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.     

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Book reviews

     

Crystal structures of two isostructural compounds: a second polymorph of dipotassium hydrogen citrate,K2HC6H5O7, and potassium rubidium hydrogen citrate,KRbHC6H5O7

The crystal structures of a new polymorph of dipotassium hydrogen citrate, 2K⁺·HC₆H₅O₇^2?, and potassium rubidium hydrogen citrate, K⁺·Rb⁺·HC₆H₅O₇^2?, have been solved and refined using laboratory powder X‐ray diffraction and optimized using density functional techniques. In the new polymorph of the dipotassium salt, KO₇ and KO₈ coordination polyhedra share corners and edges to form a three‐dimensional framework with channels parallel to the a axis and [111]. The hydrophobic methylene groups face each other in the channels. The un‐ionized carboxylic acid group forms a strong charge‐assisted hydrogen bond to the central ionized carboxylate group. The hydroxy group forms an intermolecular hydrogen bond to a different central carboxylate group. In the potassium rubidium salt, the K⁺ and Rb⁺ cations are disordered over two sites, in approximately 0.72:0.28 and 0.28:0.72 ratios. KO₈ and RbO₉ coordination polyhedra share corners and edges to form a three‐dimensional framework with channels parallel to the a axis. The un‐ionized carboxylic acid group forms a strong charge‐assisted hydrogen bond to an ionized carboxylate group. The hydroxy group forms an intermolecular hydrogen bond to the central carboxylate group. Density functional theory (DFT) calculations on the ordered cation structures suggest that interchange of K⁺ and Rb⁺ at the two cation sites changes the energy insignificantly.     

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Polymeric complexes of cornstarch and waxy cornstarch phosphates with milk casein and their performance as biodegradable materials

Complexes of phosphated cornstarch and waxy cornstarch with casein were prepared and characterised. They were prepared from casein in defatted milk and corn and waxy corn starches phosphated to degree of substitution values (DS) of 0.0637 and 0.0968, respectively. The components were blended in starch to casein ratios of 2:1, 1:1, and 1:2, then precipitated with hydrochloric acid. Aqueous solubility, water binding capacity, IR spectra, and thermal analysis (thermogravimetry, TG, and differential thermogravimetry, DTG) of the precipitates revealed that they were not simple physical mixtures of the components. The components interact with one another electrostatically with involvement of the starch phosphate groups and the peptide bonds of casein as documented by the IR spectra. Because of their insolubility in 7 M aqueous urea solution they might also be considered as complexes in which the components were chemically bound. Enzymatic studies showed that they are biodegradable materials.     

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