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A unital commutative Banach algebra is spectrally separable if for any two distinct non-zero multiplicative linear functionals φ and ψ on it there exist a and b in such that ab = 0 and φ(a)ψ(b) ≠ 0. Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra is spectrally separable if there are enough elements in such that the corresponding multiplication operators on have the decomposition property (δ). On the other hand,...

Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with ${\sigma }_T\left(e\right)={\sigma }_{\delta }\left(T\right)$, respectively $r_T\left(e\right)=r\left(T\right)$, is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.

Let $𝒳$ be an infinite dimensional complex Banach space and $B\left(𝒳\right)$ be the Banach algebra of all bounded linear operators on $𝒳$. Żelazko [1] posed the following question: Is it possible that some maximal abelian subalgebra of $B\left(𝒳\right)$ is finite dimensional? Interestingly, he was able to show that there does exist an infinite dimensional closed subalgebra of $B\left(𝒳\right)$ with all but one maximal abelian subalgebras of dimension two. The aim of this note is to give a negative answer to the original question and prove that there...

We show that for a linear space of operators $ℳ\subseteq ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ the following assertions are equivalent. (i) $ℳ$ is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map $\Psi =({\psi }_1,{\psi }_2)$ on a bilattice $\mathrm{Bil}\left(ℳ\right)$ of subspaces determined by $ℳ$ with $P\le {\psi }_1(P,Q)$ and $Q\le {\psi }_2(P,Q)$ for any pair $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$, and such that an operator $T\in ℬ({\mathscr{H}}_1,{\mathscr{H}}_2)$ lies in $ℳ$ if and only if ${\psi }_2(P,Q)T{\psi }_1(P,Q)=0$ for all $(P,Q)\in \mathrm{Bil}\left(ℳ\right)$. This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.

We study operators whose commutant is reflexive but not hyperreflexive. We construct a C₀ contraction and a Jordan block operator $S_B$ associated with a Blaschke product B which have the above mentioned property. A sufficient condition for hyperreflexivity of $S_B$ is given. Some other results related to hyperreflexivity of spaces of operators that could be interesting in themselves are proved.