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.

REFERENCES