الفهرس 
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Abstract
In this thesis, we aim to introduce the concepts of the joint centre valued ranges of an n-tuple of elements in a W*-algebra. Also we study the idea of the translation-invariant of the numerical ranges and the joint numerical ranges as well as some related topics are studied.
This thesis consists of four chapters :
In the first chapter, Chapter 1, we are concerned with some of the basic definitions and main theorems of the general theory of Banach algebras, C*-algebras and W*-algebras, that is the necessary background which will be useful in the later chapters.
In Chapter 2, we introduce the concepts of the joint centre valued range and the joint maximal centre valued range. We shall study the relationship between these two sets and the sets of the joint numerical range and the joint maximal numerical range. Some properties and applications of the two sets are studied.
In Chapter 3, we introduce the notion of the joint essential centre valued range and the joint maximal essential centre valued range. We shall study the relationship between these two sets and the sets of the joint essential numerical range and the joint maximal essential numerical range. Some properties and applications of the two sets are studied.
In the last chapter, Chapter 4, we study the idea of the translation-invariant of the numerical ranges and the joint ranges. Also we study the boundary of the joint maximal numerical range. Finally we study the norm of an operator on Calkin algebra.
Throughout this thesis R and C will denote the real and complex fields respectively, and we shall write R+ for the set of positive real numbers. The dual space; the set of all bounded (continuous) linear functionals on a Banach space X, will be denoted by X*. We shall write B(H) for the set of all bounded linear operators on a Hilbert space H, K(H) for the set of all compact linear operators on H and Q(H) for the Calkin algebra B(H)/K(H).
We wish to point out that the results in Chapter 2 are accepted for publication in “The Proceedings of the Mathematical and Physical Society of Egypt” with title :
“On the joint maximal centre valued range”.
Also the results in Chapter 3 have been submitted for publication in “The Proceedings of the Mathematical and Physical Society of Egypt” with title :
“On the joint essential and joint maximal essential centre valued range”.