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Journal of the London Mathematical Society 1999 60(2):581-588; doi:10.1112/S0024610799007814
© 1999 by London Mathematical Society
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© The London Mathematical Society

Sandwiching C0-Semigroups

J. M. A. M. Van Neerven

Department of Mathematics, Delft University of Technology PO Box 5031, 2600 GA Delft, Netherlands J.vanNeerven{at}twi.tudelft.nl

Received 11 September 1997. Revision received 8 April 1998.

Let T = {T(t)}t≥0 be a C0-semigroup on a Banach space X. The following results are proved.

(i) If X is separable, there exist separable Hilbert spaces X0 and X1, continuous dense embeddings j0:X0 -> X and j1:X -> X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that j0 {circ} T0(t) = T(t) {circ} j0 and T1(t) {circ} j1 = j1 {circ} T(t) for all t ≥ 0.

(ii) If T is {odot}-reflexive, there exist reflexive Banach spaces X0 and X1 , continuous dense embeddings j:D(A2) -> X0, j0:X0 -> X, j1:X -> X1, and C0-semigroups T0 and T1 on X0 and X1 respectively, such that T0(t) {circ} j = j {circ} T(t), j0 {circ} T0(t) = T(t) {circ} j0 and T(t) {circ} j1 = j1 {circ} T(t) for all t ≥ 0, and such that {sigma}(A0) = {sigma}(A) = {sigma}(A1), where Ak is the generator of Tk, k = 0, Ø, 1.


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