Abstract
A Banach space operatorT ∈B(χ) is said to behereditarily normaloid, denotedT ∈ ℋN, if every part ofT is normaloid;T ∈ ℋN istotally hereditarily normaloid, denotedT ∈ ℑHN, if every invertible part ofT is also normaloid. Class ℑHN is large; it contains a number of the commonly considered classes of operators. The operatorT isalgebraically totally hereditarily normaloid, denotedT ∈a — ℑHN, both non-constant polynomialp such thatp(T) ∈ ℑHN. For operatorsT ∈a − ℑHN, bothT andT* satisfy Weyl’s theorem; if also either ind(T−μ)≥0 or ind(T−μ)≤0 for all complexμ such thatT−μ is Fredholm, thenf(T) andf(T*) satisfy Weyl’s theorem for all analytic functionsf ∈ ℋ(σ(T)). For operatorsT ∈a — ℑHN such thatT has SVEP,T* satisfiesa-Weyl’s theorem.
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Duggal, B.P. Weyl’s theorem for totally hereditarily normaloid operators. Rend. Circ. Mat. Palermo 53, 417–428 (2004). https://doi.org/10.1007/BF02875734
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DOI: https://doi.org/10.1007/BF02875734