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Additional resources for An introduction to the calculus of finite differences and difference equations

Example text

Let n be an open neighbourhood of u(A). Then there exi3ts > 0 such that u(B) en for any operator B on X with IIA - BII < e. PROOF. Choose a Cauchy contour r in (1) 'Y Assume that IIA - BII :5 h. = min{II(A - n around u(A). Put A)-l 11-1 I A E r}. Then (2) AE Since A - A is invertible for A E u(B) n r = 0 and r, r. 2 in [GG] to show that II(A - A)-l _ (A _ B)-III < II(A - A)-11l2I1A - BII - l-II(A - A)-IIlIlA - BII :5 211(A - A)-11l2I1A - BII, AE Let P be the Riesz projection corresponding to the part of u(B) inside III - PII = 112~i f[(A - A)-l - (A - B)-I]dAIl r :5 2~ f II(A - A)-l - (A - B)-llidA r :5 CIIA- BII, r.

2 to show that All and A21 are similar. 2. Let Al and Az be compact operators, and assume that >. - Al and>' - A2 are equivalent at each point of C. Then Al and Az are similar. PROOF. For some open neighbourhood U of 0 we have (2) >. )(>. - AdE(>'), >. E U. ) are invertible operators on the Banach space X and EO and Fe-) are analytic on U. Let 0" be the part of the spectrum of Al outside U. 3). Since>. 4 MATRICIAL COUPLING AND EQUIVALENCE spectrum. It follows that the part of 0'(A2) outside U coincides with 0'.

RO"(A) < o. o. t ~ o. o. 1. Assume there exists a strictly positive operator Z on H such that ZA + A* Z is strictly negative. Choose 0 > 0 in such a way that (1) holds with A replaced by Z. ] by setting [x, y] = (Zx, y) (x,y E H). Let 111·111 be the corresponding norm. Note that (2) Thus the original norm II . II and the new norm III . III are equivalent. J is again a Hilbert space. It follows that A is a bounded linear operator on H endowed with the new III· III. ]. For x and yin H we have = (ZAx, y) = (x, A* Zy) [Ax, y] [x, A#y] = (Zx, A#Y) = (x, ZA#y).