Weyl’s theorem for totally hereditarily normaloid operators

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.     

Spaces hereditarily of κ-type and point κ-type

In this paper we discuss the structure of regular spaces hereditarily of κ-type and point κ-type. In particular, it is shown that a regular spaceX is hereditarily of point κ-type iff for everyp∈X there is a setE (possibly empty) of isolated points ofX such that (1)E∪{p} is compact and (2)X(E∪{p}, X)≤κ.     

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ? does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, S _r be the image of S under the translation of ? by a rational number r and F = {S _r : r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ? possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ? O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.     

k－spaces and Products of Spaces with σ－hereditarily Closure－Preserving k－networks

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