ON THE DIFFERENCE EQUATION OF THE POINCARÉ TYPE (Part 2)

L.A. Gutnik

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

Table of contents

§0. Foreword. §1. Begin of the proof of Theorem 6. §2. The general plan of the construction of the spaces V_{m}^{∨} and V_{m}^{∧}. §3. On some linear normed spaces of sequenses of elements of a linear normed space. §4. End of the proof of Theorem 6.

§0. Foreword.

Here I begin 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] - [36]). With this aim I prove here the following auxiliary Theorem 6 , which is proved in [14] as Theorem 1.

Theorem 6. Let us consider the following difference equation:

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (1)

with n ∈ ℕ, a_{k}(ν) ∈ ℂ for k = 0, …, n and ν ∈ ℕ − 1. Let

a_{k}^{∼} ∈ ℂ, a_{k}(ν) ∈ ℂ, a_{n}(ν) = 1, a_{k}(ν) − a_{k}^{∼} = O(1/(ν + 1)), (2)

where k = 0, …, n and ν ∈ ℕ − 1. Let further

q ∈ [1, n] ∩ ℤ, p = n − q, a_{q}^{∼} ≠ 0, (3)

T_{1}(z) = ∑_{k=0}^{p} a_{q+k}^{∼}z^{k} (4)

and suppose that the characteristic polynomial

T (z) = ∑_{k=0}^{n} a_{k}^{∼}z^{k} (5)

of the equation (1) satisfies the following equality:

T (z) = z^{q}T_{1}(z). (6)

For m ∈ ℕ − 1, let V_{m} denote the ℂ-linear space of solutions y = y(ν) of the equation

∑_{k=0}^{n} a_{k}(ν)y(ν + k) = 0, (7)

where ν ∈ m − 1 + ℕ, related to equation (1). Then there exist C > 0 and m ∈ ℕ such that V_{m} splits into direct sum V_{m}^{∧} ⊕ V_{m}^{∨} of two its subspaces V_{m}^{∧} and V_{m}^{∨}, which have the following propeties:

a)

V_{m}^{∧} = {y ∈ V_{m}: y(ν) = O(1)(C/ν)^{ν/q})}; (8)

b) if q = n, then

V_{m}^{∨} = {0}; (9)

c) if q < n, then V_{m}^{∨} coincides with the space of solutions of a difference equation of Poincaré type

∑_{k=0}^{p} b_{k}(ν)y(ν + k) = 0, (10)

where p = n − q, b_{k}(ν) ∈ ℂ for k = 0, …, p and ν ∈ m − 1 + ℕ,

b_{0}(ν) ≠ 0, b_{p}(ν) = 1, (11)

for ν ∈ m − 1 + ℕ,

b_{k}(ν) − a_{q+k}^{∼} = O(1/ν), (12)

where k = 0, …, p and ν ∈ m − 1 + ℕ.

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. Begin of the proof of Theorem 6.

Lemma 1. Let C_{0} ≥ 1, m ∈ ℕ − 1, m_{0} = [nC_{0}] + m and for the coefficients of the equation (7) the following inequality holds:

|a_{k}(ν)| ≤ \frac{C_{0}}{ν + 1}, (13)

where k = 0, …, n − 1 and ν ∈ m − 1 + ℕ. Let further C > enC_{0}. Then for any solution y(ν) of the equation (7) the following inequality holds:

y(ν) = O(1)(C/ν)^{ν/n}, (14)

where ν ∈ m − 1 + ℕ.

Proof. If for some ν_{0} ∈ m_{0} − 1 + ℕ and all the k = 0, …, n − 1 the following inequality holds,

|y(ν_{0} + k)| ≤ γ,

then the inequality ν_{0} > nC_{0} implies that the inequality |y(ν)| ≤ γ will be fulfilled for all the ν ∈ ν_{0} − 1 + ℕ. Therefore, if

|y(m_{0} + k)| ≤ γ_{0},

where k = 0, …, n − 1 then, in view of (13),

|y(ν)| ≤ γ_{0}

for ν ∈ m_{0} − 1 + ℕ,

|y(ν)| ≤ γ_{0}C_{0}n/m_{0}

for ν ∈ m_{0} − 1 + n + ℕ,

|y(ν)| ≤ γ_{0}(C_{0}n)^{2}/(m_{0}(m_{0} + n))

for ν ∈ m_{0} − 1 + 2n + ℕ,

|y(ν)| ≤ γ_{0}(C_{0})^{κ}/(m_{0}/n)_{κ} =

γ_{0}(C_{0})^{κ} \frac{Γ(m_{0}/n)}{Γ(m_{0}/n + κ)}

for ν ∈ m_{0} − 1 + nκ + ℕ. But κ = ν/n + O(1); therefore the equality (14) follows from the Stirling’s formula. ■

Together with Lemma 1 we have proved the Theorem 6 for the case q = n. The following result was proved in [33]

L.A.Gutnik, On the difference equations of the Poincaré type (Part 2) 4

Theorem 5. Let the functin ξ(x) is defined on [0, +∞), let ξ(x) decreases together with increasing of the variable x in [0, +∞), let lim_{x→∞}(ξ(x)) = 0 and let ξ(x) > 0 for x ∈ [0, +∞). Let ξ(x/2) = O(ξ(x)), (15) when x → ∞,

lim_{x→0}(log(ξ(x)))/x = 0. (16)

Let a_{k}(ν) − a_{k}^{∼} = O(ξ(ν)), k = 0, …, n, when ν → ∞. Let further the characteristical polynomial (5) of the equation (1) may be represented in the form T (z) = T_{1}(z)T_{2}(z), (17) where

T_{1}(z) = ∑_{α=0}^{p} b_{α}^{∼}z^{α}, T_{2}(z) = ∑_{β=0}^{q} u_{β}^{∼}z^{β}, b_{p}^{∼} = u_{q}^{∼} = a_{n}^{∼} = 1 (18)

and absolute value of each root of T_{1}(z) is greater than the absolute value of each root of T_{2}(z).

Then there exist m ∈ ℕ,

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

and

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

such that b_{α}(ν) − b_{α}^{∼} = O(ξ(ν)), α = 0, …, p, b_{p}(ν) = 1, b_{0}(ν) ≠ 0, (19)

u_{β}(ν) − u_{β}^{∼} = O(ξ(ν)), β = 0, …, q, u_{q}(ν) = 1, (20) where ν ∈ ℕ + m − 1, and, moreover, the connected with the equation (1) the equation (7) is equivqlent to the equation

∑_{α=0}^{p} b_{α}(ν)y(ν + α) = r(ν), (21)

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

∑_{β=0}^{q} u_{β}(ν)r(ν + β) = 0 (22)

with ν ∈ ℕ − 1 + m.

Lemma 2. Let the conditions of the Theorem 6 are fulfilled and q < n. Then there exist m ∈ ℕ,

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

and

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

such that

b_{α}(ν) − a_{q+α}^{∼} = O(1/ν), α = 0, …, p, b_{p}(ν) = 1, b_{0}(ν) ≠ 0, (23)

u_{β}(ν) = O(1/ν), β = 0, …, q, u_{q}(ν) = 1, (24)

where ν ∈ ℕ + m − 1, and, moreover, the connected with the equation (1) the equation (7) is equivalent to the equation (21) where ν ∈ ℕ − 1 + m and r(ν) satisfies to the equation (22) with ν ∈ ℕ − 1 + m.

Proof The Lemma is direct corollary of the Theorem 5. with T_{2}(z) = z^{q} and ξ(x) = 1/(x + 1). ■

§2. The general plan of the construction of the spaces V_{m}^{∨} and V_{m}^{∧}.

First we take on the role of m in the Theorem 6 the m of the Lemma 2. Let R_{m} be the linear over ℂ space of all the solutions of the equations (22). According the Lemma 1, there exists C > 0, such that

r(ν) = O(1)(C/ν)^{ν/n} (25)

for r(ν) ∈ R_{m} and ν ∈ m − 1 + ℕ. The connected with the equation (21) map

y(ν) → r(ν) = ∑_{α=0}^{p} b_{α}(ν)y(ν + α) (26)

is a ℂ-linear map of the space V_{m} onto R_{m} and the null-space V_{m}^{∨} of this map is a ℂ-linear subspace of V_{m}, which coincides with the space of solutions of the equation (10). The Theorem 6 will be proved, if after replacement of m in the Lemma 2 by the bigger m ∈ ℕ we will constructed a splitting monomorphism ξ_{m} of the space R_{m} into V_{m} with the property:

y(ν) = O(1)(C/ν)^{ν/n}, (27)

for y(ν) ∈ V_{m}^{∧} = ξ_{m}(R_{m}) and ν ∈ m − 1 + ℕ.

§3. On some linear normed spaces of sequenses of elements of a linear normed space.

Let K be one of the fields ℝ or ℂ and 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}|}), (28)

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}) , (29)

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 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)).

Clearly, if L is a K − algebra, then each a ∈ E_{m}(L) determines an acting on 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 E_{m}(L) acts also K-linear operator ▽𝔐^{∨}(𝔏), which turns any 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}, (30)

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}, (31)

where a_{k} ∈ E_{m}(L) for k = 0, …, deg(α) and a_{deg(α)} ≠ 0_{E_{m}(L)}. Clearly, (31) may be rewritn in the form

α = ∑_{k=0}^{∞} μ_{a_{k}} ∘ ▽^{k}, (32)

where a_{k} = 0_{E_{m}(L)} for k ∈ deg(α) + ℕ. It follows from (30) that 𝔄_{m}(L) is a graduated algebra, and if

β = ∑_{r=0}^{p} μ_{b_{r}} ∘ ▽^{r} ∈ 𝔄_{m}(L), (33)

γ = ∑_{s=0}^{q} μ_{c_{s}} ∘ ▽^{s} ∈ 𝔄_{m}(L), (34)

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}; (35)

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 (30), 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 (31) 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

(36) 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]. (37)

If α = 0_{𝔄_{m}(L)}, then we put

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

The equality (30) shows that the map

α → P (α, z) (38)

is an epimorphism of the algebra 𝔄\overrightarrow{m}(L) on on the algebra L[z]. We note that, if α ∈ 𝔄_{m}(ℂ), then Ker(α) coincides with the linear space of all the solutions of the equation (7); moreover if α ∈ 𝔄\overrightarrow{m}(ℂ), then the corresponding to α equation (7) 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 the α ∈ 𝔄_{m}(L)⧹{0_{𝔄_{m}(L)}}, which have the representation (31) with a_{deg(α)} ∈ 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 (31) 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 (31) 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}, (39)

then

r_{m,μ}^{▷}(α) = ∑_{k=0}^{n} μ_{r_{m,μ}(a_{k})} ∘ ▽^{k}, (40)

where the operator ▽ in (39) acts in E_{m}(L) and the operator ▽ in (40) 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}, (41) 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 3. 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. For m = μ we have equality Ker(r_{m,μ}) = 0_{E_{m}(L)} and assertion of the Lemma is obvious. Let μ > m. Clearly,

Ker(r_{m,μ}) = {x ∈ E_{m}(L): x(ν) = 0_{L}, ν ∈ μ − ℕ − 1}.

Let α ∈ 𝔄_{m}(L)^{∧} has the form (39) with a_{0}(ν) ∈ L^{∗}. If x ∈ Ker(r_{m,μ}), and

y = α(x), (42)

then

y(ν) = ∑_{k=0}^{n} a_{k}(ν)x(ν + k) (43)

and therefore y(ν) = 0_{L} for ν ∈ μ − 1 + ℕ, if x(ν) = 0_{L} for ν ∈ μ − 1 + ℕ. On the other hand for given y ∈ Ker(r_{m,μ}) the coordinates x(ν) in (42) must be equal to 0_{L}, if ν ∈ μ − 1 + ℕ, and the equalities

(44)x(μ − j) = (a_{0})(μ − j))^{−1} (y(μ − j) − ∑_{k=1}^{n} a_{k}(μ − j)x(μ − j + k)) .

determined successesvely and in the unique way the coordinatess x(μ − j) for j ∈ [1, μ − m] ⋂ ℤ. ■

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. Since r_{m,μ} is an epimorphism of E_{m}(L) onto E_{μ}(L), it follows that there exists z ∈ E_{m}(L) such that r_{m,μ}(z) = x In view of (41),

r_{m,μ}(α(z)) = (r_{m,μ}^{▷}(α))(r_{m,μ}(z)) = (r_{m,μ}^{▷}(α))(x) = r_{m,μ}(g).

Then g − α(z) ∈ Ker(r_{m,μ}). According to the Lemma 3, Ker(r_{m,μ}) contains an element u such that g − α(z) = α(u). Let y = z + u. Then

α(y) = g, r_{m,μ}(y) = r_{m,μ}(z) = x.

If α(y) = r_{m,μ}(y) = 0_{E_{m}(L)}, then y ∈ Ker(r_{m,μ}), and the Lemma 3 implies the equality y = 0_{E_{m}(L)}. ■

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. Let x ∈ Ker(r_{m,μ}^{▷}(α)). Clearly, the conditions of the Corollary 1 are fulfilled for g = 0_{E_{m}(L)} and x. Therefore there exist a unique y ∈ Ker(α) such that r_{m,μ}(y) = x. ■

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

a_{k} = (a_{k}(1), a_{k}(2), …, a_{k}(ν), …) ∈ E\overrightarrow{1}(ℂ), (45)

where k = 0, …, n. Moreover a_{n} = 1_{E_{1}(ℂ)}, for α in (39) Ker(α) coincides with the linear over ℂ space of all the solutions of the equation (1), polynomial (5) is equal to P (α, z) = P (r_{0,m}^{▷}(α), z), where m ∈ ℕ, and Ker(r_{1,m}^{▷}(α)) coincides with linear over ℂ space V_{m} of all the solutions of the equation (7).

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)^{≻} and let

ℑ_{m}^{≻}(L) = {α ∈ 𝔄_{m}^{≻}(L): P (α, z) = 0}.

The Lemma 2 may be reformulated now as follows:

Lemma 4. Let α ∈ 𝔄_{0}^{≻}(ℂ) ⋂ 𝔄_{0}^{∨}(ℂ), and P (α, z) coincides with the polynomial T (z) in (5) and (6).

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

r_{0,m}^{▷}(α) = ψβ (46)

such that

ψ ∈ 𝔄_{m}^{≻}(ℂ) ⋂ 𝔄_{m}^{∨}(ℂ), ψ − ▽^{q} ∈ ℑ_{m}^{≻}(ℂ), (47)

β ∈ 𝔄_{m}^{≻}(ℂ) ⋂ 𝔄_{m}^{∨∧}(ℂ), deg(β) = p = n − q, (48)

and P (β, z) coincides with the polynomial T_{1}(z) in (6).

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

w_{C,n}(ν) = (\frac{C}{ν + 1})^{ν/n} , (49)

where ν ∈ ℕ − 1. The Lemma 1 admits the following reformulation:

Lemma 5. Let m ∈ ℕ and α from (31) belongs to 𝔄_{m}^{∨}(ℂ) and

deg(α) = n, a_{k} ∈ r_{m}(𝔳)E_{m,∞},

where k = 0, …, n − 1. Let further

C_{0} ≥ 1, q_{∞}((r_{m}(𝔳))^{−1})a_{n−k} ≤ C_{0},

where k = 0, …, n − 1. Let

C > enC_{0}. (50)

Then Ker(α) ⊂ r_{0,m}(w_{C,n})E_{m,∞}(ℂ) ⊂ E_{m,0}(ℂ).

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^{−1})(ν),

where ν ∈ m − 1 + ℕ. Clearly, (A^{∨})^{−1} = (A^{−1})^{∨}. For λ > 0 let T_{m,λ}(L) denotes the element of E_{m}(𝔐^{∨}(L)), for which ((T_{m,λ}(L)(ν))y)(ν) = λ^{ν}y(ν), where ν ∈ m − 1 + ℕ. Clearly,

(T_{m,λ}(𝔐^{∨}(L))A)^{∨} = (T_{m,λ}(L))^{∨}A^{∨} = A^{∨}(T_{m,λ}(L))^{∨}, (51)

T_{m,1}(L))^{∨} = 1_{𝔐^{∨}(E_{m}(L))}, T_{m,λ_{1}λ_{2}}(L))^{∨} = (T_{m,λ_{1}}(L))^{∨})(T_{m,λ_{2}}(L))^{∨}),

where A ∈ E_{m}(𝔐^{∨}(L)), λ > 0, λ_{1} > 0 and λ_{2} > 0. Let

E_{m,λ}(L) = {y ∈ E_{m}(L): p_{m,λ}(y) = p_{m,∞}((T_{m,λ}(L))^{−1}y) < +∞}.

Then (E_{m,λ}(L), p_{m,λ}) is a Banach space, and

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

Clearly, the map T_{m,λ}(L) is an isometry of E_{m,1}(L) = E_{m,∞}(L) onto E_{m,λ}(L), and the map T_{m,λ}(L)▽^{m} is an isometry of (E_{0,1}(L), p_{0,1}) = (E_{0,∞}(L), p_{0,∞}) onto (E_{m,λ}(L), p_{m,λ}).

Lemma 6. Let m ∈ ℕ − 1 and α from (31) belongs to 𝔄\overrightarrow{m}(ℂ) Then α is bounded linear operator on the space E_{m,λ}(ℂ) and

p_{m,λ}^{∼}(α) ≤ ∑_{k=0}^{deg(α)} p_{m,1}(a_{k})λ^{k} (52)

Proof. Since α ∈ 𝔄\overrightarrow{m}(ℂ), it follows that a_{k} ∈ E_{m,1}(ℂ) for k = 0, …, n In view of (31), if y ∈ E_{m,λ}(ℂ), then

(α(y))(ν) = ∑_{k=0}^{deg(α)} a_{k}(ν)λ^{ν+k}λ^{−ν−k}y(ν + k) =

λ^{ν} ∑_{k=0}^{deg(α)} a_{k}(ν)λ^{k}λ^{−ν−k}y(ν + k)

and

| ∑_{k=0}^{deg(α)} a_{k}(ν)λ^{k}λ^{−ν−k}y(ν + k)| ≤ (53)

p_{m,λ}(y) ∑_{k=0}^{deg(α)} p_{m,1}(a_{k})λ^{k}.

The inequality (52) follows from (53). ■

Lemma 7. If λ > 0, θ > 0,

A ∈ E_{m,θ/λ}(𝔐(L)), (54)

then A^{∨} turns E_{m,λ}(L) in E_{m,θ}(L), and

(p^{∼})_{m,θ/λ}(A) = sup({p_{m,θ}(A^{∨}y): y ∈ E_{m,λ}(L), p_{m,λ}(y) ≤ 1}).

Proof. Let y ∈ E_{m,λ}(L) and z = (T_{m,λ}(L))^{−1}y. Then

z ∈ E_{m,1}(L), p_{m,λ}(y) = p_{m,1}(z).

Let B = (T_{m,θ/λ}(𝔐(L)))^{−1}A. Then

B ∈ E_{m,1}(𝔐(L)), (p^{∼})_{m,θ/λ}(A) = (p^{∼})_{m,1}(B).

Therefore, in view of (51),

B^{∨}z ∈ E_{m,1}(L), T_{m,θ}(L)(B^{∨}z) ∈ E_{m,θ}(L),

L.A.Gutnik, On the difference equations of the Poincaré type (Part 2) 14

A^{∨}y = A^{∨}T_{m,λ}z = A^{∨}T_{m,θ}(L))T_{m,λ/θ}(L)z =
T_{m,θ}(L))A^{∨}T_{m,λ/θ}(L)z =
T_{m,θ}(L)(T_{m,λ/θ}(𝔐(L))A)^{∨}z =
T_{m,θ}(L))((T_{m,θ/λ}(𝔐(L)))^{−1}A)^{∨}z ∈ E_{m,θ}(L).

Further we have

(p^{∼})_{m,θ/λ}(A) = sup({p^{∼}((θ/λ)^{−ν}A(ν))): ν ∈ m − 1 + ℕ}) = sup({p((θ/λ)^{−ν}A(ν)z(ν))): z(ν) ∈ L, p(z(ν)) ≤ 1, ν ∈ m − 1 + ℕ}) = sup({p((θ^{−ν}A(ν)y(ν))): y(ν) ∈ L, p(y(ν)) ≤ λ^{ν}, ν ∈ m − 1 + ℕ}) = sup({p((θ^{−ν}((A^{∨}y)(ν))): y(ν) ∈ L, λ^{−ν}p(y(ν)) ≤ 1, ν ∈ m − 1 + ℕ}) =

sup({p_{m,θ}(A^{∨}y): y(ν) ∈ E_{m,λ}(L), p_{m,λ}(y)) ≤ 1}).

■

Corollary. If A ∈ E_{m,1}(𝔐(L)), then A^{∨} turns E_{m,λ}(L) in E_{m,λ}(L), and (p^{∼})_{m,1}(A) = (55) sup({p_{m,λ}(A^{∨}y): y ∈ E_{m,λ}(L), p_{m,λ}(y)) ≤ 1}) = (p_{m,λ})^{∼}(A^{∨}) = (p_{m,1}^{∼})(A^{∨}).

Proof. The assertion of the Corollary follows directly from the assertion of the Lemma for θ = λ. ■

Clearly, if λ > 0, θ > 0, A ∈ E_{m,λ}(𝔐(L)), B ∈ E_{m,θ}(𝔐(L)), then AB is contained in E_{m,λθ}(𝔐(L)). Clearly, ▽ maps E_{m,λ}(L) in E_{m,λ}(L) and (p_{m,λ})^{∼}(▽) = λ. (56) Clearly, for any k ∈ ℕ − 1, A ∈ E_{m}(𝔐(L))

(▽ ∘ A^{∨})^{k} = (∏_{κ=1}^{k}(▽^{κ}A)) ∘ ▽^{κ}. (57)

Let L is a Banach space over th field K and A ∈ E_{m,1}(𝔐^{∧}(L)). Let further there exists A^{−1} ∈ E_{m,1}(𝔐^{∧}(L)), and (p^{∼})_{m,1}(A^{−1}) = ρ < 1/λ. (58) Then, clearly, 𝔐^{∧}(E_{m,λ}(L)) contains the linear operator

− (A^{−1})^{∨} ∑_{k=0}^{∞}(▽ ∘ (A^{−1})^{∨})^{k} = (59)

−(A^{−1})^{∨}(1_{𝔐^{∧}(E_{m,λ}(L))} − ▽ ∘ A^{−1})^{∨})^{−1} = (▽ − A^{∨})^{−1},

L.A.Gutnik, On the difference equations of the Poincaré type (Part 2) 15

and in view of (56) and (59), (p_{m,1})(▽ − A^{∨})^{−1}) ≤ ρ/(1 − ρλ). (60) According to (57), the equality (59) may be rewritten in the form

(▽ − A^{∨})^{−1} = −(A^{−1})^{∨} ∑_{k=0}^{∞} (∏_{κ=1}^{k}(▽^{κ}(A^{−1}))^{∨}) ∘ ▽^{k}. (61)

Lemma 8. ([21], Lemma 2, [15], Lemma 2) Let A ∈ Mat_{n}(ℂ) an let k is a maximal order of its Jordan blocks. Then there exists a constante γ^{∗}(A) > 0 with the following properties:

for any ε > 0 there exists a norm p_{A,ε} on ℂ^{n} such that p_{A,ε} ≤ γ^{∗}(A)(max(1, 1/ε)^{k−1}h, (62)

h ≤ γ^{∗}(A)(max(1, ε)^{k−1}p_{A,ε}, (63)

(p_{A,ε})^{∼} ≤ (γ^{∗}(A))^{2}(max(ε, 1/ε)^{k−1}h^{∼}, (64)

h^{∼} ≤ (γ^{∗}(A))^{2}(max(ε, 1/ε)^{k−1}(p_{A,ε})^{∼}, (65)

∥A∥_{sp} ≤ (p_{A,ε})^{∼} ≤ ∥A∥_{sp} + (sign(k − 1))ε, (66) where ∥A∥_{sp} denotes the maximum of the absolute values of eigenvalues of the matrix A. If, moreover, det(A) ≠ 0, ∥A^{−1}∥_{sp}^{−1} > (sign(k − 1))ε, (67) then ∥A^{−1}∥_{sp} ≤ (p_{A,ε})^{∼}(A^{−1}) ≤ (∥A^{−1}∥_{sp}^{−1} − (sign(k − 1))ε)^{−1} (68)

Proof. Let C ∈ Mat_{n}(ℂ), det(C) ≠ 0 and J = C^{−1}AC (69) is a Jordan form of A. Let J is composed by s Jordan k_{i} × k_{i}-blocks J_{i}, where i = 1, …, s and ∑_{i=1}^{s} k_{i} = n. Let ε > 0, and let T_{m,ε}^{∧} denotes the diagonal m × m−matrix, which i − th diagonal element is equal to ε^{i−1},

L.A.Gutnik, On the difference equations of the Poincaré type (Part 2) 16

where i = 1, …, m. Let further T_{ε}^{∨} denotes the n × n−diagonal matrix composed by the blocks T_{k_{i},ε}^{∧}, where i = 1, …, s. Let γ^{∗}(A) = max(h^{∼}(C^{−1}), h(C)), (70)

p_{A,ε}(X) = h((CT_{ε}^{∨})^{−1}X), (71) where X ∈ ℂ^{n}. Then

(72) p_{A,ε}(X) ≤ h^{∼}(C)h^{∼}((T_{ε}^{∨})^{−1})h(X) ≤ γ^{∗}(A) max(1, (1/ε)^{k−1})h(X)

for X ∈ ℂ^{n}; therefore (62) holds. Clearly, h(X) = h(CT_{ε}^{∨}(CT_{ε}^{∨})^{−1}X) ≤ h(C)h(T_{ε}^{∨})h(CT_{ε}^{∨})^{−1}X) ≤ (73)

γ^{∗}(A) max(1, ε^{k−1})p_{A,ε}(X)

for X ∈ ℂ^{n}; therefore (63) holds. In view of 71, (p_{A,ε})^{∼}(B) = sup({p_{A,ε}(BX): X ∈ ℂ^{n}, p_{A,ε}(X) ≤ 1}) = (74)

sup({h((CT_{ε}^{∨})^{−1}BX): X ∈ ℂ^{n}, h((CT_{ε}^{∨})^{−1}X) ≤ 1}) =
sup({h((CT_{ε}^{∨})^{−1}BCT_{ε}^{∨}Y): Y ∈ ℂ^{n}, h(Y) ≤ 1}) =
h^{∼}(CT_{ε}^{∨})^{−1}BCT_{ε}^{∨}),

where B ∈ Mat_{n}(ℂ). The equalities (74) imply (64) and (66). It follows from the equalities (74) that h^{∼}(B) = (p_{A,ε})^{∼}(CT_{ε}^{∨}B(CT_{ε}^{∨})^{−1}), (75) where B ∈ Mat_{n}(ℂ). The equality (75) implies (65). Let det(A) ≠ 0, and let Λ is the diagonal n × n−matrix, which diagonal elements are equal to the corresponding diagonal elements of the matrix J. If (67) holds, then (T_{ε}^{∨})^{−1})JT_{ε}^{∨} = Λ(E − N), (76) where E is the unit n × n-matrix, N is a nilpotent n × n-matrix and

h^{∼}(N) ≤ ∥A^{−1}∥_{sp}(sign(k − 1))ε, h^{∼}(Λ^{−1}) = ∥A^{−1}∥_{sp},

(T_{ε}^{∨})^{−1})J^{−1}T_{ε}^{∨} = (E − N)^{−1}Λ^{−1}, (p_{A,ε})^{∼}(A^{−1}) =

h^{∼}(CT_{ε}^{∨})^{−1}A^{−1}CT_{ε}^{∨}) = h^{∼}((T_{ε}^{∨})^{−1}J^{−1}T_{ε}^{∨}) ≤

∥A^{−1}∥_{sp} ∑_{κ=0}^{∞}(∥A^{−1}∥_{sp}sign(k − 1))ε)^{κ} =

L.A.Gutnik, On the difference equations of the Poincaré type (Part 2) 17

∥A^{−1}∥_{sp}(1 − ∥A^{−1}∥_{sp}(sign(k − 1))ε)^{−1} = ((∥A^{−1}∥_{sp})^{−1} − (sign(k − 1))ε)^{−1}. ■

Corollary. If all the eigenvalues of the matrix A are symple, then (p_{A,ε})^{∼} = ∥A∥_{sp}. (77) If, moreover, det(A) ≠ 0, (78) then (p_{A,ε})^{∼}(A^{−1}) = (∥A^{−1}∥_{sp})^{−1} . (79)

Proof. Since in this case k = 1, and, consequently, (78) implies (67), it follows that the assertion of the Lemma follows directly from (66) - (68). ■

Lemma 9. ([15], Lemma 2). Let are fulfilled all the conditions of the Lemma 8 and let B ∈ Mat_{n}(ℂ), ε_{1} > 0, (p_{A,ε})^{∼}(B − A) ≤ ε_{1}, (80) then (p_{A,ε})^{∼}(B) ≤ ∥A∥_{sp} + (sign(k − 1))ε + ε_{1}. (81) If, moreover, the inequalities (67) hold and ∥A^{−1}∥_{sp}^{−1} > (sign(k − 1))ε + ε_{1}, (82) then

(83) det(B) ≠ 0, (p_{A,ε})^{∼}(B^{−1}) ≤ (∥A^{−1}∥_{sp}^{−1} − (sign(k − 1))ε − ε_{1})^{−1}

Proof. The inequality (81) follows directly from (66) and (80). If, moreover, all the inequalities (67) and (82) hold, then let us to represent B in the form B = A(E − A^{−1}(A − B)); (84) in view of (68), (80) and (82),

(p_{A,ε})^{∼}(A^{−1}(A − B)) ≤ (∥A^{−1}∥_{sp}^{−1} − (sign(k − 1))ε)^{−1} ε_{1} < 1; therefore the matrices (E − A^{−1}(A − B))^{−1}, B^{−1} = (E − A^{−1}(A − B))^{−1}A^{−1} (85) exist and

(p_{A,ε})^{∼}(B^{−1}) = (p_{A,ε})^{∼}((E − A^{−1}(A − B))^{−1}A^{−1}) ≤

(p_{A,ε})^{∼}((E − A^{−1}(A − B))^{−1})(p_{A,ε})^{∼}(A^{−1}) ≤

(1 − \frac{ε_{1}}{(∥A^{−1}∥_{sp})^{−1} − (sign(k − 1))ε})^{−1} ×

((∥A^{−1}∥_{sp})^{−1} − (sign(k − 1))ε)^{−1} =

((∥A^{−1}∥_{sp})^{−1} − (sign(k − 1))ε − ε_{1})^{−1} .

■

Corollary 1. Let are fulfilled all the conditions of the Lemma 9, and all the eigenvalues of the matrix A are symple. Then (p_{A,ε})^{∼}(B) ≤ ∥A∥_{sp} + ε_{1}. (86) If, moreover, det(A) ≠ 0, ∥A^{−1}∥_{sp}^{−1} > ε_{1}, (87) then det(B) ≠ 0, (p_{A,ε})^{∼}(B^{−1}) ≤ (∥A^{−1}∥_{sp}^{−1} − ε_{1})^{−1} . (88)

Proof. Since in this case k = 1, and, consequently, (87) implies (82), it follows that the assertions of the Lemma follows directly from (81) - (83). ■

Corollary 2. ([14], Lemma 3). Let are fulfilled all the conditions of the Lemma 8, det(A) ≠ 0, 0 < ε < (∥A^{−1}∥_{sp})^{−1} /2 (89) and let B ∈ Mat_{n}(ℂ), (p_{A,ε})^{∼}(B − A) ≤ ε, (90) then (p_{A,ε})^{∼}(B) ≤ ∥A∥_{sp} + 2ε, (91) the matrix B^{−1} exists and (p_{A,ε})^{∼}(B^{−1}) ≤ (∥A^{−1}∥_{sp}^{−1} − 2ε)^{−1} . (92) Proof. Let us take ε_{1} = ε. Then (82) follows from (88) and (89). ■

§4. End of the proof of Theorem 6.

Let in accordance with (10)

▽^{p} + ∑_{k=0}^{p−1} μ_{b_{k}} ∘ ▽^{k}, (93)

where b_{k} ∈ ℂ + (r_{0,m}𝔳)E_{m,1}ℂ for k = 0, …, p − 1. In view of (23),

lim(b_{k}) = a_{q+k} (94)

where k = 0, …, p − 1. Let

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 \\ −a_{q}^{∼} & −a_{q+1}^{∼} & −a_{q+2}^{∼} & … & −a_{q+p−1}^{∼} \end{matrix})

We take now on the role of the matrix A in the Lemma 8 and Lemma 9 the matrix B^{∼}. Since, in view of (3) a_{q}^{∼} ≠ 0, it follows that (B^{∼})^{−1} exists. We take now on the role ε in the Lemmata 8 and 9 and their corollaries the number

ε_{0} = (∥(B^{∼})^{−1}∥_{sp})^{−1} /3, (95)

and we take

𝔮 = p_{B,ε_{0}}.

Since lim(B) = B^{∼}, it follows that we can (making use the operator r_{m,μ}) replace the number m on some bigger m, such that for C from (49) and (50) the inequality

m ≥ C max (1, (6 ∥(B^{∼})^{−1}∥_{sp})^{q}) , (96) holds and 𝔮(B(ν) − B^{∼}) ≤ ε_{0}, (97) where ν ∈ m − 1 + ℕ. It follows from (95) and (97) that for B^{∼} and B(ν) with ν ∈ m − 1 + ℕ are fulfilled all the conditions of the Corrollary 2 of the Lemma 9; therefore there exists (B(ν))^{−1} for ν ∈ m − 1 + ℕ and

𝔮^{∼}((B(ν))^{−1}) ≤ (3ε_{0} − 2ε_{0})^{−1} = 3 ∥(B^{∼})^{−1}∥_{sp} .

Consequenty, there exists B^{−1} ∈ E_{m,1}(Mat_{p}(ℂ)) and (𝔮^{∼})_{m,1}(B^{−1}) ≤ 3 ∥(B^{∼})^{−1}∥_{sp} . (98) In view of (96) and (49),

(w_{C,q}(ν))^{1/ν} = (\frac{C}{ν + 1})^{1/q} < (99)

min (1, (∥(B^{∼})^{−1}∥_{sp})^{−1} /6) = min (1, ε_{0}/2) ,

where ν ∈ m − 1 + ℕ. In accordance with (58)-(61), (96),(98) and (99), if

3 ∥(B^{∼})^{−1}∥_{sp} ≤ \frac{1}{2λ}, (100)

then

ρ = (q^{∼})_{m,1}(B^{−1}) ≤ 3 ∥(B^{∼})^{−1}∥_{sp} ≤ \frac{1}{2λ} < \frac{1}{λ}, (101)

the algebra 𝔐^{∧}(E_{m,λ}(ℂ^{p})) contains the linear operator

− ((B^{−1}))^{∨} ∑_{k=0}^{∞}(▽ ∘ (B^{−1})^{∨})^{k} = (102)

−(B^{−1})^{∨}(1_{𝔐(E_{m,λ}(ℂ))} − ▽ ∘ B^{−1})^{∨})^{−1} = (▽ − B^{∨})^{−1},

and, in view of the Lemma 7, its corollary, Lemma 6, (98),(60) (𝔮_{m,λ})^{∼}((▽ − B^{∨})^{−1}) ≤ ρ/(1 − ρλ) ≤ (103)

3 ∥(B^{∼})^{−1}∥_{sp} /(1 − 3 ∥(B^{∼})^{−1}∥_{sp} λ) ≤ 6 ∥(B^{∼})^{−1}∥_{sp} .

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)) , (104)

Y_{n,y}^{#}(ν) = ( 0 ; ⋮ ; 0 ; y(ν)) , (105)

where ν ∈ m − 1 + ℕ. Clearly, p_{λ}(y) = h_{λ}(y) = p_{λ}(Y_{n,y}^{#}) = h_{λ}(Y_{n,y}^{#}) ≤ h_{λ}(Y_{n,y}) ≤ (106)

max(1, |λ|^{n−1})h_{λ}(Y_{n,y}^{#}),

where y ∈ E_{m}(ℂ). If |λ| ≤ 1, (107) then all the inequalities (106) turn into equalities. Let

λ(ν) = (w_{C,q}(ν))^{1/ν},

where ν ∈ m − 1 + ℕ. In view of (99, for λ = λ(μ) with μ ∈ m − 1 + ℕ are fulfilled all the conditions (100) and (107). Let z ∈ (r_{o,m}(w_{C,q})E_{m,∞}(ℂ) and

p_{m,1}(r_{0,m}(w_{C,q})^{−1}z) = h_{m,1}(r_{0,m}(w_{C,q})^{−1}z) = γ

Then p_{μ,λ(μ)}(r_{m,μ}z) = h_{μ,λ(μ)}(r_{m,μ}z) = (108)

sup{(λ(μ))^{−ν}|(r_{m,μ}z)(ν)|: ν ∈ μ − 1 + ℕ} =

sup \{(\frac{λ(ν)}{λ(μ)})^{ν} (λ(ν))^{−ν}|(r_{m,μ}z)(ν)|: ν ∈ μ − 1 + ℕ\} =

sup \{(\frac{1 + μ)}{1 + ν)})^{ν} (λ(ν))^{−ν}|(r_{m,μ}z)(ν)|: ν ∈ μ − 1 + ℕ\} ≤

sup {(λ(ν))^{−ν}|(r_{m,μ}z)(ν)|: ν ∈ μ − 1 + ℕ} ≤

sup {(λ(ν))^{−ν}|(r_{m,μ}z)(ν)|: ν ∈ m − 1 + ℕ} =
h_{m,1}((r_{0,m}(w_{C,q})^{−1}z) = p_{m,1}((r_{0,m}(w_{C,q})^{−1}z) = γ,

where μ ∈ m − 1 + ℕ; consequently r_{m,μ}z ∈ E_{μ,λ(μ)}, where μ ∈ m − 1 + ℕ. In view of (108),

p_{μ,λ(μ)} (Y_{q,r_{m,μ}z}^{#}) = h_{μ,λ(μ)} (Y_{q,r_{m,μ}z}^{#}) ≤ γ,

where μ ∈ m − 1 + ℕ. Therefore, in view of (62), 𝔮_{μ,λ(μ)} (Y_{q,r_{m,μ}z}^{#}) ≤

γ^{∗}(B^{∼})(max(1, 1/ε_{0}))^{p−1}h_{μ,λ(μ)} (Y_{q,r_{m,μ}z}^{#}) ≤
γ^{∗}(B^{∼})(max(1, 1/ε_{0}))^{p−1}γ,

where μ ∈ m − 1 + ℕ. Consequently, in view of (103),

𝔮_{μ,λ(μ)} ((▽ − B^{∨})^{−1})Y_{q,r_{m,μ}z}^{#}) ≤ (109)

6γγ^{∗}(B^{∼})(max(1, 1/ε_{0}))^{p−1} ∥(B^{∼})^{−1}∥_{sp} ,

where μ ∈ m − 1 + ℕ. In view of (109) and (63),

h ((λ(μ))^{−ν} (((▽ − B^{∨})^{−1})Y_{q,r_{m,μ}z}^{#}) (ν)) ≤ (110)

6γ(γ^{∗}(B^{∼}))^{2}(max(ε_{0}, 1/ε_{0}))^{p−1} ∥(B^{∼})^{−1}∥_{sp} ,

where μ ∈ m − 1 + ℕ and ν ∈ μ − 1 + ℕ. It follows from the inequality (110) for ν = μ ∈ m − 1 + ℕ that

h ((λ(μ))^{−ν} (((▽ − B^{∨})^{−1})Y_{q,r_{m,μ}z}^{#}) (ν)) ≤ (111)

6γ(γ^{∗}(B^{∼}))^{2}(max(ε_{0}, 1/ε_{0}))^{p−1} ∥(B^{∼})^{−1}∥_{sp} .

For any X ∈ ℂ^{q} let π(X) denotes the first coordiate of the column X, and let π be the map of ℂ^{q} on ℂ, which turns each X ∈ ℂ^{q} into π(X). In view of (111),

h ((λ(ν))^{−ν} (π ((▽ − B^{∨})^{−1})Y_{q,r_{m,ν}z}^{#}) (ν))) ≤ (112)

6γ(γ^{∗}(B^{∼}))^{2}(max(ε_{0}, 1/ε_{0}))^{p−1} ∥(B^{∼})^{−1}∥_{sp} ,

where ν ∈ m−1+ℕ. Let 𝔭 denotes the norm on (r_{0,m}(w_{ℂ,q}))E_{m,∞}(ℂ), defined by means the equality 𝔭(z) = p_{m,∞}((r_{0,m}(w_{ℂ,q}))^{−1}z), and let ϕ be the map of the space (r_{0,m}(w_{ℂ,q}))E_{m,∞}(ℂ) in E_{m,∞}(ℂ), such that

(ϕ(z))(ν) = π (((▽ − B^{∨})^{−1})Y_{q,r_{m,ν}z}^{#}) (ν))

for any z ∈ (r_{0,m}(w_{ℂ,q}))E_{m,∞}(ℂ) and any ν ∈ m − 1 + ℕ. It follows now from (112) that ϕ maps (r_{0,m}(w_{ℂ,q}))E_{m,∞}(ℂ) into (r_{0,m}(w_{ℂ,q}))E_{m,∞}(ℂ), is a bounded linear operator on (r_{0,m}(w_{ℂ,q}))E_{m,∞}(ℂ), and

𝔭^{∼}(ϕ) ≤ 6γ(γ^{∗}(B^{∼}))^{2}(max(ε_{0}, 1/ε_{0}))^{p−1} ∥(B^{∼})^{−1}∥_{sp} ,

So, we can take now on the role of the mentioned in the section 2 the splitting homomorphism ξ_{m} the restriction of the map ϕ on the subspace Ker(ψ) of the space (r_{0,m}(w_{ℂ,q}))E_{m,∞}(ℂ), where ψ is a homomorpism in (46). ■

References.

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

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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