Another Note on Weyl’s Theorem

“Weyl’s theorem holds” for an operator T on a Banach space X when the complement in the spectrum of the “Weyl spectrum” coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the “Browder spectrum”, which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in T . In this note we try to explore these distinctions. Recall [2, 4, 6] that a bounded linear operator T ∈ BL(X,X) on a Banach space X is Fredholm if T (X) is closed and both T−1(0) and X/cl (TX) are finite dimensional: in this case, we define the index of T by index (T ) = dim T−1(0) − dim X/T (X). An operator T ∈ BL(X,X) is called Weyl if it is Fredholm of index zero, and is called Browder if it is Fredholm “of finite ascent and descent”: equivalently ([6], Theorem 7.9.3) if T is Fredholm and T − λI is invertible for sufficiently small λ 6= 0 in C. The (Fredholm) essential spectrum σess(T ), the Weyl spectrum ωess(T ) and the Browder spectrum ω comm ess (T ) of T are defined by σess(T ) = {λ ∈ C : T − λI is not Fredholm}, (0.1) ωess(T ) = {λ ∈ C : T − λI is not Weyl} (0.2) and ω ess (T ) = {λ ∈ C : T − λI is not Browder}; (0.3) evidently σess(T ) ⊆ ωess(T ) ⊆ ω ess (T ) = σess(T ) ∪ acc σ(T ), (0.4) where we write acc K for the accumulation points of K ⊆ C and σ(T ) for the usual spectrum of T . If we write iso(K) = K \ acc(K) and π 0 (T ) = {λ ∈ iso σ(T ) : 0 < dim (T − λI)−1(0) <∞} (0.5) for the isolated eigenvalues of finite multiplicity, and ([6], (9.8.3.4)) π00(T ) = σ(T ) \ ω ess (T ) (0.6) for the Riesz points of T , then ([6], Theorem 9.8.4) with the help of the “punctured neighbourhood theorem” iso σ(T ) \ σess(T ) = iso σ(T ) \ ωess(T ) = π00(T ) ⊆ π 0 (T ). (0.7) Received by the editors December 18, 1995. 1991 Mathematics Subject Classification. Primary 47A10.