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Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 823-824
SUPERSETS FOR THE SPECTRUM OF ELEMENTS
IN EXTENDED BANACH ALGEBRAS
MORTEZA SEDDIGHIN
Division of Natural Sciences
Lewis Clark College
Lewiston, ID 83501
(Received June I, 1987 and in revised form October 21, 1987)
ABSTRACT.
,
If A is a Banach Algebra with or without an identity, A can be always
extended to a Banach algebra A with identity, where A is simply the direct sum of A
and
In this note we find supersets for the
the algebra of complex numbers.
spectrum of elements of A.
KEY WORDS AND PHRASES.
Banach Algebra, spectrum of elements and quasi-singular
elements.
1980 AMS SUBJECT CLASSIFICATION CODE.
46H99.
INTRODUCTION.
Let A be a Banach algebra.
Then we know that the set A
(x,) (y,B)
{(x,): x
A,
complex}
(x + y, +8) and
together with the operations (x,) + (y,8)
(xy + 8x + ey, B), and norm
II
(x,)II
Ilxll
+
II
is a Banach algebra,
Although this is usually done for algebras A
whose identity element is (0,I).
without identity, to extend them to algebras with identity; we can also start with a
Banach algebra A with identity (In this case the identity of A is no more an identity
for A).
2.
MAIN RESULTS.
DEFINITION Z.I. An element x in a Banach algebra A is called quasl-regular if
xoy
yox
operation,
0 for some y
A, where xoy
x
+ y
xy.
xoy is called the circle
x is called quasl-slngular if it is not quasi-regular.
For an element x
in A, the special radius of x is defined by
r(x)-- lira
n-o
II xn II
I/n
THEOREM 2,1. Let A be the extension of A, as above and let
spectrum of (x,) in A, then
if A already has an identity, and
X
((x,a)) {:
I- cl
=< rCx)}
6
((x,)) denote the
824
M. SEDDIGHIN
if A does not have an identity.
First suppose A has an identity.
PROOF.
(x,u)
(x, a-A).
(0,l)
(xy +
(x, a- ) (y,
Now, if
6
# a and
x
+ (a- )y,l).
(x + ), then the equation
+_I
xy
+ (a
x
A) y
(2.1)
0
To see this, write (2.1) as (
has a solution.
(a
A
Let % be a complex number, then
# a, then
If
)[x +
6 (x + a).
A
]y
-x or
~{6A(X)
{a}
This implies
d;
a--(x+a-
y
+ a}
((x,a)) _c_ {a} U
X
6A(X
)
-I
2
A)xy+x+ (- A) y
(-x).
6 ((x,a)),
(x+a-
)-I-
0 or
exists since
and, therefore, we have:
+ a).
Now suppose A does not have an identity and let A # a.
If
x is quasi-irregular,
then there exists an element z in A such that:
a_-----X
If we take y
+
x
+
O.
z
z, then we have:
+--X_I
xy
Hence, ~{a}
xz
~{AI
x
+ (a
) y
0.
quasi-singular}
((x,a)). For an element
<
a in a Banach algebra, the inequality r(a)
implies a is a quasi-regular with quasiinverse a
Z
# a :/_ x
a
n
is
(Rickart [I]).
Hence, for an element
x to be quasi-
REFERENCES
I.
RICKART, C.E.
(1960).
General Theory of Banach Algebras, D. Van Nostrand, Princeton,