By Louis Auslander
Complaints of the yankee Mathematical Society
Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236
Published through: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2035904
Page count number: 7
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Additional resources for An Account of the Theory of Crystallographic Groups
2 and the simple observation that (A-wI) € ^(M,0) whenever A € ^(M,w). 4. co) implies that ||r "(A,A)|| M /(ReA-w)” satisfying IIT(t)ll ^ M e s | a € C :ReA > wt , t ^ 0, the c p(A) and that for ReA >u> . 11) Proof. For any A € C with ReA >0, define R(X)x s r '*0 T(t)x dt. Clearly , for ReA >u> , and x e X , the integral is well defined and we have IIR(A)xll ^ ( M/(ReA-i*)) )llxll. Since R(A) = R(A,A), we conclude that C It remains to prove the inequality differentiable any number of times. 11).
Equivalence r(A) is closed and D(A) both the spaces if they are so in any one of them. (i)*. 10), we have IIX"r "(A,A)x II s ||x V ( A , A ) x |^ a |x||^ a M llxll for n e N^, and X € X and hence ||r "(X,A)x || a (m /a ") llxll This proves for X 6 X and n € N q . (ii). For the sufficient condition, Then by virtue of the renormlng lemma. suppose (i) and (ii) hold. 1, ensuring topological equivalence of the spaces X and Xoo, we have (a) : r(A) is a closed subset of Xoo x Xoo, and (b) : p(A) D (0,oo) and ||AR(X,A)x||j^^ ^ D(A)^” = Xoo .
Thus we have proved that A JJ (0,oo)c p(A) and that the resolvent R(A,A) = R^^ for A > 0. 3) that ||R(A,A)|| ^ 1/A for all A > 0. This completes the proof of the lemma. 3. resolvent set By examining the proof one can easily observe that the pi A) of generators of contraction semigroups contains the entire open right half-plane, that is, p(A) D - j A € C : R e A > 0 } - . 4. Let A € £ ub R(A) Q X. Suppose A (X) with domain and range D(A), satisfies the following properties: (i) A is closed, D(A) is dense in X :ii) p(A) D (0,co) and I I Then A R(A,A)- as A s 1/A for A >0.
An Account of the Theory of Crystallographic Groups by Louis Auslander