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.     

Synthesis of (S)‐3‐heteroaryl‐2‐hydroxy‐l‐propyl benzoates by ‘ring switching’ methodology

(S)‐5‐Benzoyloxymethyl‐3‐[(E)‐(dimethylamino)methylidene]tetrahydrofuran‐2‐one (6), prepared in 5 steps from L‐glutamic acid (1), was used as precursor in a one step ‘ring switching’ synthesis of (S)‐2‐hydroxy‐3‐heteroaryl‐l‐propyl benzoates 13‐18, 23, 24. In the reaction of 6 with 2‐aminopyridine (21) and 2‐amino‐4,6‐dimethylpyrimidine (22) the corresponding dimethylamine substitution products (25, 26) were obtained.     

Matrix semigroup homomorphisms into higher dimensions

We study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all n-by-n matrices over an algebraically closed field of characteristic zero to the semigroup of m-by-m matrices over the same field. We prove that every non-degenerate homomorphism from the multiplicative semigroup of all n-by-n matrices to the semigroup of (n + 1)-by-(n + 1) matrices when n ? 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices to the semigroup of 5-by-5 matrices is reducible.     

Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup

We prove that the minimal cardinality of a semitransitive subsemigroup in the singular part $\mathcal{I}_{n}\setminus \mathcal{S}_{n}$ of the symmetric inverse semigroup $\mathcal{I}_{n}$ is 2n?p+1, where p is the greatest proper divisor of n, and classify all semitransitive subsemigroups of this minimal cardinality.     

Zur spektrochemischen Bestimmung von Aluminium in Uran

Ohne Zusammenfassung