By Jayaram C.

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On (0, T). If one also assumes that E has the Radon--Nikodym property (for example, is reflexive), then conditions a), b) are equivalent with the following: c) u is an LD-solution of the problem (6), (7), and u is absolutely continuous on any [a, b] ~ (0, T ) ; d) u i s an L D - s o l u t i o n o f t h e p r o b l e m ( 6 ) , g r a b l e on any [a, b ] c ( 0 , T). (7), and t h e f u n c t i o n t + IIIAu(t)III is inte- The e q u i v a l e n c e o f c o n d i t i o n s a) and b) f o r f = O, F = {0} g o e s b a c k t o B r e z i s and P a z y [ 2 4 6 ] , C r a n d a l l and L i g g e t t [ 2 8 9 ] , and M i y a d e r a [ 5 1 1 ] .

E. on [~, 8]. We note that in the literature the notation < , >s and < , >i is often used instead of < , >+ and < , >_, and also T(x, z) and [x, z] instead of [z, x]. Sato [603] gave explicit formulas for [z, x] (and thus for +) in various function spaces. ----llm ess sup {z (s) x (s):J x ( s ) ] > IIx II . - 8 } , el0 and _ i s o b t a i n e d b y r e p l a c i n g sup b y i n f ; +-=Iz(s)x(s)lx(s)lP-ids/llxllP-2 (x--/=O) ( i n t h e l a s t t h r e e c a s e s S i s an a r b i t r a r y space with measure).

E. on S for 0~AxEF+. Bratteli, Kishimoto, and Robinson [226, 421] showed that if the self-adjoint operator A in Li(S) generates a contracting positivity-preserving semigroup U(t), and f:R+ § R+ is such that/6C=(0, ~), (--I),/t"+')(r)>O for r > 0, n = 0, I .... 0 and f ~ const, then V(t) strengthens positivity if and only if U(t) strengthens positivity. 29. Comparison of Semigroups. Let A, B be complete infinitesimal operators of semigroups U(t), V(t) in E. We shall compare the solutions of the equations du/dt = Au, dv/dt = Bv with common initial condition x.

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2-Join decomposition lattices by Jayaram C.

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