Advances in Pure Mathematics
Vol.2 No.5(2012), Article ID:22798,4 pages DOI:10.4236/apm.2012.25044
On the Generality of Orthogonal Projections and e-Projections in Banach Algebras
Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran
Email: moh.asgari@iauctb.ac.ir, s_karimizad@yahoo.com, rahimi@iauctb.ac.ir
Received March 2, 2012; revised April 18, 2012; accepted April 28, 2012
Keywords: generalized orthogonal projection; orthogonal projection; generalized e-projection; e-projection
ABSTRACT
In this paper we develop the orthogonal projections and e-projections in Banach algebras. We prove some necessary and sufficient conditions for them and their spectrums. We also show that the sum of two generalized orthogonal projections and
is a generalized orthogonal projection if,
. Our results generalize the results obtained for bounded linear operators on Hilbert spaces.
1. Introduction
Orthogonal projections on Hilbert spaces play important roles in many applications in mathematics, science and engineering including signal and image processing, integral equations and many other areas. In this article we introduce generalized orthogonal projections, generalized e-projections in Banach algebras and we show that they share many useful properties with their corresponding notions in C*-algebras. For more information we refer to the articles by Berkson [1], Schmoeger [2], Du et al. [3], Grob et al. [4] and Lebtahi et al. [5].
The paper is organized as follows: Section 1, contains a few elementary definitions and results from Banach algebras theory. In this section we introduce the concepts of numerical range and the spectrum and the spectral radius of an element and investigate their properties. In section 2, we introduce the generalized orthogonal projections, generalized e-projections in Banach algebras and we study some necessary and sufficient conditions for them and their spectrums.
Throughout this paper, will denote a complex unital Banach algebras (with unit 1) and
denote the dual space of
. For
,
, define the support set at x
Then for all define the sets
and their union, the numerical range of
We also denote the spectrum and the spectral radius of 𝔞 by and
respectively.
Lemma 1.1. [6]. Let then 1)
is a compact convex subset of ℂ.
2)
3)
4)
5)
The fundamental link between the numerical range of and the group
is as follows:
By Lemma 5.2 of [6], an element is said to be hermitian if
or equivalently
, equivalently
.
We denote the set of all hermitian elements of by
. It is well-known that if
then the convex hull of the spectrum satisfies
and
. Also
is closed real subspace of
and
and if
then
Furthermore, if is a
-algebra, then by Example 5.3 of [6],
if and only if
.
An element is called positive if
. We denote the set of all positive elements of
by
. By Theorem 5.14 of [6],
if and only if
and
. In the real Banach space
, the set
is a normal closed cone in which 1 is an interior point. Let
. Since
, hence each element of
has a unique representation of the form
with
. If we define a linear involution
from
to itself by
then
with the norm of
is a complex Banach space and
is a continuous linear involution on
. In the general case
is not an algebra and
is not an involution because in particular
. However, if
and for every
with
,
, then
is a complex unital
-algebra with continuous involution
and
is its set of self-adjoint elements [6].
We say that is normal if
with
and
. Observe that
is normal if and only if
and
. An element
satisfying
is called a partial isometry.
Definition 1.2. Let be a complex unital
-algebra, then
is called an orthogonal projection if
. Moreover
is called a Moore-Penrose invertible if there exists some
such that
In this case is the Moore-Penrose inverse of
and usually denoted by
. If
is Moore-Penrose invertible, then
is unique.
Definition 1.3. Let be a complex unital Banach algebra. An element
is called an orthogonal projection if
and
. Moreover
is called a Moore-Penrose invertible if there exists
such that
then the element is called the Moore-Penrose inverse of
, and it also will be denoted by
. The MoorePenrose inverse of
is unique in the case when it exists.
If is Moore-Penrose invertible then the equality
does not hold in general. Hence it is interesting to distinguish such elements.
Definition 1.4. An element of a unital Banach algebra
is said to be e-projection if there exists
and
.
If , we define the centralizer of
by
We say that commutes if any two elements of
commute with each other. If
is commutes and
then by Theorem 11.22 of [7]
is a commutative Banach algebra (with unit 1),
and
for every
.
Lemma 1.5. [7]. Let be a complex unital Banach algebra, let
be a normal element,
. If
is the set of all nontrivial complex homomorphisms of
. Then 1)
2)
3) for all
and
4) If, then
2. g-Orthogonal Projections and Generalized e-Projections
Definition 2.1. An element is called generalized orthogonal projection or simply a g-orthogonal projection if there exists a natural number
such that
Also
is said to be generalized e-projection if there exists
and
Theorem 2.2. Suppose that is a g-orthogonal projection. Then 1)
is normal.
2)
3) If for all
then
and
Proof.
1) Since hence we have
2) Let, then by the Lemma 1.5 there is a
such that
, thus we have
Now if, then
and hence
which implies that
with
3) Since hence
Using the Lemma 1.6 (4) we have
and
This yields
Thus we have
Since
hence
which shows that
Therefore
The second implication is obvious.
Theorem 2.3. Let be a generalized e-projection. Then 1)
and
is an e-projection.
2)
Proof.
1) Since hence we have
and
.
2) This follows immediately from Theorem 2.2.
Theorem 2.4. Let and
Then the following statements hold:
1) If is normal, then
2) If for all
, then
is a g-orthogonal projection.
Proof.
1) Put Since
is normal hence
and so
Now suppose that
, then there exists some
with
.
Let, since
thus
. This shows that
, and so
From this we have
.
2) By the Murphy’s Theorem [8], so
is normal. Now from
we obtain
Using the Lemma 1.6(4) and applying (1) we have
Since for all
, hence
This shows that
Since, thus
, which implies that
.
Theorem 2.5. An element is a g-orthogonal projection if and only if u is normal and
Proof. If u is a g-orthogonal projection then the implication follows from the Theorem 2.2. Conversely suppose that u is normal and
For every we define the Reiesz projection of u associated with
by
where is a smooth closed curve which
interior to
and
exterior to
. Then by Proposition VII.4.11 of [9], u has the representation as folslows:
where for all
and and
and for
and
. Now we compute
Theorem 2.6. Suppose that and
If u has the representation
where
is a Riesz projection of u associated with and
is a smooth closed curve which
interior to
and
exterior to
. Then u is a generalized eprojection.
Proof. Since for all we have
hence
In the general case if then it does not follows that
.
Example 2.7. Let with pointwise multiplication and let
be defined by
Define the norm on
by
Then is a complex commutative Banach algebra with unit
If
then the following properties are shown in [6].
and but
and each element of
is normal. Furthermore if
, then
Lemma 2.8. Let as in Example 2.7. Then
is a g-orthogonal projection if and only if
Proof. For all we have
hence if and only if
Now if
then by Theorem 2.4, we have which implies that
. The converse implication follows immediately from Theorem 2.2.
Lemma 2.9. Let as in Example 2.7 and
. Then the following conditions are equivalent:
1)
2)
3)
4)
Proof.
1) 2) follows from the Lemma 2.8.
2) 3) Let
Since
hence
If then
thus
with
or
Let
Since
hence
and
It follows
therefore
This shows that
with
.
(3) (4): Clear.
(4) (1): follows from the spectral mapping theorem.
Theorem 2.10. Let be g-orthogonal projections such that
. Then
is a g-orthogonal projection.
Proof. By the hypotheses
for all hence we have
3. Acknowledgements
The author expresses his gratitude to the referee for carefully reading of the manuscript and giving useful comments.
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