NEW BROWDER AND WEYL TYPE THEOREMS

Abstract

In this paper we introduce and study the new properties (W �), (UWa), (UW E) and (UW �). The main goal of this paper is to study relationship between these new properties and other Weyl type theorems. Moreover, we reconsider several earlier results obtained respec- tively in (11), (18), (14), (1) and (13) for which we give stronger versions. by σa(T) the approximate point spectrum of T. If the range R(T) of T is closed and α(T) < ∞ (resp. β(T) < ∞), then T is called an upper semi-Fredholm (resp. a lower semi-Fredholm) operator. If T ∈ L(X) is either upper or lower semi Fredholm, then T is called a semi-Fredholm operator, and the index of T is defined by ind(T) = α(T) − β(T). If both of α(T) and β(T) are finite, then T is called a Fredholm operator. An operator T ∈ L(X) is called a Weyl operator if it is a Fredholm operator of index zero. The Weyl spectrum σW(T) of T is defined by σW(T) = {λ ∈ C | T − λI is not a Weyl operator}. For a bounded linear operator T and a nonnegative integer n, define T(n) to be the restriction of T to R(T n ), viewed as a map from R(T n ) into R(T n ) (in particular T(0) = T). If for some integer n the range space R(T n ) is closed and T(n) is an upper (resp. a lower) semi-Fredholm operator, then T is called an upper (resp. a lower) semi-B-Fredholm operator. A semi-B-Fredholm operator T is an upper or a lower semi-B-Fredholm operator, and in this case the index of T is defined as the index of the semi-Fredholm operator T(n), see (12). Moreover if T(n) is a Fredholm operator, then T is called a B-Fredholm operator, see