# THE JUMP OF A SEMI-FREDHOLM OPERATOR

• Lee, Dong-Hak (Department of Mathematics, Kang Won University, Choon Chun 200-701) ;
• Lee, Woo-Young (Department of Mathematics, Sung Kyun Kwan University, Suwon 440-746)
• Published : 1994.07.01

#### Abstract

In this note we give some results on the jump (due to Kato [5] and West [7]) of a semi-Fredholm operator. Throughout this note, suppose X is an Banach space and write L(X) for the set of all bounded linear operators on X. A operator $T \in L(x)$ is called upper semi-Fredholm if it has closed range with finite dimensional null space, and lower semi-Fredholm if it has closed range with its range of finite co-dimension. It T is either upper or lower semi-Fredholm we shall call it semi-Fredholm and Fredholm it is both. The index of a (semi-) Fredholm operator T is given by $$index(T) = n(T) = d(T),$$ where $n(T) = dim T^{-1}(0)$ and d(T) = codim T(X).