On the structure of polynomially compact operators

Abstract

Throughout this note, let H denote an infinite dimensional separable Hilbert space, let L(H) denote the algebra of bounded linear operators on H and let K(H) denote the ideal of compact operators on H. If T ∈ L(H) write σ(T ) for the spectrum of T . If T ∈ L(H) is a Fredholm operator, that is, T has finite dimensional null space and its range of finite co-dimension, then the index of T , denoted ind (T ), is given by ind (T ) = dim T−1(0) − dim T (H)⊥ (= dim T−1(0) − dim T ∗−1(0)). An operator T ∈ L(H) is called a Weyl operator if it is Fredholm of index zero. An operator T ∈ L(H) is called a Browder operator if it is Fredholm “of finite ascent and descent”: equivalently ([5, Theorem 7.9.3]) if T is Fredholm and T − λI is invertible for sufficiently small λ 6= 0 in C. The essential spectrum σe(T ), the Weyl spectrum ω(T ) and the Browder spectrum σb(T ) of T ∈ L(H) are defined by