the classical case, one can show the existence of constants ω ≥ 0 and M ≥ 1 such

that




T (s)
α⊗ S(t)




 ≤ Meω(t+s) for s, t ≥ 0.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 894

4. The Hille-YosidaTheorem for T.P.S’.

Definition 4. Let X and Y be Banach spaces and A be a linear transformation that maps R+
2

into L
�

X
α⊗ Y

�
given by A = (A1⊗ I , I ⊗ A2), where A1, ,A2 are linear operators on X and Y

respectively, satisfying:

a. For any (a, b) ∈R+2

A

�
a

b

��
x ⊗ y

�
= (aA1 ⊗ I ,+bI ⊗ A2)

�
x ⊗ y

�
,

x ∈D�A1

�
, y ∈D�A2

�
.

b. A is the infinitesimal generator of a C0 T.P.S. (T (s)⊗ S(t))s,t≥0.

Then we call the linear transformation B = (A1⊗ I , I ⊗A2) the pseudo-infinitesimal generator

of (T (s)⊗ S(t))s,t≥0.

We should remark that uniqueness of the closure of a linear operator, and uniqueness of

the infinitesimal generator of a T.P.S. imply that the pseudo-infinitesimal generator of a T.P.S.

is unique.

Now, we are ready to prove one of the main results of this section (A Hille-Yosida Theorem

for T.P.S.’).

Theorem 4. Let X , Y be Banach spaces. A linear transformation A from R
+2

into L
�

X
α⊗ Y

�

is the pseudo-infinitesimal generator of a C0 T.P.S. (T (s)⊗ S(t))s,t≥0 on X
α⊗ Y satisfying

‖T (s)⊗ S(t)‖ ≤ Meω(s+t), for all s, t ≥ 0, for some constants M ≥ 1,ω ≥ 0, if and only if the

followings hold

(i)
�

A
�0

1

���
x ⊗ y

�
=
�
A1 x
�⊗ y, and

�
A
�1

0

���
x ⊗ y

�
= x⊗�A2 y

�
, x ∈D�A1

�
, y ∈D�A2

�

for some linear operators A1,A2 (not necessarily bounded) on X , Y respectively.

(ii) A1,A2 in part (i) are closed and densely defined on X , Y respectively.

(iii) ρ(Ai) contains (ω,∞), i = 1,2, and for every λ > ω



�Rλ(Ai)
�n


≤ Mi

(λ−ω)n , n= 1,2,3, . . . , for some Mi ≥ 1, i = 1,2.

Proof. Let the conditions (i), (ii) and (iii) hold. From (ii), (iii) and Theorem 1.7 in

[1] A1,A2 are the infinitesimal generators of one parameter C0 semigroups say (T (s))s≥0,

(S(t))t≥0 on X , Y respectively, satisfying

‖T (s)‖ ≤ M1eωs for all s ≥ 0, and ‖S(t)‖ ≤ M2eωt for all t ≥ 0.

By Theorem 1 (T (s)⊗ S(t))s,t≥0 is a C0 T.P.S. on X
α⊗ Y satisfying

‖T (s)⊗ S(t)‖ ≤ ‖T (s)‖‖S(t)‖



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 895

≤ M1M2eω(s+t)

= Meω(s+t), for all s, t ≥ 0.

By Theorem 2, the transformation
�
A1 ⊗ I , I ⊗ A2

�
is the pseudo-infinitesimal generator of

T (s)⊗ S(t). Let (a, b) ∈R+2

, x ∈D�A1

�
, y ∈D�A2

�
. Then

��
A1 ⊗ I , I ⊗ A2

��a

b

���
x ⊗ y

�
=
�
aA1 ⊗ I + bI ⊗ A2

��
x ⊗ y

�
,

which is by (i), �
aA

�
0

1

�
+ bA

�
1

0

���
x ⊗ y

�
=

�
A

�
a

b

���
x ⊗ y

�
.

Therefore A
�a

b

�
coincides with

�
A1 ⊗ I , I ⊗ A2

��a

b

�
on D

�
A1

�⊗D�A2

�
for every (a, b) ∈R+2

,

thus their closures coincide.

But the transformation mapping (a, b) ∈ R
+2

into A
�a

b

�
=
�
A1 ⊗ I , I ⊗ A2

��a

b

�
is the in-

finitesimal generator of (T (s)⊗ S(t))s,t≥0 (See Theorem 2, and Corollary 2). In other words,

A
�a

b

�
is the pseudo-infinitesimal generator of (T (s)⊗ S(t)).

Conversely, let A be as in the statement. Since T (s)⊗ S(t) is a C0 T.P.S., then by Theorem

1 there exist unique β 6= 0, and unique one parameter C0 semigroups
�bT (s)

�
s≥0

,
�bS(t)

�
t≥0

on X , Y respectively, such that (1) holds. Let A1,A2 be their generators. Then by Theorem 2,

A1,A2 satisfy (i) and (ii). By Theorem 2 and Corollary 2, the transformation

(a, b) 7→ �A1 ⊗ I , I ⊗ A2

��a

b

�
=
�

A1 ⊗ I , I ⊗ A2

��a

b

�

is the infinitesimal generator of bT (s)⊗bS(t). But bT (s)⊗bS(t) = T (s)⊗S(t). Thus
�
A1 ⊗ I , I ⊗A2

�

is the pseudo-infinitesimal generator of T (s)⊗ S(t).

Uniqueness of the pseudo-infinitesimal generator, implies that the linear transformation�
A1 ⊗ I , I ⊗ A2

�
= A. That is A

�a

b

�
=
�
A1 ⊗ I , I ⊗ A2

��a

b

�
for all (a, b) in R

+2

. In particular, for

(a, b) = (0,1), and (a, b) = (1,0). Hence (i) is fulfilled.

Theorem 5. Let X , Y be Banach spaces, and (T (s)⊗ S(t))s,t≥0 be a C0 T.P.S. on the Banach space

X
α⊗ Y with infinitesimal generator A = (A1⊗ I , I ⊗ A2). If λ ∈ ρ

�
(A1⊗ I , I ⊗ A2)

�a

b

��
, where

(a, b) ∈R+2

, and λ > (a+ b)max
i=1,2

�
ω(Ai)

�
, where 0<ω(Ai) ∈ ρ(Ai), for i = 1,2, then

�
Rλ

�
(A1⊗ I , , I ⊗ A2)

�
a

b

����
x ⊗ y

�
=

∞∫

0

e−λt (T (at)⊗ S(bt))
�

x ⊗ y
�

d t. (3)

Proof. Let x ∈ X , y ∈ Y , (a, b) ∈R+2

, and λ be as given. Define

R (λ)
�

x ⊗ y
�
=

∞∫

0

e−λt (T (at)⊗ S(bt))
�

x ⊗ y
�

d t.



R. Khalil, R. Al-Mirbati, D. Drissi / Eur. J. Pure Appl. Math, 3 (2010), 881-898 896

Since the map t 7→ (T (at)⊗ S(bt))
�

x ⊗ y
�

is continuous and λ > (a+ b)max
i=1,2

�
ω(Ai)

�
, the

integral exists as an improper Riemann integral and defines a bounded linear operator on

X ⊗ Y . Further, for h> 0

T (ah)⊗ S (bh)− I ⊗ I

h
R(λ)

�
x ⊗ y

�

=
1

h

∞∫

0

e−λt
�
(T (a(t + h)⊗ S (b (t + h))

�
x ⊗ y

�− (T (at)⊗ S (bt))
�

x ⊗ y
��

d t

=
1

h




∞∫

h

e−λ(t−h) (T (at)⊗ S (bt))
�

x ⊗ y
�

d t

−
∞∫

0

e−λt (T (at)⊗ S (bt))
�

x ⊗ y,
�

d t




=
eλh

h

∞∫

h

e−λt (T (at)⊗ S (bt))
�

x ⊗ y
�

d t − 1

h

∞∫

0

e−λt (T (at)⊗ S (bt))
�

x ⊗ y,
�

d t

=
eλh− 1

h

∞∫

0

e−λt (T (at)⊗ S (bt))
�

x ⊗ y
�

d t − eλh

h

h∫

0

e−λt (T (at)⊗ S (bt))
�

x ⊗ y
�

d t.

Taking the limit of both sides as h→ 0+ yields
�
(A1⊗ I , I ⊗ A2)

�
a

b

���
R(λ)

�
x ⊗ y

��
= λR(λ)

�
x ⊗ y

�− �x ⊗ y
�

.

This implies that

R(λ)
�

x ⊗ y
� ∈D

�
(A1⊗ I , I ⊗ A2)

�
a

b

��
for all x ⊗ y ∈ X ⊗ Y,

and �
λI ⊗ I − (A1⊗ I , I ⊗A2)

�
a

b

��
R(λ) = I ⊗ I on X ⊗ Y.

Now, for x ⊗ y ∈D
�
(A1 ⊗ I , I ⊗ A2)

�a

b

��⊆D

�
(A1 ⊗ I , I ⊗ A2)

�a

b

��
we have

R(λ)

��
(A1 ⊗ I , I ⊗ A2)

�
a

b

���
x ⊗ y

��

=

∞∫

0

e−λt (T (at)⊗ S (bt))

��
(A1 ⊗ I , I ⊗ A2)

�
a

b

���
x ⊗ y,

��
d t

=

∞∫

0

e−λt

��
A1 ⊗ I , I ⊗ A2

��a

b

���
(T (at)⊗ S (bt))

�
x ⊗ y

��
d t,



REFERENCES 897

by Theorem 3. Since (A1⊗ I , I ⊗ A2)
�a

b

�
is closed by Corollary 2, it follows that the right-hand

side of (3) is

��
A1 ⊗ I , , I ⊗ A2

��a

b

�� ∞∫

0

e−λt (T (at)⊗ S (bt))
�

x ⊗ y
�

d t

= (A1 ⊗ I , I ⊗ A2)

�
a

b

��
R(λ)

�
x ⊗ y

��
.

Hence,

R(λ)

�
λI −

�
(A1⊗ I , I ⊗A2)

�
a

b

����
x ⊗ y

�
=
�

x ⊗ y
�

,

for all x ⊗ y ∈D
�
(A1 ⊗ I , I ⊗ A2)

�a

b

��∩ (X ⊗ Y ). Since by Corollary 1 the domain

D

�
(A1 ⊗ I , I ⊗ A2)

�a

b

��
is dense in X

α⊗ Y , then

R(λ)

�
λI − (A1 ⊗ I , I ⊗ A2)

�
a

b

��
= I ,

where I is the identity map on X
α⊗ Y .

References

[1] W. Arendt, Grabosch, A. Greiner, G., Groh, U., Lotz, H. P., Moustakas, U., Nagel, R.,

Neubrander, F., and Schlotterbeck, U. One Parameter Semigroups of Positive Operators

(edited by R. Nagel). Lecture Notes in Mathematics, 1184. Springer-Verlag. 1986.

[2] K.J. Engel, and R. Nagel. One Parameter Semigroups for Linear Evolution Equations.

New York: Springer-Verlag. 2000.

[3] J. Goldstein. Semigroups of Linear Operators and Applications. New York: Oxford Uni-

versity Press. 1985.

[4] E. Hille, and R. S. Phillips. Functional Analysis and Semigroups. Rhode Island: Amer.

Math. Soc. Colloq. Publi. 31, Providence. 1957.

[5] T. Ichinose. On the spectra of tensor products of linear operators in Banach spaces. J.

Reine Angew. Math., (244): 119 - 153 .1970.

[6] T. Ichinose. Operators on tensor product of Banach spaces. Transactions of The American

Mathematical Socociety, 1(70): 197 - 219. 1972.

[7] R. Khalil, and S. Al-Sharif. Two parameter semigroups. Journal of Applied Mathematics

and Computation, (156): 403 - 414. 2004.



REFERENCES 898

[8] W. A. Light, and E. W. Cheney. Approximation theory in tensor product spaces, Lecture

notes in maths., 1169. Springer-Verlag .1985.

[9] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equa-

tions, Lecture notes, University of Maryland. 1974.

[10] A.Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equa-

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