Synthesis of o-sulfamidotriazobenzenes from 1,1'-sulfonylbis(benzotriazole)

Easily accessible 1,1'-sulfonylbis(benzotriazole) (Bt2SO2, 1) reacts with secondary amines at room temperature to afford (i) the corresponding o-sulfamidotriazobenzenes 2a-d (53-75%) via concurrent substitution of the first and ring opening of the second benzotriazolyl group and (ii) N-sulfonylbenzotriazoles 3b,c,e,f (7-73%). 1-(Morpholine-4-sulfonyl)-1H-benzotriazole 3c reacts with piperidine, pyrolidine, and N-methylpiperazine under microwave irradiation (120 W) at 120 degrees C for 10 min to give the unsymmetrical sulfamides 4a-c (80-90%).     

On exponential stability of contraction semigroups

A semigroup [T(t)] on a Hilbert space is exponentially stable if there exist real constants M≥1 and α>0 such that ∥T(t)∥≤Me ^−αt for every t≥0. If [T(t)] is a strongly continuous contraction semigroup, then it is proved that we can set M=1 in the definition of exponential stability if and only if the generator A of [T(t)] is boundedly strict dissipative (just a strict dissipative A is not enough).     

On the reduction of a contraction semigroup to a completely non unitary semigroup

According to a Theorem of B. Sz.-Nagy and C. Foia?, every strongly continuous semigroup of contraction operators on a Hilbert space, can be decomposed into a completely non unitary part and a unitary part. In this note we wish to show that by appropriately perturbing its generator, a contraction semigroup can be reduced to a completely non unitary one. In control theory, such a perturbation is related to the so called state feedback, and the reduction presented here has application in the problem of stabilizing linear control systems on a Hilbert space. This will be briefly discussed.     

Stabilities of Hilbert space contraction semigroups revisited

A necessary and sufficient condition for exponential stability of Hilbert space contraction semigroups is obtained in terms of an inequality involving the dissipative norm associated with the generator of the semigroup and with the Hilbert space norm.