Weyl Spectra and Weyl’s Theorem

“Weyl’s theorem” for an operator on a Hilbert space is a statement that the complement in the spectrum of the “Weyl spectrum” coincides with the isolated eigenvalues of finite multiplicity. In this paper we consider how Weyl’s theorem survives for polynomials of operators and under quasinilpotent or compact perturbations. First, we show that if T is reduced by each of its finite-dimensional eigenspaces then the weyl spectrum obeys the spectral mapping theorem, and further if T is reduction-isoloid then for every polynomial p, Weyl’s theorem holds for p(T ). The results on perturbations are as follows. If T is a “finite-isoloid” operator and if K commutes with T and is either compact or quasinilpotent then Weyl’s theorem is transmitted from T to T +K. As a non-commutative perturbation theorem, we also show that if the spectrum of T has no holes and at most finitely many isolated points, and if K is a compact operator then Weyl’s theorem holds for T +K when it holds for T . Introduction. H. Weyl [22] examined the spectra of all compact perturbations T +K of a hermitian operator T and discovered that λ ∈ σ(T + K) for every compact operator K if and only if λ is not an isolated eigenvalue of finite multiplicity in σ(T ). Today this result is known as Weyl’s theorem, and it has been extended from hermitian operators to hyponormal operators and to Toeplitz operators by L. Coburn [7], to several classes of operators including seminormal operators by S. Berberian [2],[3], and to a few classes of Banach space operators [15],[17]. Weyl’s theorem may fail for even the square of T when it holds for T (see [18, Example 1]). In [14], it was shown that Weyl’s theorem holds for polynomials of hyponormal operators. The first aim of this paper is to extend this result via “Berberian” spectra. On the other hand, Weyl’s theorem is liable to fail under “small” perturbations if “small” is interpreted in the sense of compact or quasinilpotent. Recently Weyl’s theorem under small perturbations has been considered in [11],[12],[13], and [18]. The second aim of this paper is to explore how Weyl’s theorem survives under quasinilpotent or compact perturbations. Throughout this paper 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 closed ideal of compact operators on H. If T ∈ L(H) write ρ(T ) for the resolvent set of T ; σ(T ) for the spectrum of T ; π0(T ) for the set of eigenvalues of T ; π0f (T ) for the eigenvalues of finite multiplicity; π0i(T ) for the eigenvalues of infinite multiplicity. An operator T ∈ L(H) is said to be Fredholm if T−1(0) and T (H)⊥ are both finite-dimensional. The index of a Fredholm operator T ∈ L(H), denoted ind (T ), is given by ind (T ) = dimT−1(0)− dimT (H)⊥ (= dimT−1(0)− dimT ∗−1(0)). 2000 Mathematics Subject Classification. Primary 47A10,47A53,47A55