The essential spectrum

Abstract

If B(H) is the algebra of bounded linear operators on a Hilbert space (H) and K is the ideal of compact operators on (H), one forms the Calkin algebra B(H)/K and the natural map π: B(H) → B(H)/K. Recall that A∈B(H) is Fredholm if π(A) is invertible in B(H)/K. A well-known theorem [19, p. 356] says that A is Fredholm precisely when Rng A is closed and both ker A and H/RangA are finite dimensional. An operator A is semi-Fredholm if π(A) is either right or left invertible in B(H)/K. Equivalently. A is semi-Fredholm if and only if RngA is closed and either ker(A) or H/RngA is finite dimensional. We also use the notation $$ \sigma (A): = \{ \lambda \in \mathbb{C}:\lambda I - A is not invertible\} (spectrum of A),$$ $$ \sigma (A): = \{ \lambda \in \mathbb{C}:\lambda I - A is not Fredholm\} (essential spectrum of A).$$ Note that σ e (A) ⊂ σ(A). For a semi-Fredholm operator A let $$ ind(A): = dim ker A - dim (H/Rng A)$$ be the index of A. When the set ℤ∪{±∞| is endowed with the discrete topology, the map A→ ind(A) (from the set of semi-Fredholm operators to ℤ∪{±∞| is continuous [19, p. 361].