On the Generality of Orthogonal Projections and e-Projections in Banach Algebras ()
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.