R-bounded approximating sequences and applications to semigroups

Abstract

It is shown that on certain Banach spaces, including C[0, 1] and L 1 [0, 1], there is no strongly continuous semigroup (T t ) 0<t <1 consisting of weakly compact operators such that (T t ) 0<t <1 is an R-bounded family. More general results concerning approximating sequences are included and some variants of R-boundedness are also discussed.  2004 Elsevier Inc. All rights reserved.

Rademacher bounded families of operators on 𝐿₁

“…Precisely, we show that if A is R-sectorial and > 0, then there is an invertible operator U : L 1 → L 1 with U −I < such that for some positive Borel function w we have U (D(A)) ⊃ L 1 (w). This refines both the results of [5] and [8].…”

“…In [8] it was shown that if A has an H ∞ -calculus, then A is bounded on any reflexive subspace of D(A) (with the graph norm); this had the implication that there are very few examples of sectorial operators with an H ∞ -calculus on L 1 and, in particular, essentially no reasonable differential operator can have this property. In [5] it was shown that there are no R-bounded strongly continuous semigroups on L 1 consisting of weakly compact operators; it also follows from the results of [5] that if A is an R-sectorial operator on L 1 , then the resolvent R(ζ, A) can never be a weakly compact operator.…”

“…It follows from results of [5] that if −A is the generator of a semigroup such that e −tA is weakly compact for t > 0, or if the resolvents R(z, A) are weakly compact operators, then A cannot be R-sectorial. The next theorem strengthens this conclusion.…”

“…Therefore we can use the results of [5] to show that they cannot be R-sectorial. In contrast, resolvents of differential operators on unbounded domains are, in general, not compact.…”

On R-boundedness of Unions of Sets of Operators

“…During the past few years a theory of L p -multipliers for operator valued functions has been developed by means of the notion of R-boundedness of sets of operators, see for example [1,2,3,4,5,6,7,8,10,12,13,17,18,19]. This theory has been applied to study maximal regularity of certain abstract evolution equations.…”