By Hellmut Baumgartel

ISBN-10: 3764316640

ISBN-13: 9783764316648

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**Additional info for Analytic perturbation theory for matrices and operators**

**Sample text**

4. Obvious by 3. 3. Let H be atomic. 1. H is factorial if and only if c(H) = 0. 2. If H is not factorial, then 2+sup ∆(H) ≤ c(H). In particular, if c(H) < ∞, then ∆(H) is finite. 3. If c(H) = 2, then H is half-factorial. 4. If c(H) = 3, then every L ∈ L(H) is an arithmetical progression with difference 1. Proof. , 3. and 4. 2. 2. If H is not factorial, then there exists some a ∈ H such that |Z(a)| ≥ 2. 3. 3 is not true. 11). 11 we discuss a simple domain R with c(R) = 3 and ρ(R) = ∞. Our next concept is that of tameness which we introduce in a local and in a global version.

Let d1 , . . , dt ∈ ∆ and k1 , . . , kt ∈ Z be such that d = k1 d1 + . . + kt dt . Without restriction we may suppose that k1 , . . , ks , −ks+1 , . . , −kt ∈ N for some s ∈ [1, t]. For every i ∈ [1, t] there exists an element ai ∈ H and some li ∈ N such that {li , li + di } ⊂ L(ai ). Thus we get t s |ki | ai L ⊃ i=1 t s ki li + i=1 t (−ki )(li + di ) , i=s+1 ki (li + di ) + i=1 (−ki )li , i=s+1 and since the difference between these two lengths equals d, we obtain d ≥ min ∆(H), hence d = min ∆(H), and the other assertions follow easily.

It is not known whether the power series ring over an atomic domain is atomic. The situation changes completely if we consider BFdomains. 7 below we prove that the polynomial ring, the power series ring and the ring of integer-valued polynomials over a BF-domain are again BF-domains. We recall the definition of the ring of integer-valued polynomials. For an integral domain R, we define the ring of integer-valued polynomials over R by Int(R) = {f ∈ q(R)[X] | f (R) ⊂ R}. For the theory of integer-valued polynomials we refer to the books of W.

### Analytic perturbation theory for matrices and operators by Hellmut Baumgartel

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