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R), and the biorthogonal relationships I are fulfilled. Pro05 Define the CSRF {y, ,. . 2. 2) of T - holds. Since { v , ,. . 3 yields that { $ J ) : 1 j r, 0 1 mi - I } is a CSEAV of T* at p . An easy calculation gives that < < < < is the principal part of at p . 6. I), and the uniqueness of the v j implies the uniqueness of the v?. For i = 1,. . , r and h = 1,. . 4. Since Tyi has a zero of order 2 mi, we infer Hence For i = 1 , . . ,r, j = 1,. . , r, h = mi, and 1 = 0 , . . 4) yield 30 I.

Then - y j ( ( - p ) n ' ~ ( v J. - C j )=) D - D j= 1 is holomorphic in a neighbourhood of p. Since yl ( p ) ,. . , y r ( p ) are linearly independent, the functions (. 3. As v j and C j are polynomials of degree less than m i , we obtain v j - C j = 0. 4). 9) idE = j= 1 + ( A - /L)-"'1yj(h)O ( T * v j ) ( A ) ( D T ) ( A ) for A E U \ { p ) . Hence the sum is holomorphic in U . Since yl ( p ) ,. . 3. 9) to y,(A) and obtain for il E U \ { p ) . At p the vector function DTyi has a zero of order at least m i , whence is holomorphic there.

R ) . S+J This completes the proof since det S = (. - p ) m with respect to this basis, where Sx s+ J m= r j= 1 . x J mi is the algebraic multiplicity of T at p. 9. The completion of biorthogonal systems of root functions Let R be a domain in C and E and F be Banach spaces. 9 we have seen that two systems of root functions { y l ,. . ,y,) and {v,, . . ,v,) of T and T* at p are canonical systems of root functions if r = nu1 T ( p ) and if they are biorthogonal. 38 I. Operator functions in Banach spaces Now assume that there are biorthogonal systems of root functions { y l ,.

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Algebraic solutions of ODE using p-adic numbers by Katz N.M.

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