ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 3)

L.A. Gutnik

Dedicated to the memory of Professor A.O. Gelfond.

Table of contents

§0. Foreword. §1. The proof of the Theorem 7 in the case s = 1, q = 0, where q is defined in Theorem 6 [37]. §2. Begin of the proof of the Theorem 7 in the general case. §3. On the some linear normed spaces. §4. End of the proof of the Theorem 7. §5. Exactness of the obtained results.

§0. Foreword.

Here I complete the presentation of the proof of Theorem 4, which was formulated in the Part 1 ([33]) and proved in [15]; this theorem plays important role in my work ([18] - [38]) and is stated in [39] as Lemma 15. I use the results and notations of the previous parts [33], [37]. I will prove the following Theorem 7, which is more exact than Theorem 4.

Theorem 7. Let s ∈ ℕ − 1, n ∈ ℕ,

a_{i}^{∼} ∈ ℂ, a_{i}(ν) ∈ ℂ,

a_{n}(ν) = 1, a_{i}(ν) − a_{i}^{∼} = O(1/ν) (1)

for ν ∈ ℕ and i = 0, …, n. Let us consider the following difference equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (2)

where ν ∈ ℕ − 1. For m ∈ ℕ let V_{m} denotes the linear over ℂ space of solutions y = y(ν) of the equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (3)

where ν ∈ m + ℕ − 1. Let the absolute values of all the roots of the characteristical polynomial

T (z) = ∑_{k=0}^{n} a_{k}^{∼}z^{k} (4)

are among the numbers {ρ_{i}: 1 ≤ i ≤ 1 + s}, such that ρ_{s+1} = 0 and ρ_{j} < ρ_{i} for 1 ≤ i < j ≤ s + 1. Let e_{i} and k_{i} denote respectively the sum of and the maximum of the multiplicities of those roots, whose absolute value is equal to ρ_{i}, where i = 1, …, s + 1, and let k^{∗} = k_{s+1}. We suppose that, if s > 0, then

e_{i} > 0 (5)

for i = 1 …, s. For given y = y(ν) in ℂ^{m−1+ℕ}, let

ω_{n,y}(ν) = max(|y(ν)| , …, |y(ν + n − 1)|).

Then there exist A > 0, m ∈ ℕ, α^{∧}(ν) > 0 with ν ∈ m + ℕ − 1 and the subspaces V_{m,1}^{∨}, …, V_{m,s+1}^{∨} such that

lim_{ν→∞} α^{∧}(ν) = 0,

V_{m} = V_{m,1}^{∨} ⊕ …, ⊕ V_{m,s+1}^{∨}, dim_{ℂ}(V_{m,i}^{∨}) = e_{i}, 1 ≤ i ≤ s + 1,

and, if y ∈ V_{m,θ}^{∨} for some θ ∈ {1, ..., s}, then

exp(−A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}ω_{n}(y)(m) ≤ ω_{n,y}(ν)

for ν ∈ m + ℕ − 1; moreover, the spaces

V_{m,j}^{∧} = V_{m,j}^{∨} ⊕ … ⊕ V_{m,s+1}^{∨},

where j = 1 …, s + 1, and, if s ≥ 1 natural projections π_{j},

V_{m,j}^{∧} ↦ V_{m,j}^{∨},

where j = 1 …, s, have the folloving properties:

if y ∈ V_{m,θ}^{∧} for some θ ∈ {1, ..., s}, then

ω_{n,y}(ν) ≤ exp(A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}ω_{n,y}(m), (6)

(ω_{nπ_{theta(y)}}(m) − α(ν)ω_{n,y}(m))(ρ_{θ})^{ν} × (7)

exp(−A(ln(ν) + ν^{1−1/k_{θ}})) ≤ ω_{n,y}(ν),

where ν ∈ m + ℕ − 1;

if

k^{∗} > 0, (8)

and y ∈ V_{m,s+1}^{∨} (= V_{m,s+1}^{∧}), then

|y(ν)| ≤ (A/ν)^{ν/k^{∗}}ω_{n,y}(m),

where ν ∈ m + ℕ − 1.

Corollary. (cf. Theorem 4). If

y ∈ V_{m,θ}^{∧}⧹V_{m,θ+1}^{∧}, (9)

then

exp(−A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}β ≤ ω_{n,y}(ν),

where β = β(y) > 0 does not depend from ν.

Proof. The condition (9) implies the the inequality π_{θ}(y) ≠ 0_{V_{m,θ}^{∨}}. If ω_{n,π_{θ}(y)}(m) = 0, then π_{θ}(y) = 0_{V_{m,θ}}^{∨}. Consequently, ω_{n,π_{θ}(y)}(m) > 0 and there exists ν_{0} ∈ m + ℕ − 1 such that

(ω_{n,π_{θ}(y)}(m)/2 < (ω_{n,π_{θ}(y)}(m) − α(ν)ω_{n,y)}(m))

for ν ∈ ν_{0}+ℕ−1. Therefore 0 < ω_{n,y)}(ν) for ν ∈ m+ℕ−1, and, consequently there exists β = β(y) > 0 such that

exp(−A(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}β(y) ≤ ω_{n,y}(ν)

for ν ∈ m + ℕ − 1. ■

So, Theorem 4 is a corollary of the Theorem 7. Acknowledgment. I express my deepest thanks to Professor B.Z.Moroz, to Professor I.I. Piatetski-Shapiro, to Professor A.G.Aleksandrov, to Professor P.Bundshuh, and to Professor S.G.Gindikin for the support of my work.

§1. The proof of the Theorem 7 in the case s = 1, k^{∗} = 0.

Lemma 1. Let a ∈ ℕ, b ∈ ℕ − 1 + a, C > 0 Then

0 < ∑_{κ=a}^{b} ln(1 + C/κ) ≤ (10)

ln(1 + C/a) + b ln(1 + C/b) − a ln(1 + C/a) + C ln((b + C)/(a + C)).

Proof. Since the function ln(1+C/x) decreases together with increasing of the variable x ∈ [1, +∞), it follows that

∑_{κ=a}^{b} ln(1 + C/κ) ≤ (11)

ln(1 + C/a) + ∫_{x=a}^{b} ln(1 + C/x)dx =

ln(1 + C/a) + b ln(1 + C/b) − a ln(1 + C/a) + ∫_{x=a}^{b} (C/(C + x))dx.

■ Corollary. If a ∈ ℕ, b ∈ ℕ + a − 1, b < 2a, C > 0, then

∑_{κ=a}^{b} ln(1 + C/κ) ≤ 3C.

Proof. The assertion of the Corollary directly follows from (11).

Lemma 2. If s = 1, k^{∗} = 0, then assertion of the Theorem 7 is true.

Proof. The condition k^{∗} = 0 implies the inequality

a_{0}^{∼} = T (0) ≠ 0, (12)

and the condition s = 1 implies for p in Theorem 6 of the paper [37] the equality p = n . Let

A(ν) = (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −a_{0}(ν) & −a_{1}(ν) & −a_{2}(ν) & … & −a_{n−1}(ν) \end{matrix}) , (13)

where ν ∈ ℕ − 1, and let

A^{∼} = (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −a_{0}^{∼} & −a_{1}^{∼} & −a_{2}^{∼} & … & −a_{n−1}^{∼}. \end{matrix}) (14)

Let λ_{j}, where j = 1, …, r is the sequence of all the mutually distinct roots of the polynomial (4) and k_{j}^{∗} is the multiplicity of the root λ_{j}. Let further k = sup k_{j}^{∗}: j = 1, …, r. Clearly,

∑_{j=0}^{r} k_{j}^{∗} = n,

0 < |λ_{j}| ≤ ρ_{1} = ∥A∥_{sp} < R = h^{∼}(A^{∼}) + 1,

where j = 1, …, r. In view of (2) in [37], there exists C_{1} > 0 such that

h^{∼}(A(ν) − A^{∼}) ≤ C_{1}/(ν + 1). (15)

Therefore, according to the Lemma 8 in [37], for any ε > 0 the following inequalities holds

p_{A^{∼},ε}^{∼}(A(ν) − A^{∼}) ≤ (16)

(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}h^{∼}(A(ν) − A^{∼}) ≤
(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}C_{1}/(ν + 1),

p_{A^{∼},ε}^{∼}(A(ν)) ≤ ρ_{1} + sign(k − 1)ε + (17)

(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1}C_{1}/(ν + 1) ≤ (max(ε, 1/ε))^{k−1} ≤

(ρ_{1} + (sign(k − 1))ε) (1 + (γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1} \frac{C_{1}}{ρ_{1}(ν + 1)}) ,

ln(p_{A^{∼},ε}^{∼}(A(ν))) ≤ ln(ρ_{1}) + ln(1 + (sign(k − 1))ε/ρ_{1}) + (18)

ln (1 + (γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1} \frac{C_{1}}{ρ_{1}(ν + 1)}) ,

where ν ∈ ℕ − 1. We consider first the case k > 1. For given ν ∈ ℕ + 1 we take d ∈ ℕ and a_{i} ∈ [1, ν] ∩ ℕ, where i = 0, …, d, in such a way that

a_{0} = 1, a_{d} = ν, a_{i−1} < a_{i} ≤ 2a_{i−1}, (19)

where i = 1, …, d, and

d ≤ \frac{ln(κ)}{ln(2)} + 1. (20)

According to the Corollary of the Lemma 1 and (19) – (20),

∑_{κ=a_{i−1}}^{a_{i}−1} ln(p_{A^{∼},ε}^{∼}(A(κ))) ≤ (21)

(a_{i} − a_{i−1}) ln(ρ_{1}) + (a_{i} − a_{i−1}) ln(1 + ε/ρ_{1}) +

∑_{κ=a_{i−1}}^{a_{i}−1} ln (1 + (γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1} \frac{C_{1}}{ρ_{1}(κ + 1)}) ≤

(a_{i} − a_{i−1})(ln(ρ_{1}) + ln(1 + ε/ρ_{1})) + 3(γ^{∗}(A^{∼}))^{2}(max(ε, 1/ε))^{k−1} \frac{C_{1}}{ρ_{1}} ,

where i = 1, …, d. We take now in (21 ε = ε_{i} = (a_{i−1})^{−1/k}. Then we obtain the inequality

∑_{κ=a_{i−1}}^{a_{i}−1} ln(p_{A^{∼},ε}^{∼}(A(ν))) ≤ (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}),

where i = 1, …, d. Therefore

ln(p_{A^{∼},ε_{i}}^{∼} ( ∏_{κ=1}^{a_{i}−a_{i−1}} A(a_{i} − κ)) ≤ (22)

(a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)},

ln(h^{∼} ( ∏_{κ=1}^{a_{i}−a_{i−1}} A(a_{i} − κ)) ≤ (23)

ln((γ^{∗}(A^{∼}))^{2})(max(ε, 1/ε))^{k−1}) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}) ≤

2 ln(γ^{∗}(A^{∼})) + (i − 1)(1 − 1/k)ln(2) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}) =

(i − 1)(1 − 1/k)ln(2) + (a_{i} − a_{i−1}) ln(ρ_{1}) + O(2^{(i−1)(1−1/k)}),

where i = 1, …, d. Consequently,

ln (h^{∼} (∏_{κ=1}^{ν} A(ν − κ))) = (24)

ln (h^{∼} ((∏_{i=1}^{d} ∏_{κ=1}^{a_{i}−a_{i−1}} A(a_{i} − κ)) A(0))) =

= ν ln(ρ_{1}) + O((ν + 1)^{(1−1/k)}),

where ν ∈ ℕ − 1. If k = 1, then, according to (18),

ln(p_{A^{∼},1}^{∼}(A(ν))) ≤ ln(ρ_{1}) + ln (1 + (γ^{∗}(A^{∼}))^{2} \frac{C_{1}}{ρ_{1}(ν + 1)}) ≤

ln(ρ_{1}) + (γ^{∗}(A^{∼}))^{2} \frac{C_{1}}{ρ_{1}(ν + 1)},

ln (h^{∼} (∏_{κ=1}^{ν} A(ν − κ))) ≤ (25)

ln ((γ^{∗}(A^{∼}))^{2}p_{A^{∼},1}^{∼} (∏_{κ=1}^{ν} A(ν − κ))) ≤

ν ln(ρ_{1}) + O(ln(e(ν + 1))).

where ν ∈ ℕ − 1. In view of (24) – (25),

ln (h^{∼} (∏_{κ=1}^{ν} A(ν − κ))) = (26)

= ν ln(ρ_{1}) + O((ν + 1)^{(1−1/k)}) + O(ln(e(ν + 1))),

where ν ∈ ℕ − 1.

As in Section 3 of [37], let K denotes one of the fields ℝ or ℂ and L denotes a linear normed space over K with norm p = p(x). In the case L = K^{n} we fix as p = p(x), wehre x ∈ K^{n}, the maximum of the absolute values of coordinates of the element x in the standard basis, i.e.

p(x) = h(x) = sup({|x_{1}|, …, |x_{n}|}), (27)

where

x = (x_{1} ; ⋮ ; x_{n}) .

If L is a Banach space with the norm p, then K−algebra of all the linear continuous operators acting in L will be denoted by 𝔐^{∧}(L), and the norm on 𝔐^{∧}(L), associated with the norm p will be denoted by p^{∼}. So,

p^{∼}(A) = sup({p(AX): X ∈ L, p(X) ≤ 1}).

It is well known that the associciated with h norm on Mat_{n}(ℂ) is defined as follows

h^{∼}(A) = sup ({∑_{k=1}^{n} |a_{i,j}|: i = 1, …, n}) , (28)

where A = (a_{i,k}) ∈ Mat_{n}(ℂ). The norms h and h^{∼} coincide respectiely with with the norms q_{∞} and q_{∞}^{∼} considered in section 6 of the paper [33].

Let m ∈ ℕ, and let E_{m}(L) be the set L^{m−1+ℕ} of all the maps of the set m − 1 + ℕ into L. The set E_{m}(L) is a linear space over K, where the muliplication of the elements by the number from K and addition of the elements are defined coordinate-wise. The subspace of E_{m}(L) composed by all the constant maps is isomorphic to L, and we identify this subspace with L.

As in Section 4 of [37], for any y ∈ E_{m}(ℂ) and n ∈ ℕ let Y_{n,y} and Y_{n,y}^{#} denote the elements in the space E_{m}(ℂ^{n}), wich are determined respectively by means the following equalities:

Y_{n,y}(ν) = ( y(ν) ; ⋮ ; y(ν + n − 1)) , (29)

Y_{n,y}^{#}(ν) = ( 0 ; ⋮ ; 0 ; y(ν)) , (30)

where ν ∈ m − 1 + ℕ. Let is fixed m ∈ ℕ. If y = y(ν) is a solution of the equation (3) for ν ∈ ℕ + m − 1, then

Y_{n,y}(ν) = ( ∏_{κ=1}^{ν−m} A(ν − κ)) Y_{n,y}(m),

where ν ∈ ℕ, and, in view of (26),

ω_{n,y}(ν) = h (Y_{n,y}(ν)) ≤ (31)

h^{∼} ( ∏_{κ=1}^{ν−m} A(ν − κ)) h (Y_{n,y}(m)) =

exp (O(1) (ν^{1−1/k} + ln(ν))) (ρ_{1})^{ν} ω_{n,y}(m), where ν ∈ ℕ + m − 1. So, with m = 0 the asserted by the Theorem 7 the upper estimate of the value ω_{n,y}(ν) (ρ_{θ})^{−ν} = ω_{n,y}(ν) (ρ_{1})^{−ν} is obtained in the case s = 1, k^{∗} = 0, because in this case θ = 1. Let

m_{1} = [2C_{1}h^{∼} ((A^{∼})^{−1})] + 1,

and let is fixed m ∈ m_{1} − 1 + ℕ Then

h^{∼} ((A^{∼})^{−1} (A(ν) − A^{∼})) ≤ h^{∼} ((A^{∼})^{−1}) C_{1}/(ν + 1) ≤
h^{∼} ((A^{∼})^{−1}) C_{1}/([2C_{1}h^{∼} ((A^{∼})^{−1})] + 1) ≤ 1/2,

if ν ∈ m − 1 + ℕ, the matrix E + ((A^{∼})^{−1}(A(ν) − A^{∼})) is invertible,

h^{∼} ((E + ((A^{∼})^{−1}(A(ν) − A^{∼})))^{−1}) ≤ 2,

if ν ∈ m − 1 + ℕ, there exists the matix

(A(ν))^{−1} =
(E + ((A^{∼})^{−1} (A(ν) − A^{∼})))^{−1} (A^{∼})^{−1} ,

if ν ∈ m − 1 + ℕ, moreover,

h^{∼} ((A(ν))^{−1}) ≤ h^{∼} ((E + ((A^{∼})^{−1}(A(ν) − A^{∼})))^{−1}) h^{∼} ((A^{∼})^{−1}) ≤

2h^{∼} ((A^{∼})^{−1})

and, finally,

h^{∼} ((A(ν))^{−1} − (A^{∼})^{−1}) =
h^{∼} ((A(ν))^{−1} (A^{∼} − A(ν)) (A^{∼})^{−1}) ≤
h^{∼} ((A(ν))^{−1}) h^{∼} ((A^{∼} − A(ν))) h^{∼} ((A^{∼})^{−1}) ≤
2 (h^{∼} ((A^{∼})^{−1}))^{2} C_{1}/(ν + 1),

where ν ∈ m − 1 + ℕ. Let C_{2} = 2 (h^{∼} ((A^{∼})^{−1}))^{2} C_{1}. We take ν ∈ m − 1 + ℕ. We take κ ∈ [m, ν − 1] ∩ ℤ on the role ν − κ in (24) with κ ∈ [1, ν − m] ∩ ℤ; we take further (A(κ))^{−1} , where m ≤ κ ≤ ν −1, on the role A(ν −κ) in (24), where 1 ≤ κ ≤ ν − m, we take A_{1}^{∼} = (A^{∼})^{−1} on the role A^{∼}, we take C_{2} on the role C_{1} and, finally, we see that 1/ρ_{1} will play the role 1/ρ_{1}. Then we see that

ln (h^{∼} ( ∏_{κ=m}^{ν−1} (A(κ))^{−1})) ≤ (32)

−ν ln(ρ_{1}) + O (ν^{1−1/k} + ln(ν)) ,

where ν ∈ m − 1 + ℕ ⊂ m_{1} − 1 + ℕ. Since

Y_{n,y}(ν) = ( ∏_{κ=1}^{ν−m} A(ν − κ)) Y_{n,y}(m_{1}),

Y_{n,y}(m) = ( ∏_{κ=1}^{ν−m} A(ν − κ))^{−1} Y_{n,y}(ν) =

( ∏_{κ=m}^{ν−1} (A(κ))^{−1}) Y_{n,y}(ν),

it follows that

ω_{n,y}(m) = h (Y_{n,y}(m)) ≤ (33)

h^{∼} ( ∏_{κ=m}^{ν−1} (A(κ))^{−1}) h (Y_{n,y}(ν)) =

exp (O(1) (ν^{1−1/k} + ln(ν))) (ρ_{1})^{−ν} ω_{n,y}(ν),

where ν ∈ ℕ. So, with m ∈ m_{1} − 1 + ℕ the asserted by the Theorem 7 the lower estimate of the value ω_{n,y}(ν) (ρ_{θ})^{−ν} = ω_{n,y}(ν) (ρ_{1})^{−ν} is obtained in the case s = 1, k^{∗} = 0, because in this case θ = 1. ■

Lemma 3. If s = 0, k^{∗} = n, (see notations in [33], §0, formulation of the Theorem 4) then assertion of the Theorem 7 is true.

Proof is given in [37], Theorem 6.

§2. Begin of the proof of the Theorem 7 in the general case.

For the proof we use induction on σ, where σ = s, if k^{∗} = 0, and σ = s+1, if k^{∗} > 0, Since for σ = 1 Theorem 7 is proved in the section 1 (Lemma 2 and Lemma 3) we suppose that

σ ≥ 2 (34)

and the assertions of the Theorem 7 are true for σ−1. In view of (34), s ≥ 1, and, in view of (5), e_{1} > 0. Clearly,

∑_{j=1}^{s+1} e_{j} = n. (35)

If s = 1, then, in view of (34) and (35), e_{1} < n; if s ≥ 2, then, (5) and (35) imply the inequality e_{1} < n. Let p = e_{1}, and let q = n − p. Since e_{1} < n, it follows that p < n and q > 0. As in (9) of the paper [33], let

T_{1}(z) = ∑_{α=0}^{p} b_{α}^{∼}z^{k}, T_{2}(z) = ∑_{β=0}^{q} u_{β}^{∼}z^{k}. (36)

Then for the polynomial (4) the following equlity holds

T (z) = T_{1}(z)T_{2}(z), (37)

i.e. the equality (8) of the paper [33] holds, b_{p}^{∼} = u_{q}^{∼} = a_{n}^{∼} = 1 and absolute value of each root of the polynomial T_{1}(z) is greater than the absolute value of each root of the polynomial T_{2}(z). Therefore with ξ(x) = 1/(x + 1) all the conditions of the Theorem 5 in the paper [33] are fulfilled. Consequently, the following assertion is true:

Assertion 1. There exist m ∈ ℕ, the numbers b_{α}(ν) ∈ ℂ, u_{β}(ν) ∈ ℂ, where α = 0, …, p, β = 0, …, q, ν ∈ ℕ + m − 1, such that

b_{α}(ν) − b_{α}^{∼} = O(1/ν), (38)

where

α = 0, …, p, b_{p}(ν) = 1, b_{0}(ν) ≠ 0, ν ∈ ℕ + m − 1, (39)

u_{β}(ν) − u_{β}^{∼} = O(1/ν), (40)

where

β = 0, …, q, u_{q}(ν) = 1, ν ∈ ℕ + m − 1, (41)

and the connected with the equation (3) the equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (42)

where ν ∈ ℕ − 1 + m, is equivalent to the equation

∑_{α=0}^{p} b_{α}(ν)y(ν + α) = r(ν), (43)

where ν ∈ ℕ − 1 + m and r(ν) satisfies to the equation

∑_{β=0}^{q} u_{β}(ν)r(ν + β) = 0 (44)

with ν ∈ ℕ − 1 + m.

We denote by R_{m} the space of all the sollutions of the equation (44). The map

y(ν) → ∑_{α=0}^{p} b_{α}(ν)y(ν + α) = r(ν) (45)

is the homomorphism of the space V_{m} of solutions of the equation (42) into the space R_{m}; we denote this homomorphism by ϕ_{m}.

Let

C_{3} = sup {∑_{α=0}^{p} |b_{α}(ν)|: ν ∈ m − 1 + ℕ}; (46)

C_{4} = sup {∑_{β=0}^{q} |u_{β}(ν)|: ν ∈ m − 1 + ℕ}; ; (47)

C_{5} = sup {∑_{k=0}^{n} |a_{k}(ν)|: ν ∈ m − 1 + ℕ}; ; (48)

according to (38) – (41) and (1), 1 ≤ C_{k} < +∞ < +∞ for k = 1, 2 3. In view of (46),

|(ϕ_{m}(y)(ν) ≤ C_{3}ω_{p+1,y}(ν),

ω_{q,ϕ_{m}(y)}(ν) ≤ C_{3}ω_{n+1,y}(ν) ≤ C_{3}C_{5}ω_{n,y}(ν), (49) where ν ∈ m − 1 + ℕ.

Together with the equation (43) we will consider the corresponding homogeneous equation

∑_{α=0}^{p} b_{α}(ν)y(ν + α) = 0, (50)

where ν ∈ ℕ − 1 + m. If y_{1}(ν) is a solution of the equation (50), then, clearly, ω_{p,y_{1}}(ν + i) ≤ C_{3}^{i}ω_{p,y_{1}}(ν), (51) where i ∈ ℕ − 1, ω_{p,y_{1}}(ν) ≤ ω_{n,y_{1}}(ν), ω_{n,y_{1}}(ν) ≤ C_{3}^{q}ω_{p,y_{1}}(ν), (52) where ν ∈ ℕ − 1 + m.

If r(ν) is a solution of the equation (44), then, clearly, ω_{q,r}(ν + i) ≤ C_{4}^{i}ω_{q,r}(ν), (53) where i ∈ ℕ − 1, ω_{q,r}(ν) ≤ ω_{n,r}(ν), ω_{n,r}(ν) ≤ C_{4}^{p}ω_{q,r}(ν), (54) where ν ∈ ℕ − 1 + m. Let C > 0, n ∈ ℕ,

w_{C,n}(ν) = (\frac{C}{ν} )^{ν/n} , (55)

where ν ∈ ℕ; let further ρ > 0, for which v_{C,n,ρ}(ν) = ρ^{ν} exp (C ((ν)^{1−1/n} + ln(ν))) , (56) where ν ∈ ℕ.

According to inductive hypothesis, there exist A^{∨} > 0, m ∈ ℕ, α^{∨}(ν) > 0 with ν ∈ m + ℕ − 1 and the subspaces R_{m,2}^{∨}, …, R_{m,s+1}^{∨} such that

lim_{ν→∞} α^{∨}(ν) = 0,

R_{m} = R_{m,2}^{∨} ⊕ …, ⊕ R_{m,s+1}^{∨}, dim_{ℂ}(R_{m,i}^{∨}) = e_{i}, 2 ≤ i ≤ s + 1, if r ∈ R_{m,θ}^{∨} (57)

L.A.Gutnik, On the difference equations of the Poincaré type (Part 3) 12

for some θ ∈ {2, ..., s}, then exp(−A^{∨}(ln(ν) + ν^{1−1/k_{θ}}))ω_{q,r}(m)(ρ_{θ})^{ν} ≤ ω_{q,r)}(ν), (58) where ν ∈ ℕ + m − 1; moreover, the spaces

R_{m,j}^{∧} = R_{m,j}^{∨} ⊕ … ⊕ R_{m,s+1}^{∨},

where j = 2, …, s + 1 and natural projections π_{j}^{∨} of the space R_{m,j}^{∧} onto the space R_{m,j}^{∨}, where j = 2, …, s have the folloving properties:

if s ≥ 2 r ∈ R_{m,θ}^{∧} (59) for some θ ∈ {2, ..., s}, then ω_{q,r}(ν) ≤ (ρ_{θ})^{ν} exp(A^{∨}(ln(ν) + ν^{1−1/k_{θ}}))ω_{q,r}(m), (60)

ω_{q,r}(ν) ≥ (ω_{q,π_{θ}^{∨}(r)}(m) − α^{∨}(ν)ω_{q,r)}(m)) × (61)

(ρ_{θ})^{ν} exp(−A^{∨}(ln(ν) + ν^{1−1/k_{θ}})),

where ν ∈ ℕ + m − 1; if (8) holds and r ∈ R_{m,s+1}^{∨} (= R_{m,s+1}^{∧}), (62) then |r(ν)| ≤ (A^{∨}/ν)^{ν/k^{∗}}ω_{q,r}(m), (63) where ν ∈ ℕ + m − 1.

Lemma 4. Let for sufficient big m ∈ ℕ the splitting monomorphism

ψ_{m}: R_{m} → V_{m}

is specified, having the folloing properties:

a) ϕ_{m}ψ_{m} is the ientity map R_{m} → R_{m};

b) there exists a constant C_{6} > 0 such that if (59) holds, then sup{ω_{n,ψ_{m}(r)}(ν)/v_{A^{∨},n,ρ_{θ}}(ν): ν ∈ ℕ − 1 + m} ≤ (64)

C_{6} sup{ω_{q,r}(ν)/v_{A^{∨},k_{θ},ρ_{θ}}(ν): ν ∈ ℕ − 1 + m},

and if (62) holds, then sup{ω_{n,ψ_{m}(r)}(ν)/w_{A^{∨},k^{∗}}(ν): ν ∈ ℕ − 1 + m} ≤ (65)

C_{6} sup{ω_{q,r}(ν)/w_{A^{∨},k^{∗}}(ν): ν ∈ ℕ − 1 + m},

Then assertions of the Theorem 4 are true for σ.

Proof. We denote by I the identity map V_{m} → V_{m}. Let further V_{m,1}^{∨} denotes the space of the solutions of the equation (50). Let us to prove that V_{m} = V_{m,1}^{∨} ⊕ ψ_{m} (R_{m}) , i.e that each element y ∈ V_{m} may be uniquely represented in the form

y = y_{1} + y^{∧}, (66)

where y = y_{1} ∈ V_{m,1}^{∨}, y^{∧} ∈ ⊕ψ_{m} (R_{m}) . Let

y^{∧} = (ψ_{m}ϕ_{m})(y) = (ψ_{m}(ϕ_{m}(y)), (67)

y_{1}^{∧} = y − y^{∧} = (I − ψ_{m}ϕ_{m})(y). (68)

In view of (67), y^{∧} ∈ ψ_{m}(R_{m}),

ϕ_{m}(y^{∧}) = ϕ_{m}(ψ_{m}(ϕ_{m}(y)) = (ϕ_{m}ψ_{m})(ϕ_{m}(y)) = ϕ_{m}(y), (69)

because ϕ_{m}(y) ∈ R_{m} and ψ_{m}ϕ_{m} is the ientity map R_{m} → R_{m}. Therefore we have the equality ϕ_{m}(y_{1}) = ϕ_{m}(y)−ϕ_{m}(y^{∧}) = 0, and, consequently, y_{1} ∈ V_{m,1}^{∨}. If y_{2} ∈ V_{m,1}^{∨}, y_{3} ∈ R_{m}, y_{4} = ψ_{m}(y_{3}) and y = y_{2} + y_{4}, then

y^{∧} = (ψ_{m}ϕ_{m})(y) = (ψ_{m}(ϕ_{m}(y_{2})) + ψ_{m}(ϕ_{m}(y_{4})) =
ψ_{m}(ϕ_{m}(ψ_{m}(y_{3}))) = ψ_{m}((ϕ_{m}ψ_{m})(y_{3})) =
ψ_{m}(y_{3}) = y_{4}, y_{1}^{∧} = y − y^{∧} = y − y_{4} = y_{2}.

So, the existence and the uniqueness of the representation (66) is proved.

Let V_{m,i}^{∨} = ψ_{m}(R_{m,i}^{∨}) for i = 2, …, s + 1. Monomorphism ψ_{m} is an isomorphism of the space

R_{m} = R_{m,2}^{∨} ⊕ … ⊕ R_{m,s+1}^{∨}

on the space

ψ_{m}(R_{m}) = ψ_{m}(R_{m,2}^{∨}) ⊕ … ⊕ ψ_{m}(R_{m,s+1}^{∨}) =
V_{m,2}^{∨} ⊕ … ⊕ V_{m,s+1}^{∨}.

Then

V_{m} = V_{m,1}^{∨} ⊕ ψ_{m} (R_{m}^{∨}) = V_{m,1} ⊕ (V_{m,2}^{∨} ⊕ … ⊕ V_{m,s+1}^{∨}) = (70)

V_{m,1}^{∨} ⊕ … ⊕ V_{m,s+1}^{∨};

moreover dim_{ℂ} (V_{m,1}^{∨}) = p = e_{1},

dim_{ℂ} (V_{m,i}^{∨}) = dim_{ℂ} (R_{m,i}^{∨}) = e_{i},

where i = 2, …, s + 1. The natural projection π_{1} of the space V_{m,1}^{∧} = V_{m} onto the space V_{m,1}^{∨} coincides with I − ψ_{m}ϕ_{m} and, if j ∈ [2, s] ∩ ℕ,then the natural projection π_{j} of the space V_{m,j}^{∧} onto V_{m,j}^{∨} coincides with ψ_{m}π_{j}^{∨}. Therefore π_{j}^{∨} = ϕ_{m}π_{j}. Let g ∈ q − 1 + ℕ, and let us consider the map

Y_{g,r}(m) → Y_{g,ψ_{m}(r)}(m),

where r ∈ R_{m}. This map defines a linear isomorphism of q−dimesional linear over ℂ space R_{m} onto the q−dimensinal linear over ℂ space V_{m,2}^{∧}. Consequently, there exists two matrices T_{1,g} ∈ Mat_{g}(ℂ) and T_{2,g} ∈ Mat_{g}(ℂ) such that

Y_{g,r}(m) = T_{1,g}Y_{g,ψ_{m}(r)}(m), Y_{g,ψ_{m}(r)}(m) = T_{2,g}Y_{g,r}(m).

(Clearly, T_{2,q} = (T_{1,q})^{−1} , if g = q).

Therefore there exist C_{7,g} > 0 and C_{8,g} > 0 such that

ω_{g,r}(m) ≤ C_{7,g}ω_{g,ψ_{m}(r)}(m), ω_{g,ψ_{m}(r)}(m) ≤ C_{8,g}ω_{g,r}(m). (71)

Let y ∈ V_{m,θ}^{∧} for some θ ∈ {2, ..., s + 1}. Then, since ψ_{m} isomorphically maps R_{m,θ}^{∧} onto V_{m,θ}^{∧}, it follows that y = ψ_{m}(r), r = ϕ_{m}(y), where r ∈ R_{m,θ}^{∧}. If 2 ≤ θ ≤ s, then in view of (60), (64), (49),

sup{ω_{n,y}(ν)/v_{A^{∨},k_{θ},ρ_{θ}}(ν): ν ∈ ℕ − 1 + m} ≤
sup{ω_{n,ψ_{m}(r)}(ν)/v_{A^{∨},k_{θ},ρ_{θ}}(ν): ν ∈ ℕ − 1 + m} ≤
C_{6} sup{ω_{q,r}(ν)/v_{A^{∨},k_{θ},ρ_{θ}}(ν): ν ∈ ℕ − 1 + m} ≤
C_{6}ω_{q,r}(m) = C_{6}ω_{q,ϕ_{m}(y)}(m) ≤ C_{6}C_{3}C_{5}ω_{n,y}(m).

Therefore

ω_{n,y}(ν) ≤ v_{A^{∨},_{θ},ρ_{θ}}(ν)C_{6}C_{3}C_{5}ω_{n,y}(m) = (72)

ρ^{ν} exp (O(1) ((ν)^{1−1/k_{θ}} + ln(ν))) ω_{n,y}(m).

If k^{∗} > 0 and θ = s + 1, then in view of (63), (65), (49),

sup{ω_{n,y}(ν)/w_{A^{∨},k^{∗}}(ν): ν ∈ ℕ − 1 + m} ≤
sup{ω_{n,ψ_{m}(r)}(ν)/w_{A^{∨},k^{∗}}(ν): ν ∈ ℕ − 1 + m} ≤
C_{6} sup{ω_{q,r}(ν)/w_{A^{∨},k^{∗}}(ν): ν ∈ ℕ − 1 + m} ≤
C_{6}ω_{q,r}(m) = C_{6}ω_{q,ϕ_{m}(y)}(m) ≤ C_{6}C_{3}C_{5}ω_{n,y}(m).

Therefore

ω_{n,y}(ν) ≤ w_{A^{∨},k^{∗}}(ν)C_{6}C_{3}C_{5}ω_{n,y}(m). (73)

If 2 ≤ θ ≤ s, then ϕ_{m}(π_{θ}(y)) = π_{θ}^{∨}(r) ∈ V_{m,θ}^{∨}, where r = ϕ_{m}(y), and, in view of (58),

exp(−A^{∨}(ln(ν) + ν^{1−1/k_{θ}}))ω_{q,π_{θ}(r)}(m)(ρ_{θ})^{ν} ≤ ω_{q,π_{θ}(r)}(ν); (74)

in view of (71), (54),

ω_{n,y}(m) = ω_{n,ψ_{m}(r)}(m) ≤ C_{8,n}ω_{n,r}(m) ≤ C_{8,n}C_{4}^{p}ω_{q,r}(m),
ω_{n,π_{θ}(y)}(m) = ω_{g,ψ_{m}(π_{θ}^{∨}(r))}(m) ≤ C_{8,n}ω_{n,π_{θ}^{∨}(r)}(m) ≤ C_{8,n}C_{4}^{p}ω_{q,π_{θ}^{∨}(r)}(m).

Therefore, in view of (74), (49),

(C_{8,n}C_{4}^{p})^{−1}ω_{n,π_{θ}(y)}(m) exp(−A^{∨}(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν} ≤

L.A.Gutnik, On the difference equations of the Poincaré type (Part 3) 15

ω_{q,π_{θ}^{∨}(r)}(m) exp(−A^{∨}(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν} ≤
ω_{q,π_{θ}^{∨}(r)}(ν) = ω_{q,ϕ_{m}(π_{θ}(y))}(ν) ≤
C_{3}C_{5}ω_{n,π_{θ}(y)}(ν).

Therefore ω_{n,π_{θ}(y)}(m) exp(−O(1)(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν} ≤ ω_{n,π_{θ}(y)}(ν). (75)

Let d_{θ} = ∑_{j=θ}^{s+1} e_{j} Clearly, y − π_{θ}(y) ∈ V_{m,1+θ}. The map π_{θ} is a linear map of d_{θ}− dimensional linear over ℂ space V_{m,θ}^{∧} onto onto the e_{θ}− dimensinal linear over ℂ space V_{m,θ}^{∨}. Consequently, there exists T_{0,θ} ∈ Mat_{n}(ℂ) such that

Y_{n,π_{θ}(y)}(m) = T_{0,θ}Y_{n,y}(m).

Therefore there exists C_{0,θ} > 0 such that ω_{n,π_{θ}(y)}(m) ≤ C_{0,θ}ω_{n,y}(m). (76) Clearly, y − π_{θ}(y) ∈ V_{m,1+θ}, ω_{n,y−π_{θ}(y)}(m) ≤ (1 + C_{0,θ})ω_{n,y}(m). (77)

In view of (75), (72), (77) ω_{n,π_{θ}(y)}(ν) ≥ ω_{n,π_{θ}(y)}(ν) − ω_{n,y−π_{θ}(y)}(ν) ≥ (78)

exp(−O(1)(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}×

(ω_{n,π_{θ}(y)}(m) − exp(O(1)(ln(ν) + ν^{1−1/k_{θ}} + ν^{1−1/k_{θ+1}} (\frac{ρ_{θ+1}}{ρ_{θ}} )^{ν} ω_{n,y}(m)) ,

if 2 ≤ θ < s and ν ∈ m − 1 + ℕ; moreover

exp(O(1)(ln(ν) + ν^{1−1/k_{θ}} + ν^{1−1/k_{θ+1}})) (\frac{ρ_{θ+1}}{ρ_{θ}} )^{ν} = o(1), (79)

when ν → ∞ and 2 ≤ θ < s.

In view of (75), (73), (77) ω_{n,π_{θ}(y)}(ν) ≥ ω_{n,π_{θ}(y)}(ν) − ω_{n,y−π_{θ}(y)}(ν) ≥ (80)

exp(−O(1)(ln(ν) + ν^{1−1/k_{θ}}))(ρ_{θ})^{ν}×

(ω_{n,π_{θ}(y)}(m) − (\frac{O(1)}{ν})^{ν/n} ω_{n,y}(m)) ,

if 2 ≤ θ = s and ν ∈ m − 1 + ℕ; moreover

(\frac{exp(O(1))}{ν})^{ν/n} = o(1), (81)

when ν → ∞.

L.A.Gutnik, On the difference equations of the Poincaré type (Part 3) 16

Let y ∈ V_{m}. We apply (33) in Lemma 2 to y_{1} = π_{1}(y). In view of (52), we have (C_{3})^{−q}ω_{n,π_{1}(y)}(m) exp (−O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν} ≤ (82)

ω_{p,π_{1}(y)}(m) exp (−O(1) (ν^{1−1/k_{1}} + ln(ν))) ≤
(ρ_{1})^{−ν} ω_{p,π_{1}(y)}(ν) ≤ (ρ_{1})^{−ν} ω_{n,π_{1}(y)}(ν),

where ν ∈ m − 1 + ℕ and O(1) depends only from the equation (3). If (59) holds for r = ϕ_{m}(y), then in view of (64), (60), (49),

sup{ω_{n,ψ_{m}(ϕ_{m}(y))}(ν)/v_{A^{∨},k_{θ},ρ_{θ}}(ν): ν ∈ ℕ − 1 + m} ≤

C_{6} sup{ϕ_{m}(y)(ν)/v_{A^{∨},k_{θ},ρ_{θ}}(ν): ν ∈ ℕ + m − 1} ≤ C_{6}ω_{q,ϕ_{m}(y)}(m) ≤
C_{6}C_{3}C_{5}ω_{n,y}(m);

and therefore ω_{n,ψ_{m}(ϕ_{m}(y))}(ν) ≤ C_{6}C_{3}C_{5}ω_{n,y}(m)v_{A^{∨},k_{θ},ρ_{θ}}(ν) = (83)

C_{6}C_{3}C_{5}ω_{n,y}(m)(ρ_{1})^{ν}v_{A^{∨},k_{θ},ρ_{θ}}(ν)/(ρ_{1})^{ν},

where ν ∈ ℕ + m − 1. In view of (82) and (83), ω_{n,y}(ν) = ω_{n,π_{1}(y)+ψ_{m}(ϕ_{m}(y))}(ν) ≥ (84)

ω_{n,π_{1}(y)}(ν) − ω_{n,ψ_{m}(ϕ_{m}(y))}(ν) ≥

(C_{3})^{−q}ω_{n,π_{1}(y)}(m) exp (−O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν}−

C_{6}C_{3}C_{5}ω_{n,y}(m)(ρ_{1})^{ν}v_{A^{∨},k_{θ},ρ_{θ}}(ν)/(ρ_{1})^{ν} =

(ω_{p,π_{1}(y)}(m) exp (−O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν}−

ω_{n,y}(m) (exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) \frac{v_{A^{∨},k_{θ},ρ_{θ}}(ν)}{(ρ_{1})^{ν}}) ,

where ν ∈ ℕ + m − 1 and

(85) lim_{ν→∞} (exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) \frac{v_{A^{∨},k_{θ},ρ_{θ}}(ν)}{(ρ_{1})^{ν}}) = 0.

If for r = ϕ_{m}(y) (62) holds, then in view of (65), (63), (49),

sup{ω_{n,ψ_{m}(ϕ_{m}(y))}(ν)/w_{A^{∨},k^{∗}}(ν): ν ∈ ℕ − 1 + m} ≤

C_{6} sup{ϕ_{m}(y)(ν)/w_{A^{∨},k^{∗}}(ν): ν ∈ ℕ + m − 1} ≤ C_{6}ω_{q,ϕ_{m}(y)}(m) ≤
C_{6}C_{3}C_{5}ω_{n,y}(m);

and therefore ω_{n,ψ_{m}(ϕ_{m}(y))}(ν) ≤ C_{6}C_{3}C_{5}ω_{n,y}(m)w_{A^{∨},k^{∗}}(ν), (86) where ν ∈ ℕ + m − 1. In view of (82) and (86), ω_{n,y}(ν) = ω_{n,π_{1}(y)+ψ_{m}(ϕ_{m}(y))}(ν) ≥ (87)

L.A.Gutnik, On the difference equations of the Poincaré type (Part 3) 17

ω_{n,π_{1}(y)}(ν) − ω_{n,ψ_{m}(ϕ_{m}(y))}(ν) ≥

(C_{3})^{−q}ω_{n,π_{1}(y)}(m) exp (−O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν}−

C_{6}C_{3}C_{5}ω_{n,y}(m)w_{A^{∨},k^{∗}}(ν) =

ω_{p,π_{1}(y)}(m) exp (−O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν}−

ω_{n,y}(m) (exp (O(1) (ν^{1−1/k_{1}} + ln(ν + 1))) \frac{w_{A^{∨},k^{∗}}(ν)}{(ρ_{1})^{ν}}) ,

where ν ∈ ℕ + m − 1, and

(88) lim_{ν→∞} (exp (O(1) (ν^{1−1/k_{1}} + ln(ν + 1))) w_{A^{∨},k^{∗}}(ν)/(ρ_{1})^{ν}) = 0.

We apply (31) in Lemma 2 to y_{1} = π_{1}(y). Then, in view of (52) , we have

(C_{3})^{−q}ω_{n,π_{1}(y)}(ν) ≤ ω_{p,π_{1}(y)}(ν) ≤

exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν} ω_{p,π_{1}(y)}(m) ≤
exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν} ω_{n,π_{1}(y)}(m) ≤
exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν} ω_{p,π_{1}(y)}(m) ≤
exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν} ω_{n,y−ψ_{m}(ϕ_{m}(y))}(m) ≤
exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν} ω_{n,y}(m)+
exp (O(1) (ν^{1−1/k_{1}} + ln(ν))) (ρ_{1})^{ν} ω_{n,ψ_{m}(ϕ_{m}(y))}(m).

If (59) holds for r = ϕ_{m}(y), then, in view of (83),

ω_{n,ψ_{m}(ϕ_{m}(y))}(m) ≤ C_{6}C_{3}C_{5}ω_{n,y}(m)v_{A^{∨},k_{θ},ρ_{θ}}(m).

If (62) holds for r = ϕ_{m}(y), then, in view of (86),

ω_{n,ψ_{m}(ϕ_{m}(y))}(m) ≤ C_{6}C_{3}C_{5}ω_{n,y}(m)w_{A^{∨},k^{∗}}(m).

Therefore

ω_{n,π_{1}(y)}(ν) ≤ exp (O(1) (ν^{1−1/k_{1}} + ln(ν + 1))) (ρ_{1})^{ν} ω_{n,y}(m).

In view of (83),(86) ω_{n,(y)}(ν) = ω_{n,π_{1}(y)+ψ_{m}(ϕ_{m}(y))}(ν) ≤ (89)

exp (O(1) (ν^{1−1/k_{1}} + ln(ν + 1))) (ρ_{1})^{ν} ω_{n,y}(m),

where ν ∈ ℕ + m − 1. The assertions of the Theorem 7 for σ are contained in (75), (78), (79), (80), (81), (82), (84), (85), (87), (88), (89). ■

So, we must point to the spliting monomorphism ψ_{m} of the Lemma 4.

§3. On some linear normed spaces.

Let K be one of the fields ℝ or ℂ. Let further L be a linear normed space over K with norm p(x). In the case L = K^{n} we fix as p(x), wehre x ∈ K^{n}, the maximum of the absolute values of coordinates of x in the standard basis, i.e.

p(x) = h(x) = sup({|x_{1}|, …, |x_{n}|}), (90)

where

x = (x_{1} ; ⋮ ; x_{n}) .

If L is a Banach space with the norm p, then K−algebra of all the linear continuous operators acting in L will be denoted by 𝔐^{∧}(L), and the norm on 𝔐^{∧}(L), associated with the norm p will be denoted by p^{∼}. So,

p^{∼}(A) = sup({p(AX): X ∈ L, p(X) ≤ 1}).

It is well known that the associciated with h norm on Mat_{n}(ℂ) is defined as follows

h^{∼}(A) = sup ({∑_{k=1}^{n} |a_{i,j}|: i = 1, …, n}) , (91)

where A = (a_{i,k}) ∈ Mat_{n}(ℂ). The norms h and h^{∼} coincide respectiely with the norms q_{∞} and q_{∞}^{∼} considered in [33], section 6. Let m ∈ ℕ − 1, and we denote by E_{m}(L) be the set L^{m−1+ℕ} of all the maps of the set m − 1 + ℕ into L. The set E_{m}(L) is a linear space over K, where the muliplication of the elements by the number from K and addition of the elements is defined coordinate-wise. The subspace of E_{m}(L) composed by all the constant maps is isomorphic to L, and we identify this subspace with L.

We denote by 𝔐^{∨}(L) the space of all the K−linear maps of the space L in L. If ϕ ∈ 𝔐^{∨}(L) and ψ ∈ 𝔐^{∨}(L), then ϕ ∘ ψ denotes the composition of operators ϕ and ψ, so that (ϕ∘ψ)f = ϕ((ψf)) for each f ∈ L. For x ∈ E_{m}(L) let

p_{m,∞}(x) = sup({p(x(ν)): ν ∈ m − 1 + ℕ}.

Let further E_{m,∞}(L) = {x ∈ E_{m}(L): p_{m,∞}(x) ≠ ∞},

E_{m,0}(L) = {x ∈ E_{m}(L): lim_{ν→∞} p(x(ν)) = 0},

E\overrightarrow{m}(L) = L + E_{m,0}(L).

Clearly, the space E\overrightarrow{m}(L) consists of all the y ∈ E_{m}(L), for which there exists

lim(y) = lim_{ν→∞}(y(ν)).

Let m ∈ ℕ − 1, μ ∈ m − 1 + ℕ and ler r_{m,μ} be the operator of restriction of the elements y ∈ E_{m}(L) on te set m − 1 + ℕ. Clearly, the map r_{m,μ} is an epimorphism of the space E_{m}(L) onto the space E_{μ}(L). If L is a K-algebra, then E_{m}(L) is a K-algebra, where the muliplication and addition of the elements is defined coordinate-wise; so, in this case r_{m,μ} is an epimorphism of K-algebra E_{m}(L) onto K-algebra E_{μ}(L). If L be an algebra with unity, let L^{∗} denotes the group of all its invertible elements. Then

(L^{∗})^{m−1+ℕ} ⊂ L^{m−1+ℕ};

we denote below (L^{∗})^{m−1+ℕ} by E_{m}(L^{∗}). Clearly,

E_{m}(L^{∗}) = (E_{m}(L))^{∗}.

Let L = ℂ^{n}, y ∈ E_{m}(L), and let y_{i}(ν) denotes the i-th coordinate of the element y(ν), where i = 1, …, n, ν ∈ m − 1 + ℕ; then the space (E_{m}(ℂ))^{n} contains an element ω(y), which has y_{i}(ν) as the value of its i-th coordinate at the point ν ∈ m − 1 + ℕ. So we obtain the natural isomorphism ω of the algebra E_{m}(ℂ^{n}) onto (E_{m}(ℂ))^{n}. This map ω induces an isomorphism of the algebra E_{m}(Mat_{n}(ℂ)) onto Mat_{n}(E_{m}(ℂ)). If L is a K − algebra, then each element a ∈ E_{m}(L) determines an acting on the space E_{m}(L) K-linear operator μ_{a} ∈ 𝔐^{∨}(E_{m}(L)), which turns any y ∈ E_{m}(L) into μ_{a}y = ay. On the space E_{m}(L) acts also K-linear operator ▽ ∈ 𝔐^{∨}(L), which turns any element y ∈ E_{m}(L) in the ▽y ∈ E_{m}(L) such that

(▽y)(ν) = y(ν + 1)

for any ν ∈ m − 1 + ℕ. Let us consider the subring 𝔄_{m}(L) of the ring 𝔐^{∨}(𝔏) generated by the operator ▽ and by all the operators μ_{a}, where a ∈ E_{m}(L). Clearly,

μ_{a} ∘ ▽^{r} ∘ μ_{b} ∘ ▽^{s} = μ_{a▽^{r}b_{k}} ∘ ▽^{r+s}, (92)

where {r, s} ⊂ ℕ − 1, {a, b} ⊂ E_{m}(L). For each α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}} are uniquelly defined the number deg(α) and representation of α in the form

α = ∑_{k=0}^{deg(α)} μ_{a_{k}} ∘ ▽^{k}, (93)

where a_{k} ∈ E_{m}(L) for k = 0, …, deg(α) and a_{deg(α)} ≠ 0_{E_{m}(L)}. Clearly, (93) may be rewritten in the form

α = ∑_{k=0}^{∞} μ_{a_{k}} ∘ ▽^{k}, (94)

where a_{k} = 0_{E_{m}(L)} for k ∈ deg(α) + ℕ. It follows from (92) that 𝔄_{m}(L) is a graduated algebra, and if

β = ∑_{r=0}^{p} μ_{b_{r}} ∘ ▽^{r} ∈ 𝔄_{m}(L), (95)

γ = ∑_{s=0}^{q} μ_{c_{s}} ∘ ▽^{s} ∈ 𝔄_{m}(L), (96)

then

βγ = ∑_{k=0}^{p+q} ∑_{\begin{matrix}≤r≤p \\ 0≤s≤q \\ r+s=k \end{matrix}} μ_{b_{r}▽^{r}c_{s}} ∘ ▽^{r+s}; (97)

clearly, deg(βγ) = deg(β)+deg(γ), if b_{p}(ν)^{r}c_{q}(ν+p) is different from 0 at least for one ν ∈ m−1+ℕ. Let 𝔄\overrightarrow{m}(L) be the ring generated by the operator ▽ and by all the operators μ_{a}, where a ∈ E\overrightarrow{m}(L). Since ▽a ∈ E\overrightarrow{m}(L), if a ∈ E\overrightarrow{m}(L), it follows, in view of (92), that 𝔄\overrightarrow{m}(L) is a graduated subalgebra 𝔄\overrightarrow{m}(L) of the algebra 𝔄_{m}(L), each α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}} admits a representation in the form (33) with a_{k} ∈ E\overrightarrow{m}(L) for k = 0, …, deg(α) and a_{deg(α)} ≠ 0_{E_{m}(L)}; to each such α corresponds the limit operator

(98) lim(α) = ∑_{k=0}^{deg(α)} μ_{lim(a_{k})} ∘ ▽^{k},

and polynomial

P (α, z) = ∑_{k=0}^{deg(α)} lim(a_{k})z^{k} ∈ L[z]. (99)

If α = 0_{𝔄_{m}(L)}, then we put

lim(α) = 0_{𝔄_{m}(L)}, P (α, z) = 0_{L[z]}.

The equality (92) shows that the map

α → P (α, z) (100)

is an epimorphism of the algebra 𝔄\overrightarrow{m}(L) on on the algebra L[z] (if the algebra L is noncommutative, then we can treat L[z] as a semigroup ring of the semigroup (ℕ−1, +) over the algebra L). We note that, if α ∈ 𝔄_{m}(ℂ), then, clearly, Ker(α) coincides with the linear space of all the solutions of the equation (3); moreover if α ∈ 𝔄\overrightarrow{m}(ℂ), then the corresponding to α equation (3) is an equation of the Poincar’e type and P (α, z) is its characterictical polynomial. Let L be an algebra with unity. The set of all α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}}, which have the representation (93) with a_{deg(α)} ∈ E_{m}(L^{∗}) will be denoted further by 𝔄_{m}(L)^{∘}. The set of all the the elements α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}}, which have the representation (93) with a_{deg(α)} = 1_{E_{m}(L)} will be denoted further by 𝔄_{m}(L)^{∨}. The set of all the α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}}, which have the representation (93) with a_{0} ∈ E_{m}(L^{∗}), will be denoted further by 𝔄_{m}(L)^{∧}. Let further

𝔄_{m}(L)^{∘∧} = 𝔄_{m}(L)^{∘} ⋂ 𝔄_{m}(L)^{∧}, 𝔄_{m}(L)^{∨∧} = 𝔄_{m}(L)^{∨} ⋂ 𝔄_{m}(L)^{∧},

𝔄_{m}(L)^{∘→} = 𝔄_{m}(L)^{∘} ⋂ 𝔄_{m}(L)^{→}, 𝔄_{m}(L)^{∨→} = 𝔄_{m}(L)^{∨} ⋂ 𝔄_{m}(L)^{→},

𝔄_{m}(L)^{∘∧→} = 𝔄_{m}(L)^{∘∧} ⋂ 𝔄_{m}(L)^{→}, 𝔄_{m}(L)^{∨∧→} = 𝔄_{m}(L)^{∨∧} ⋂ 𝔄_{m}(L)^{→}, Clearly, 𝔄_{m}(L)^{∘}. consists of epimorphisms of the space E_{m}(L) onto E_{m}(L). The above map r_{m,μ} induces epimorphism r_{m,μ}^{▷} of the algebra 𝔄_{m}(L) on the algebra 𝔄_{μ}(L) defined as follows:

if α ∈ 𝔄_{m}(L),

α = ∑_{k=0}^{n} μ_{a_{k}} ∘ ▽^{k}, (101)

then

r_{m,μ}^{▷}(α) = ∑_{k=0}^{n} μ_{r_{m,μ}(a_{k})} ∘ ▽^{k}, (102)

where the operator ▽ in (101) acts in E_{m}(L) and the operator ▽ in (102) acts in E_{μ}(L). Clearly, r_{m,μ}^{▷} surjectively maps

𝔄_{m}(L)^{∘} onto 𝔄_{μ}(L)^{∘}, 𝔄_{m}(L)^{∨} onto 𝔄_{μ}(L)^{∨},

𝔄_{m}(L)^{∧} onto 𝔄_{m}u(L)^{∧}, 𝔄_{m}(L)^{∘∧} onto 𝔄_{μ}(L)^{∘∧},

𝔄_{m}(L)^{∨∧} onto 𝔄_{μ}(L)^{∨∧}, 𝔄_{m}(L)^{∘→} onto 𝔄_{μ}(L)^{∘→},

𝔄_{m}(L)^{∨→} onto 𝔄_{μ}(L)^{∨→}, 𝔄_{m}(L)^{∘∧→} onto 𝔄_{μ}(L)^{∘∧→},

𝔄_{m}(L)^{∨∧→} onto 𝔄_{μ}(L)^{∨∧→}. Since the diagram

E_{m}(L) ⟶^{r_{m,μ}} E_{μ}(L)
α↓  ↓r_{m,μ}^{▷}(α)
E_{m}(L) ⟶_{r_{m,μ}} E_{μ}(L)

is commuative and therefore

r_{m,μ}α = r_{m,μ}^{▷}(α)r_{m,μ}, (103)

it follows that r_{m,μ} surjectively maps Ker(r_{m,μ}α) onto

Ker(r_{m,μ}^{▷}(α)) ⊃ r_{m,μ}Ker(α).

Lemma 5. If μ ∈ m − 1 + ℕ and α ∈ 𝔄_{m}(L)^{∧}, then the operator α bijectively maps Ker(r_{m,μ}) onto Ker(r_{m,μ}).

Proof. Proof is given in (37), Lemma 3.

Corollary 1. Let μ ∈ m − 1 + ℕ and let α ∈ 𝔄_{m}(L)^{∧}. If

g ∈ E_{m}(L), x ∈ E_{μ}(L), m ≤ μ, α ∈ 𝔄_{m}(L)^{∧},
r_{m,μ}(g) = (r_{m,μ}^{▷}(α))(x),

then there exists a unique y ∈ E_{m}(L) such that

α(y) = g, r_{m,μ}(y) = x;

Proof. Proof is given in (37), Corollary 1 to the Lemma 3.

Corollary 2. Let μ ∈ m − 1 + ℕ and α ∈ 𝔄_{m}(L)^{∧}. Then and r_{m,μ} bijectively maps Ker(α) onto Ker(r_{m,μ}^{▷}(α)) = r_{m,μ}(Ker(α)).

Proof. Proof is given in (37), Corollary 2 to the Lemma 3.

If for the equation (2) are fulfilled the conditions (1) then

a_{k} = (a_{k}(0), a_{k}(1), …, a_{k}(ν), …) ∈ E\overrightarrow{0}(ℂ), (104)

where k = 0, …, n. Moreover, a_{n} = 1_{E_{0}(ℂ)}, for α in (101) Ker(α) coincides with the linear over ℂ space of all the solutions of the equation (2), polynomial (4) is equal to the polynomial P (α, z) = P (r_{0,m}^{▷}(α), z), where m ∈ ℕ − 1, and the set Ker(r_{0,m}^{▷}(α)) coincides with linear over ℂ space V_{m} of all the solutions of the equation (3).

Let 𝔳 be the element in E_{0,0}, for which

𝔳(ν) = \frac{1}{ν + 1},

where ν ∈ ℕ − 1. Clearly, r_{0,m}(𝔳)E_{m,∞}(ℂ) ⊂ E_{m,0}(ℂ) for any m ∈ ℕ − 1. Let

E_{m,0}^{≻}(L) = r_{0,m}(𝔳)E_{m,∞}(L), E_{m}^{≻}(L) = L + E_{m,0}^{≻}(L).

Let us consider the ring 𝔄_{m}^{≻}(L) generated by the operator ▽ and by all the operators μ_{a}, where a ∈ E_{m}(L)^{≻}. The Assertion 1 may be reformulated as follows:

Lemma 6. Let α ∈ 𝔄_{0}^{≻}(ℂ) ⋂ 𝔄_{0}^{∨}(ℂ), and P (α, z) coincides with the polynomial T (z) in (4) and (37).

Then there exist m ∈ ℕ and representation of the oprator r_{0,m}^{▷}(α) in the form

r_{0,m}^{▷}(α) = ηβ (105)

such that

η ∈ 𝔄_{m}^{≻}(ℂ) ⋂ 𝔄_{m}^{∨}(ℂ), deg(η) = q (106)

β ∈ 𝔄_{m}^{≻}(ℂ) ⋂ 𝔄_{m}^{∨∧}(ℂ), deg(β) = p = n − q, (107)

and the polynomials P (β, z) , P (η, z) coincide respectively with the polynomials T_{1}(z) , T_{2}(z) in (36) .

Let L be a Banach space with the norm p. Let a = a(ν), where ν ∈ ℕ − 1, be an arbitrary sequence of positive real numbers. Let E_{m,a}(L) = aE_{m,∞}(L). Then E_{m,a}(L) consists of all such y ∈ E_{m}(L), for which

p_{m,a}(y) = p_{m,∞} (a^{−1}y) = (108)

sup {p ((a(ν))^{−1} y(ν)) : ν ∈ ℕ + m − 1} < +∞.

Then p_{m,a} is a norm on E_{m,a}(L) and E_{m,a}(L) is a Banah space relatively the norm p_{m,a}. Let λ > 0 and let 𝔞_{m,λ} be a sequence, for which

𝔞_{m,λ}(ν) = λ^{ν},

where ν ∈ ℕ + m − 1. Then

E_{m,𝔞_{m,1}}(L) = E_{m,∞}(L), p_{m,𝔞_{m,1}} = p_{m,∞},

the map, which turns any element z ∈ E_{0,𝔞_{m,1}}(L) into y ∈ E_{m}(L) such that y(ν) = a(ν)z(ν − m), where ν ∈ ℕ + m − 1, is a isometry of the space E_{0,𝔞_{m,1}}(L) = E_{0,∞}(L) on the space E_{m,a}(L). Clearly, p_{m,𝔞_{m,1}} and p_{m,∞} coincide on the space E_{m,𝔞_{m,1}}(L) = E_{m,∞}(L). Clearly,

{y ∈ E_{m}(L): p_{m,a}(y) ≤ 1} = {at: p_{m,∞}(t) ≤ 1}. (109)

Let C > 0, n ∈ ℕ. Let w_{C,n} denotes the element in E_{0,0}(ℂ), for which

w_{C,n}(ν) = (\frac{C}{ν + 1})^{ν/n} , (110)

where ν ∈ ℕ − 1; let further ρ > 0 and v_{C,n,ρ} denotes the element in E_{0}(ℂ), for which

v_{C,n,ρ}(ν) = ρ^{ν} exp (C(ν + 1)^{1−1/n} + ln(ν + 1)) , (111)

where ν ∈ ℕ − 1. Let C is bigger than all the absolute values of O(1) in (31),

(33) and A^{∨} in (58), (63). In view of (31), the sequence

Y_{n,y} = Y_{n,y}(ν),

where ν ∈ ℕ + m − 1, contains in E_{m,v_{C,n,ρ_{1}}}(ℂ^{n}), and

h_{m,v_{C,n,ρ_{1}}}(Y_{n,y}) ≤ h(Y_{n,y}(m)).

In view of (60), if s ≥ 2, r ∈ R_{m,θ}^{∧}\R_{m,θ+1}^{∧} for some θ ∈ {2, ..., s}, then

Y_{q,r} = Y_{q,r}(ν),

where ν ∈ ℕ + m − 1, contains in E_{m,v_{C,k_{θ},ρ_{θ}}} (ℂ^{q}), and

h_{m,v_{C,k_{θ},ρ_{θ}}} (Y_{q,r}) ≤ h(Y_{q,r}(m)) = ω_{q,r}(m)).

In view of (63), if k^{∗} > 0, r ∈ R_{m,s+1}^{∧}, then

Y_{q,r} = Y_{q,r}(ν),

where ν ∈ ℕ + m − 1, contains in E_{m,w_{C,k^{∗}}}(ℂ^{q}), and

h_{m,w_{C,k^{∗}}}(Y_{q,r}) ≤ h(Y_{q,r}(m)) = ω_{q,r}(m)).

Lemma 7. The operaor ▽ turns the space E_{m,a}(L) into E_{m,a}(L) if and only if, when

h_{m,a}(▽a) = (112)

sup{a(ν + 1)/a(ν): ν ∈ ℕ + m − 1} < +∞.

If (112) holds then ▽ is continuous on E_{m,a}(L) and

p_{m,a}^{∼}(▽) = h_{m,a}(▽a) (113)

Proof. Let f ∈ L⧹{0}, g = (p(f))^{−1} f,

y_{0}(ν) = g, y_{1} = ay_{0}.

L.A.Gutnik, On the difference equations of the Poincaré type (Part 3) 24

Then p_{m,a}(y_{1}) = 1,

p ((a(ν))^{−1}(▽y_{1})(ν)) = a(ν + 1)/a(ν),

where ν ∈ ℕ + m − 1; if ▽ turns E_{m,a}(L) into E_{m,a}(L) then h_{m,a}(▽a) = sup{a(ν + 1)/a(ν): ν ∈ ℕ + m − 1} = (114)

sup {p ((a(ν))^{−1}(▽y_{1})(ν)) : ν ∈ ℕ + m − 1} = p_{m,a}(▽y_{1}) < +∞,

and therefore (112) holds. If (112) holds, y ∈ E_{m,a}(L) and p_{m,a}(y) ≤ 1, then p ((a(ν))^{−1}(▽y)(ν)) = (115)

p ((a(ν))^{−1}a(ν + 1)(a(ν + 1))^{−1}y(ν + 1)) =
(a(ν))^{−1}a(ν + 1)p ((a(ν + 1))^{−1}y(ν + 1)) ≤
h_{m,a}(▽a)p_{m,a}(y).

Therefore ▽ is continuous on E_{m,a}(L) and p_{m,a}^{∼}(▽) ≤ h_{m,a}(▽a)p_{m,a}(y) (116) The equality (113) follows from (116) and from (114). ■

Each A ∈ E_{m}(𝔐^{∨}(L)) defines a linear operator A^{∨} ∈ 𝔐^{∨}(E_{m}(L)), such that

(A^{∨}y)(ν) = (A(ν))(y(ν)),

where ν ∈ m − 1 + ℕ, y ∈ E_{m}(L). The operator A^{∨} is invertible if and only if, when the operator A(ν) is invertible for any ν ∈ m − 1 + ℕ; in this case the map

ν ↦ (A(ν))^{−1},

where ν ∈ m − 1 + ℕ, will be denoted by A^{−1}. So,

A^{−1}(ν) = (A(ν))^{−1},

where ν ∈ m − 1 + ℕ. Clearly, (A^{∨})^{−1} = (A^{−1})^{∨}. If A ∈ E_{m,∞}(𝔐(L)), (117) then, clearly, (p^{∼})_{m,∞}(A) = sup{p^{∼}(A(ν)): ν ∈ m − 1 + ℕ} = (118)

sup{sup{p(A(ν)y(ν)): : y(ν) ∈ L, p(y(ν)) ≤ 1}: ν ∈ m − 1 + ℕ} =
sup{p_{m,∞}(A^{∨}(y)): y ∈ E_{m,∞}(L), p_{m,∞}(y) ≤ 1} =
(p_{m,∞})^{∼}(A^{∨}).

In particular, if a ∈ E_{m,∞}(K) (119) and we take on the role of A in (117) the we take the sequence of the (left) multiplications by a(ν) of the elements in L, then the role of A^{∨} will play the above operator μ_{a}, and

p_{m,∞}(a) = (p_{m,∞})^{∼}(μ_{a}). (120)

If a = a(ν) and d = d(ν), where ν ∈ ℕ − 1, are two sequences of te positive real mumbers and

A ∈ E_{m,a/d}(𝔐(L)), (121)

then A = μ_{a/d}B, where B ∈ E_{m,∞}(𝔐(L)), the operator A^{∨} maps the space E_{m,d}(L) into the space E_{m,a}(L), and

(p^{∼})_{m,a/d}(A) = (p^{∼})_{m,∞}(B) = (p_{m,∞})^{s}im(B^{∨}) = (122)

sup{p_{m,∞}(B^{∨}(y)): y ∈ E_{m,∞}(L), p_{m,∞}(y) ≤ 1} =
sup{p_{m,a}(B^{∨}(μ_{a}(y))): y ∈ E_{m,∞}(L), p_{m,d}(μ_{d}y) ≤ 1} =
sup{p_{m,a}(B^{∨}(μ_{a/d}(z))): z ∈ E_{m,d}(L), p_{m,d}(z) ≤ 1} =
sup{p_{m,a}((μ_{a/d}B)^{∨}(z)): z ∈ E_{m,d}(L), p_{m,d}(z) ≤ 1} =
sup{p_{m,a}(A^{∨}(z)): z ∈ E_{m,d}(L), p_{m,d}(z) ≤ 1};

in particular, if a = d the it follows from (122) that A^{∨} maps E_{m,a}(L) into E_{m,a}(L), and

(p_{m,a})^{∼}(A^{∨}) = (123)

sup{p_{m,a}(A^{∨}(z)): z ∈ E_{m,a}(L), p_{m,a}(z) ≤ 1} =
(p^{∼})_{m,a/a}(A) = (p^{∼})_{m,∞}(A).

Lemma 8. Let m ∈ ℕ − 1 and α from (93) belongs to 𝔄\overrightarrow{m}(ℂ). Let further a = a(ν), where ν ∈ ℕ − 1, be an arbitrary sequence of positive real numbers, λ > 0 and

h_{m,a}(▽a) = sup{a(ν + 1)/a(ν): ν ∈ ℕ + m − 1} ≤ λ. (124)

Then α is bounded linear operator on the space E_{m,a}(ℂ), and

(p_{m,a})^{∼}(α) ≤ ∑_{k=0}^{deg(α)} p_{m,∞}(a_{k})λ^{k} (125)

Proof. In view of (124) and Lemma 7, (113) holds. The asserion of the Lemma now follows from (120). ■

Corollary. If α from (93) belongs to 𝔄\overrightarrow{m}(ℂ), then α is bounded linear operator on the space E_{m,𝔞_{m,λ}}(ℂ), and (125) holds.

Proof. In fact,

𝔞_{m,λ}(ν + 1)/𝔞_{m,λ}(ν) = λ

So, all the conditions of the Lemma 8 are fulfilled. ■

Clearly, if a_{1}(ν) and a_{2}(ν), where ν ∈ ℕ − 1, are two arbitrary sequences of positive real numbers, and A_{i} ∈ E_{m,a_{i}}(𝔐(L)), (126) where i = 1, 2 then A_{1}A_{2} ∈ E_{m,a_{1}a_{2}}(𝔐(L)), and

(p^{∼})_{m,a_{1}a_{2}}(A_{1}A_{2}) = (p^{∼})_{m,∞} ((a_{1}a_{2})^{−1}A_{1}A_{2}) =

(p^{∼})_{m,∞} (((a_{1})^{−1}A_{1}) ((a_{2})^{−1}A_{2})) ≤
(p^{∼})_{m,∞} ((a_{1})^{−1}A_{1}) (p^{∼})_{m,∞} ((a_{2})^{−1}A_{2}) =
(p^{∼})_{m,a_{1}}(A_{1})(p^{∼})_{m,a_{2}}(A_{2}).

Clearly,

(▽A^{∨})^{k} = (∏_{κ} = 1^{k}(▽^{κ}A)^{∨} ▽^{k}, (127)

where k ∈ ℕ − 1, A ∈ E_{m}(𝔐(L))

Let L is a Banach space over th field K and A ∈ E_{m,∞}(𝔐^{∧}(L)). Let further there exists A^{−1} ∈ E_{m,∞}(𝔐^{∧}(L)), λ > 0 and (p^{∼})_{m,1}(A^{−1}) = ρ < 1/λ. (128) Let further a = a(ν), where ν ∈ ℕ − 1, be an arbitrary sequence of positive real numbers, and (refeq:cde1) holds. As we see above, (see text after (122) the operator (A^{−1})^{∨} maps E_{m,a}(L) into E_{m,a}(L), and, in view of (123) and (128), (p_{m,a})^{∼}((A^{−1})^{∨}) = (p^{∼})_{m,∞}(A^{−1}) = ρ < 1/λ. (129) According to the (129), (113) and (124) (p_{m,a})^{∼}(▽ ∘ (A^{−1})^{∨}) ≤ ρλ < 1, (130) 𝔐(E_{m,a}(L)) contains the operator

− ∑ k = 0^{∞}(▽ ∘ (A^{−1})^{∨})^{k} =

(▽ ∘ (A^{−1})^{∨} − 1_{𝔐(E_{m,a}(L)})^{−1} = A^{v}ee(▽ − A^{∨})^{−1},

and (p_{m,a})^{∼} (A^{∨}(▽ − A^{∨})^{−1}) ≤ 1/(1 − ρλ). (131) Therefore, 𝔐(E_{m,a}(L)) contains the linear operator

− (A^{−1})^{∨} ∑_{k=0}^{∞}(▽ ∘ (A^{−1})^{∨})^{k} = (▽ − A^{∨})^{−1}, (132)

and in view of (132) and (129), (p_{m,a})^{∼}((▽ − A^{∨})^{−1}) ≤ ρ/(1 − ρλ). (133) According to (127), the equality (132) may be rewritten in the form

(▽ − A^{∨})^{−1} = −(A^{−1})^{∨} ∑_{k=0}^{∞} (∏_{κ=1}^{k}(▽^{κ}(A^{−1}))^{∨}) ∘ ▽^{k}. (134)

§4. End of the proof of the Theorem 7.

Let, in accordance with (50), (105),

▽^{p} + ∑_{k=0}^{p−1} μ_{b_{k}} ∘ ▽^{k}, (135)

where b_{k} ∈ E_{m}^{≻}ℂ ⊂ E\overrightarrow{m}ℂ for k = 0, …, p − 1, and b_{k}(ν) are coefficients of the equation (50) for k = 0, …, p − 1 and ν ∈ m − 1 + ℕ. In view of (38),

b_{k}^{∼} = lim(b_{k}), (136)

where k = 0, …, p − 1,

B_{1} = (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −b_{0} & −b_{1} & −b_{2} & … & −b_{p−1} \end{matrix}) ,

B = ω^{−1}(B_{1}), where ω is the above isomorphism of the algebra E_{m}(Mat_{p}(ℂ)) onto Mat_{p}(E_{m}(ℂ)), and let

B^{∼} = (\begin{matrix}0 & 1 & 0 & … & 0 \\ 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & … & ⋮ \\ 0 & 0 & 0 & … & 1 \\ −b_{0}^{∼} & −b_{1}^{∼} & −b_{2}^{∼} & … & −b_{p−1}^{∼}. \end{matrix})

We use now the Lemmata 8 and 9 in [37] and we take on the role of the matrix A in these Lemmata the matrix B^{∼}. In view of (36), b_{0}^{∼} = T_{1}(0) ≠ 0. Therefore (B^{∼})^{−1} exists. We take on the role of ε and ε_{1} in the specified Lemma 8 and Lemma 9 of the paper [37] the number

ε_{0} = (ρ_{1} − ρ_{2})/4,

and we take on ℂ^{p} the norm p_{B^{∼},ε_{0}}. In view of (110),

w_{C,n}(ν + 1)/w_{C,n}(ν) ≤ (\frac{C}{ν + 1})^{1/n} , (137)

where ν ∈ ℕ. In view of (111),

v_{C,n,ρ}(ν + 1)/v_{C,n,ρ}(ν) ≤ ρ exp ((2C)(ν)^{−1/n}) , (138)

where ν ∈ ℕ. Making use the equality lim(B) = B^{∼}, (137), (138) and the inequalities ρ_{θ} ≤ ρ2, k_{θ} ≤ n if 2 ≤ θ ≤ s + 1, the above m may be replaced by means the map r_{m,μ}^{#} on some bigger m, such that

p^{∼}(B(ν) − B^{∼}) ≤ ε_{0}, (139) if ν ∈ ℕ − 1 + m, w_{C,n}(ν + 1)/w_{C,n}(ν) ≤ ρ_{2} + ε_{0}, (140) if ν ∈ ℕ − 1 + m, v_{C,k_{θ},ρ_{θ}}(ν + 1)/v_{C,n,ρ_{2}}(ν) ≤ ρ_{θ} + ε_{0}, (141) if 2 ≤ θ ≤ s and ν ∈ ℕ − 1 + m, C/(ν + 1) < 1, (142) if ν ∈ ℕ − 1 + m. Then, in view of (142 and the inequalities ρ_{θ} ≤ ρ2, k_{θ} ≤ n if 2 ≤ θ ≤ s + 1, the following inequalities hold:

\frac{w_{C,k^{∗}}(ν + 1)}{w_{C,k^{∗}}(ν)} ≤ \frac{w_{C,n}(ν + 1)}{w_{C,n}(ν)} ≤ ρ_{2} + ε_{0}, (143)

if k^{∗} > 0 and ν ∈ ℕ − 1 + m,

\frac{v_{C,k_{θ},ρ_{θ}}(ν + 1)}{v_{C,n,ρ_{2}}(ν)} ≤ ρ_{θ} + ε_{0} ≤ ρ_{2} + ε_{0}, (144)

if 2 ≤ θ ≤ s and ν ∈ ℕ−1+m. In view of the inequality (139) and according to the Lemma 9 in [37], the matrix (B(ν))^{−1} exists, if ν ∈ ℕ − 1 + m, and p^{∼}(B(ν)^{−1}) ≤ (ρ_{1} − 2ε_{0})^{−1}, (145) where ν ∈ ℕ − 1 + m. Cosequently, there exists B^{−1} ∈ E_{m,∞} and (p^{∼})_{m,∞} (B^{−1}) ≤ (ρ_{1} − 2ε_{0})^{−1}. (146) Making use of (129) - (134) with ρ = (ρ_{1} − 2ε_{0})^{−1}, λ = ρ_{2} + ε_{0}, a = r_{1,m}^{#}v_{C,k_{θ},ρ_{θ}}, (147) if 2 ≤ θ ≤ s, and ρ = (ρ_{1} − 2ε_{0})^{−1}, λ = ρ_{2} + ε_{0}, a = w_{C,k^{∗}}, (148) if k^{∗} > 0, θ = s + 1, we see that on E_{m,a}(ℂ^{p}) acts a linear operator

− (B^{−1})^{∨} ∑_{k=0}^{∞}(▽ ∘ (B^{−1})^{∨})^{k} = (▽ − B^{∨})^{−1}, (149)

and (p_{m,a})^{∼}(▽ − B^{∨})^{−1}) ≤ ρ/(1 − ρλ) = 1/ε_{0}. (150) For any y ∈ E_{m}(ℂ) and n ∈ ℕ let Y_{n,y} and Y_{n,y}^{#} denote the elements in the space E_{m}(ℂ^{n}), wich are determined respectively by means the following equalities:

Y_{n,y}(ν) = ( y(ν) ; ⋮ ; y(ν + n − 1)) , (151)

Y_{n,y}^{#}(ν) = ( 0 ; ⋮ ; 0 ; y(ν)) , (152)

where ν ∈ m − 1 + ℕ. Clearly, if (124) holds, then h_{m,a}(y) = h_{m,a}(Y_{n,y}^{#}) ≤ h_{m,a}(Y_{n,y}) ≤ (153)

max(1, λ^{n−1})h_{m,a}(y),

where y ∈ E_{m,a}(ℂ); terefore the maps y → Y_{n,y}^{#}, (154)

y → Y_{n,y}, (155) where

y ∈ E_{m,a}(ℂ)

are isomorphisms of the Banach space E_{m,a}(ℂ) onto closed subspaces of the Banach space E_{m,a}(ℂ^{n}) and the functions

y → h_{m,a}(Y_{n,y}^{#})

and

y → h_{m,a}(Y_{n,y}),

where y ∈ E_{m,a}(ℂ), are the equivalent to h_{m,a}(y) norms on the space E_{m,a}(ℂ). For any X ∈ ℂ^{p} let π(X) denotes the first coordiate of the column X, and we denote by π the map of ℂ^{p} on ℂ, which turns each X ∈ ℂ^{p} into π(X). The equation (43) is equivalent to the equation ((▽ − B^{∨})U)(ν) = Y_{p,r}^{#}(ν), ν ∈ m − 1 + ℕ : (156) the column U(ν), ν ∈ m − 1 + ℕ is a solution of the equation (156) if and only if, when y(ν) = (π(U))(ν) with ν ∈ m − 1 + ℕ is a solution of the equation (43). Let a is defined by means (147) – (148), and let ξ is the map of E_{m,a}(ℂ) into E_{m}(ℂ) such that ξ(z)(ν) = π ((▽ − B^{∨})^{−1}Y_{p,z}^{#}) (ν), (157) for all the z ∈ E_{m,a}(ℂ) and ν ∈ m − 1 + ℕ. In view of (147) – (149), (153) and (157), if y = ξ(z), for some z ∈ E_{m,a}(ℂ) then h_{m,a}(y) = h_{m,a}(ξ(z)) = h_{m,a} (π ((▽ − B^{∨})^{−1}Y_{p,z}^{#})) . (158) According to the Lemma 8 in the paper [37], and in view of (150), h_{m,a} (π ((▽ − B^{∨})^{−1}Y_{p,z}^{#})) ≤ (159)

γ^{∗}(B^{∼}) (max(1, ε_{0}))^{p−1} p_{m,a} ((▽ − B^{∨})^{−1}Y_{p,z}^{#}) ≤

L.A.Gutnik, On the difference equations of the Poincaré type (Part 3) 30

γ^{∗}(B^{∼}) (max(1, ε_{0}))^{p−1} (p_{m,a})^{∼} ((▽ − B^{∨})^{−1}) p_{m,a} (Y_{p,z}^{#}) ≤
γ^{∗}(B^{∼}) (max(1, ε_{0}))^{p−1} (ε_{0})^{−1}p_{m,a}(z) ≤
(γ^{∗}(B^{∼}))^{2} (max ((ε_{0})^{−1}, ε_{0}))^{p−1} (ε_{0})^{−1}h_{m,a}(z).

In view of (158), (159), (153) , (147), (148), h_{m,a}(Y_{n,y}) ≤ max (1, (ρ_{2} + ε_{0})^{n−1}) h_{m,a}(y) ≤ (160)

max (1, (ρ_{2} + ε_{0})^{n−1}) (γ^{∗}(B^{∼}))^{2} (max ((ε_{0})^{−1}, ε_{0}))^{p−1} (ε_{0})^{−1}h_{m,a}(z).

We see that for ξ are fulfilled all the conditions reqired in the Lemma 4 for the splitting monomorphism ψ_{m}. Therefore, in view of (105), the role of the splitting monomorphism ψ_{m} of the Lemma 4 may play the restriction of the above homomorphism ξ on Ker(η). ■

§5. Exactness of the obtained results.

The simple example y(ν) = ν^{α} (161) shows that the results of Theorem 7 are exact. if k_{θ} = 1. We shall show that, if k_{θ} = n > 1, then the results of Theorem 7 are exact. We take γ ∈ ℝ⧹{0}, put y_{1}(x) = exp(γx^{1/n}), (162) where x > 0, and define α(x) from the equation (△^{n}(y_{1}))(x) = (γ/|γ|)^{n}α(x)y_{1}(x). (163) Let us consider the function f(t) = f_{x}(t) = y_{1}(x + t)/y_{1}(x), (164) where t ∈ [0, n]. Since (x + t)^{1/n} − x^{1/n} ≤ x^{1/n−1}, if x > 0, t ∈ [0, n], it follows that f_{x}(t) = 1 + O (x^{1/n−1}) , (165) if x > 0, t ∈ [0, n]. Clearly,

α(x) = (γ/|γ|)^{n}(△^{n}(f_{x}))|_{t=0}.

We take t_{0} = 0, and, making use of the Lagrange theorem, define t_{k} for k = 1, …, n from the equalities (△^{n+1−k} ((f_{x})^{(k−1)})) |_{t=t_{k−1}} = (166)

(△ (△^{n−k} ((f_{x}))^{(k−1)}))) |_{t=t_{k−1}} =

(△^{n−k} ((f_{x})^{(k)})) |_{t=t_{k}},

where 0 ≤ t_{k−1} < t_{k} < t_{k−1} + 1 ≤ n for k = 1, …, n. Then α(x) = (γ/|γ|)^{n}(△^{n}(f_{x}))|_{t=0} = (167)

(γ/|γ|)^{n}f_{x}^{(}n)(t_{n}) =
(γ/|γ|)^{n}(γx^{−1/n})^{n}(1 + O(1/x)(1 + O(x^{1/n−1})) =
x^{−1}|(γ|/n)^{n} + O(x^{1/n−2}),

when x → ∞. So, the Poincaré equation

(△^{n}(y))(ν) = (γ/|γ|)^{n}α(ν)y(ν),

with ν ∈ ℕ, and α(ν) = O(1/ν) has a solution

y = y_{1}(ν) exp(γν^{1/n})

Therefore in the case k_{θ} = n > 1 the results of Theorem 7 are exact.

The following result is proved in [38]:

Theorem . ([38], Theorem 2) Let is given the equation y(ν + 2) − 2y(ν + 1) + y(ν) = a(ν)y(ν), (168) and are fulfilled all the conditions 0 < a(ν) < 1, (169)

a(ν + 1) < a(ν), (170)

a(ν) = α^{2}/(4ν) + βν^{−3/2} + O(ν^{−2}), (171) where ν ∈ ℕ, α > 0, β ∈ ℝ.

Then the equation (168) have a solutions y_{1}(ν) and y_{2}(ν) such that

y_{1}(ν) = exp(αν^{1/2} + O(ln(ν)), y_{2}(ν) = exp(−αν^{1/2} + O(ln(ν)). Remark. Taking in (171) β = 0, we see that estimates of the Theorem 7

are exact for equations of second order with coefficients contained in the set of rational functions of ν.

References.

[1] O.Perron, Über die Poincaresche Differenzengleichumg, Journal für die reine und angewandte mathematik, 1910, 137, 6 – 64;

[2] A.O.Gelfond, Differenzenrechnung (in Russian), 1967, Nauka, Moscow.

[3] A.O.Gelfond and I.M.Kubenskaya, On the theorem of Perron in the theory of differrence equations (in Russian), IAN USSR, math. ser., 1953, 17, 2, 83 – 86.

[4] M.A.Evgrafov, New proof of the theorem of Perron (in Russian),IAN USSR, math. ser., 1953, 17, 2, 77 – 82.

[5] G.A.Frejman, On theorems of of Poincaré and Perron (in Russian), UMN, 1957, 12, 3 (75), 243 – 245.

[6] N.E.Nörlund, Differenzenrechnung, Berlin: Springer Verlag, 1924.

[7] I.M.Vinogradov, Foundtions of the Number Theory (in Russian), 1952, GIT-TL.

[8] CH.-J. de la Vallée Poussin, Course d’analyse infinitésimale, Russian translation by G.M.Fikhtengolts, GT-TI, 1933.

[9] L.A.Gutnik, On the decomposition of the difference operators of Poincaré type (in Russian), VINITI, Moscow, 1992, 2468 - 92, 1 – 55.

[10] , On the decomposition of the difference operators of Poincaré type in Banach algebras (in Russian), VINITI, Moscow, 1992, 3443 - 92, 1 – 36.

[11] , On the difference equations

of Poincaré type (in Russian),

VINITI, Moscow 1993, 443 -B93, 1 – 41.

[12] , On the difference equations of Poincaré type in normed algebras (in Russian), VINITI, Moscow, 1994, 668 -B94, 1 – 44.

[13] , On the decomposition of the difference equations of Poincaré type (in Russian), VINITI, Moscow, 1997, 2062 -B97, 1 – 41.

[14] , The difference equations

of Poincaré type with characteristic polynomial having roots equal to zero (in Russian), VINITI, Moscow, 1997, 2418 - 97, 1 – 20.

[15] , On the behavior of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 1997, 3384-B97, 1 - 41.

[16] , On the variability of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 1999, 361-B99, 1 – 9.

[17] , To the question of the variability of solutions of difference equations of Poincaré type (in Russian), VINITI, Moscow, 2000, 2416-B00, 1 – 22.

[18] , On linear forms with coefficients in ℕζ(1 + ℕ), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2000, 3, 1 – 13.

[19] , On the Irrationality of Some Quantities Containing ζ(3), Eleven papers translated from the Russian, American Mathematical Society, 1988, 140, 45 - 56.

[20] , Linear independence over ℚ of dilogarithms at rational points (in Russian), UMN, 37 (1982), 179-180; english transl. in Russ. Math. surveys 37 (1982), 176-177.

[21] , On a measure of the irrationality of dilogarithms at rational points (in Russian), VINITI, 1984, 4345-84, 1 – 74.

[22] , To the question of the smallness of some linear forms (in Russian), VINITI, 1993, 2413-B93, 1 – 94.

[23] , About linear forms, whose coefficients are logarithms of algebraic numbers (in Russian), VINITI, 1995, 135-B95, 1 – 149.

[24] , About systems of vectors, whose coordinates are linear combinations of logarithms of algebraic numbers with algebraic coefficients (in Russian), VINITI, 1994, 3122-B94, 1 – 158.

[25] , About linear forms, whose coefficients are ℚ-proportional to the number log 2, and the values of ζ(s) for integer s (in Russian), VINITI, 1996, 3258-B96, 1 – 70.

[26] , The lower estimate for some linear forms, coefficients of which are proportional to the values of ζ(s) for integer s (in Russian), VINITI, 1997, 3072-B97, 1 – 77.

[27] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 1), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2001, 15, 1 – 20.

[28] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 2), Max-Plank-Institut für Mathematik, Bonn, Preprint Series,2001, 104, 1 – 36.

[29] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version,part 3), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 57, 1 – 33.

[30] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 4), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 142, 1 – 27.

[31] , On the dimension of some linear spaces over finite extension of ℚ (part 3), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, 107, 1 – 37.

[32] , On the dimension of some linear spaces over ℚ (part 3), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 16, 1 – 45.

[33] ,On the difference equation of Poincaré type (Part 1).

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 52, 1 – 43.

[34] , On the dimension of some linear spaces over ℚ, (part 4)

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 73, 1 – 37.

[35] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 5), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 83, 1 – 13.

[36] , On linear forms with coefficients in ℕζ(1 + ℕ)

(the detailed version, part 6), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 99, 1 – 33.

[37] ,On the difference equation of Poincaré type (Part 2).

Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2003, 107, 1 – 25.

[38] , On the asymptotic behavior of solutions of difference equation (in English). Chebyshevskij sbornik, 2003, v.4, issue 2, 142 – 153.

[39] , On linear forms with coefficients in ℕζ(1 + ℕ), Bonner Mathematishe Schriften Nr. 360, Bonn, 2003.

Moscow State Institute of Electronics and Mathematics Bolshoi Trekhsvyatitelskij pereulok, 1-3/12, str.8 Moscow 109028 Russian Federation E-mail: gutnik@gutnik.mccme.ru