ABSTRACT
Let X be a Banach spaces , and T a linear operator from a linear subspace D(T) of X to X, where D (T) denotes the domain of T. Denote by R(T) , N(T) the range of T and the null space of T, respectively. For a complex number A , set T>. := AI - T, where I is the identity. The resolvent set , p(T) , is the set of A such that N(T>. ) = {O} , R(T>. ) = X , and T;:l i s continuous, where R(T>. ) denotes the closure of R(T>. ) . The set a (T) := C \ p(T) is called the spectrum of T. It is classified into three disjoint subsets : the continuous spectrum Cu (T) = {>. E a(T) : N(T>. ) = {O} , R (T>.) = X, and T;l i s not continuous; the residual spectrum Ru (T) = {A E a (T) : N(T>. ) = {O} , R(T>. ) :j:. X} ; the point spectrum Pu (T) = {A E a(T) : N(T>. ) :j:. {O} } . If T is a closed operator, A E p(T) if and only if N(T>. ) = {O} , R(T>. ) = X .
