Invariant Subspaces for Semigroups of Algebraic Operators

T. Laffey showed (Linear and Multilinear Algebra6(1978), 269–305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of Hthis result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair {A, B} of arbitrary bounded operators satisfying rank (AB−BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable.     

On the West Decomposition of Riesz Operators

Let T be a Riesz operator on a Banach space and let {_n} be anenumeration of the nonzero eigenvalues of T, each repeated accordingto algebraic multiplicity. Then a sufficient condition thatT admit a West decomposition T = C + Q, where C is compact andQ is quasinilpotent, is that n[_n] < . Furthermore, C andQ can be chosen to have a quasinilpotent commutator.     

Burnside’s theorem in the setting of general fields

Archiv der Mathematik - We study the mean of the values of the zeta-function on a generalized arithmetic progression on the critical line.     

Domino 'Staudinger-aza-Wittig-1,5-phosphonium-rearrangement-fragmentation' reactions of 1-azido-2-hydroxy-4,6-dioxohexanes

The domino 'Staudinger-aza-Wittig-1,5-phosphonium-rearrangement-fragmentation' reaction of 1-azido-2-hydroxy-4,6-dioxohexanes allows a convenient synthesis of functionalized 1-acetamido-2-alkylidenecyclopentanes.     

Electrochemically controlled in-tube solid phase microextraction

We report a new in-tube solid phase microextraction approach named electrochemically controlled in-tube solid phase microextraction (EC in-tube SPME). This approach, which combined electrochemistry and in-tube SPME, led to decrease in total analysis time and increase in sensitivity. At first, pyrrole was elctropolymerized on the inner surface of a stainless steel tube. Then, the polypyrrole (PPy)-coated in-tube SPME was coupled on-line to liquid chromatography (HPLC) to achieve automated in-tube SPME–HPLC analysis. After the completion of EC-in-tube SPME–HPLC setup, the PPy-coated tube was used as working electrode for uptake of diclofenac as target analyte. Extraction ability of the tube in presence and in absence of applied electrical field was investigated. It was found that, under the same extraction conditions, the extraction efficiency could be greatly enhanced by using the constant potential. Important factors are also optimized. The detection limit (S/N = 3) and precision were 0.1 μg L⁻¹ and 4.4%, respectively.     

An operator not satisfying Lomonosov's hypothesis

An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosov's invariant subspace theorem does not apply to every operator.     

Limitations on the size of semigroups of matrices

If a nonzero linear functional has finite, countable, or bounded range when restricted to an irreducible semigroup of complex matrices, it is shown that itself has the same property. Similar results are proven under the hypothesis that a nontrivial ideal of is finite, countable, or bounded.     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.