Spectral theory of automorphic functions by Alexei B. Venkov

By Alexei B. Venkov

Venkov A.B. Spectral thought of automorphic features (AMS, 1983)(ISBN 0821830783)

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Spectral theory of automorphic functions

Venkov A. B. Spectral concept of automorphic features (AMS, 1983)(ISBN 0821830783)

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1 1 and the definition of a canonical set of Jordan chains, taking into consideration that ({Ji O F ;'o(Ao)tfJ i O ' i 1, . , r, and therefore l/I I O ' . . 6. 35 CANONICAL SET OF JORDAN CHAINS CPI O ' . . , CPrO are linearly independent. In particular, the lengths of Jordan chains in a canonical set corresponding to ilo for L(il) and D are the same, and Proposition 1 . 1 3 follows. 0 ;'o(il) Corollary 1 . 14. The sum L �= 1 K i of the lengths ofJordan chains in a canon­ ical set corresponding to an eigenvalue ilo of a monic matrix polynomial L(il) coincides with the multiplicity of ilo as a zero of det L(il).

K p } c { I , 2, . . l"kJ ' then the rank of x is equal to min{kj l xj ¥- OJ. = Proof Let where is the local Smith form of L(A) and EAO(A) and FAo(A) are matrix polynomials invertible at A Ao (Theorem S U O). By Proposition 1 . 1 1 , Yo is an eigenvector of L(A) if and only if Zo F Ao(Ao)Yo is an eigenvector of DAo(A) and their ranks coincide. So it is sufficient to prove Theorem 1 . 1 2 for DAO(A). Denote by e; the ith coordinate vector in Then it is easy to see that e;, 0, . . ), provided K; 2:: 1 .

Yij = (0, ' , . , 0, CP; o , ' . , , CPij)' j = 0, ' . , , Jl i - 1 , i = 1 , . . 45) where the number of zero vectors preceding CP; o in Yij is Jl - (j + 1 ),form a basis in . k'. Proof Again, we shall use the reduction to a local Smith form. 43). Similarly, A, B will denote the corresponding matrices formed from matrix polynomials 1. 36 LINEARIZATION AND STANDARD PAIRS A(A), B(A) at ,10 ' respectively. 43) where L(A) is replaced by £(,1) = A(A)L(A)B(A) . Then % = B%. 46) (Note that according to Proposition 1 .

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