By Jack K. Hale

ISBN-10: 0821849344

ISBN-13: 9780821849347

ISBN-10: 1551986981

ISBN-13: 9781551986982

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**Extra resources for Asymptotic behavior of dissipative systems**

**Example text**

D)LdA (22) for j = 0, 1, .... If c = 0 then d # 0 and (22) follows from property (ii'). )dA 21ri (23) Y where z = -d/c lies in the exterior of y. Since both sides are analytic functions of : in the exterior of ;, it suffices to prove (23) for large IzI. , IAI), then and (A - z)- ` = -z-'(l + Al: + A2/z2 + .. ), with both series converging absolutely for Iz1 > R. Substitution of these series into left- and right-hand sides of (23) establishes the formula (again by property (ii') for triple (X+, T, Y+)).

28 Let (X0, JO) be a pair of matrices, where X0 is a p x u matrix and J0 is a u x u block diagonal Jordan matrix with unique eigenvalue 20. 0 of multiplicity u, (ii) ker col(XOJQ)I I = (0), (iii) Proof Suppose that (XO, J0) is a Jordan pair of L(A) corresponding to A0. 11. Conversely, suppose that (i)-(iii) hold. Let Xo' _ [xv.. x;;;_1] denote the part of X0 corresponding to each Jordan block JJo) in JO (i = 1, . . , m). ) corresponding to A0. Condition (ii) ensures that the eigenvectors x0'"'....

And Y B;X-T = M_ (13) and its coefficients are given by 1-j-1 1-j-1 Bj = M+ Y_ k=0 T+Y+Aj+k+1 + M- Y_ Proof Let X = [X+ X_], T = T'_Y_Aj+k+1 (14) k=0 (T+ ) and Y = (). By the first _ Tcorollary above, (X, T, Y) is a finite spectral triple of L(A), that is, a F -spectral triple where r is a contour having all of sp(L) in its interior. B,_1] co1(XTj)j=01 then, in view of the fact that [Y. = [M+ M-] T'-' Y]3° is a left inverse of col(XTT)j=o, there is the solution [B0. B,_1] = [M+ M-]-[Y... T'-'Y]Z, that is, 1-j-1 Bj = [M+ M_] Y_ T'`YAj+k+1, k=0 which is equal to (14).

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