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

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 v is a generalized orthogonal projection if, u 0 uv vu   . Our results generalize the results obtained for bounded linear operators on Hilbert spaces.


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  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 Then for all define the sets a   , , , We also denote the spectrum and the spectral radius of by   a  and   r a respectively.Lemma 1.1.[6].Let , , a b The fundamental link between the numerical range of a   and the group   : ta e t  is as follows: We denote the set of all hermitian elements of by      .It is well-known that if then the convex hull of the spectrum satisfies   has a unique representation of the form with .If we define a linear involution from . However, if and for every , then is a complex unital -algebra with continuous involution and   is its set of self-adjoint elements [6].

 
is called an orthogonal projection if .Moreover is called a Moore-Penrose invertible if there exists some such that then the element is called the Moore-Penrose inverse of , and it also will be denoted by .The Moore-Penrose 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 .
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 Lemma 1.5.[7].Let be a complex unital Banach algebra, let for every .

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 , then by the Lemma 1.5 there is a The second implication is obvious.Theorem 2.3.Let be a generalized e-projection.Then and is an e-projection.

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: there exists some 2) By the Murphy's Theorem [8], so u is normal.Now from Using the Lemma 1.6(4) and applying (1) we have 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 Then by Proposition VII.4.11 of [9], u has the representation as folslows: where for all Theorem 2.6.Suppose that and  is a Riesz projection of u associated with  and C  is a smooth closed curve which  interior to C  and exterior to Then u is a generalized eprojection.
Proof.Since for all we have In the general case if then it does not follows that If then the following properties are shown in [6].

Acknowledgements
The author expresses his gratitude to the referee for carefully reading of the manuscript and giving useful comments.

 
This shows that with .