American Journal of Applied Sciences 7 (3): 386-389, 2010
ISSN 1546-9239
© 2010Science Publications
Corresponding Author: Sabra Ramadan, Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097,
Jazan-KSA
386
Sequences of Closed Operators and Correctness
Sabra Ramadan and Abdulla Al-Hossain
Department of Mathematics, Faculty of Science,
Jazan University, P.O. Box 2097, Jazan-KSA
Abstract: In applications and in mathematical physics equations it is very important for mathematical
models corresponding to the given problem to be correct given. In this research we will study the
relationship between the sequence of closed operators A
n
→A and the correctness of the equation Ax = y.
Also we will introduce the criterion for correctness.
Key words: Kvazi method, closed operator, correctness, Banach-Shtengaus theorem
INTRODUCTION
The operator A:U V → may be defined as the
limit of some sequence of operators
n
A :U V → . For
example the differential operator [ ] [ ]
1
d
:C a,b C a,b
dt
→
is defined as the limit of the sequence of the operators
n
n 1
1
A f(t) (f(t ) f (t)) n
n
∞
=
= + -
, where [ ]
1
n
A :C a,b
[ ] C a,b → n N 2200 ∈ . And in many places in Applied and
theoretical mathematics the properties of the operator A
implies directly from the properties of some sequence
of operators { }
n
1
A
∞
with some restrictions.
There arise the following question: Is it possible to
know the important property as correctness of the
equation Ax y = knowing properties of the sequence of
operators that converges to the operator A?
Also during present research (Radyno et al., 1993a;
1993b; Tkan and Ramadan, 1992; Ramadan and Tkan,
1992; Ramadan and Jehad, 2000; Ramadan, 1993;
1997; 1998; 1999; 2007a; 2007b; 2007c) in many
places were arisen questions about relationship between
a special sequences of operators and their limits. For
example relationship between the sequences { }
n
F( f) ϕ ,
{ }
n
f ϕ and f, where F is the Fourier Transform, and f-be
a generalized function (distribution).
Definition 4: The linear operator A:U V → is called
closed if and only if the sequence { }
n
n 1
x D(A)
∞
=
⊂ ,
and
n n
(x ,A(x )) (x,y) → , then x D(A) ∈ and y Ax = (the
graph of the operator A is closed in XxY).
In (Ramadan and Jehad, 2000) we proved the
following theorems:
Let X, Y-be Banach spaces; and
n
A,A :X Y → are
closed operators such that
n
D(A) D(A ) = and
n
n
lim A x Ax x D(A)
→∞
= 2200 ∈ (A
n
strongly converges to A).
And suppose that all operators
1
n
A :Y X
-
→ exist
and continuous and there is a positive constant C>0
such that for all n and for all
1
n
y Y,A C
-
∈ ≤ then:
Theorem 1: For each y R(A) ∈ there is a unique
x D(A) ∈ such that Ax y = .
Theorem2: For each
1 1
n
y R(A), A y A y
- -
∈ → , where
Ax y = , then the operator
1
A y
-
is defined and
bounded on R(A) .
Theorem 3: If
n n n
Ax y = and
n
y y → , then
n
x x → ,
where Ax y = .
Theorem 4: If
n n n
Ax y = and
n
x x → ,
n
A C ≤ , Ax y =
then
n
y y → .
Preliminaries: In this study we will use the following
conventional notations and definitions:
L(X,Y) The set of all linear continuous operators
A:X Y →
D(A) The domain of the operator A
R(A) The range of the operator A
A
-1
The inverse of the operator A