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Introduction
Weighted shift operators have been studied by many authors in different contexts, for instance the work by N. K. Nikol’skiĭ in the spaces spaces ℓp, [15–18], R. Gellar in Banach spaces, [2–4] and Grabiner in Banach algebras of power series spaces, [5–7].
Forward weighted operators (multiplication and integration operators) play a remarkable role in the study of bases in spaces of analytic functions and have been considered by many Russian mathematicians, [11]. The Gončarov polynomials, that under certain conditions are a basis in analytic spaces [1, 9], are related to the backward weighted operator (derivation operator).
We work with Köthe spaces and weighted shifts on them (generalized integration and derivation operators). We characterize the forward shift-invariant isomorphisms and then determine some some quasi-power bases. Our results include, as particular cases, those of Nagnibida for the multiplication and integration operators on the space of analytic functions on a disc, [11] and Prada for the multiplication operator on infinite power series spaces, [19, 20]. Using the backward shift operator we get conditions for the Gončarov polynomials to be a basis.
Basic results
Denote by λp(A), 1 ≤ p < ∞, the Köthe (echelon) space given by the matrix $\begin{array}{}
\displaystyle
A=(a_{n}^{k})_{n=0}^{\infty }
\end{array}$, $\begin{array}{}
\displaystyle
0<a_{n}^{k}\leq a_{n}^{k+1}
\end{array}$ for all n, k, that is
The canonical basis in the spaces λp(A), p = 0, 1 ≤ p < ∞, is denoted by $\begin{array}{}
\displaystyle
\delta _{n}=(\delta_{n,k})_{k=0}^{\infty }
\end{array}$, where dn,k is the Kronecker delta.
The dual space of λp(A), 1 ≤ p < ∞, p = 0, $\begin{array}{}
\displaystyle
\frac{1}{p}+\frac{1}{q}=1
\end{array}$ is given by
$$\begin{array}{}
\displaystyle
\begin{aligned}
\left( \lambda^{p}(A)\right) ^{\times} & =\left\{(x_n)_{n=0}^\infty,x_n\in\mathbb{C}:
\left(\sum_{n=0}^\infty \frac{|x_n|^q}{(a_n^k)^q}\right)^{\frac{1}{q}}<\infty,
\text{ for a suitable }k \right\}, \ 1<p<\infty.\\\\
\left( \lambda^{1}(A)\right) ^{\times}& =\left\{(x_n)_{n=0}^\infty,x_n\in\mathbb{C}:
\sup_{n\ge0}\left\{ \frac{|x_n|}{a_n^k}\right\}<\infty,
\text{ for a suitable }k \right\}, \ p=1.\\\\
\left( \lambda^{0}(A)\right) ^{\times} & =\left\{(x_n)_{n=0}^\infty,x_n\in\mathbb{C}:
\sum_{n=0}^\infty \frac{|x_n|}{a_n^k}<\infty,
\text{ for a suitable } k\right\},\ p=0.
\end{aligned}
\end{array}$$
Recall that the coordinate operators are continuous, [14].
λp(A), p ∈ [1, ∞), p = 0 is the projective limit of the Banach spaces ℓp(ak), c0(ak), diagonal transformations of ℓp, c0, with the usual topology:
is called the generalized integration operator. If $\begin{array}{}
\displaystyle
\lambda_{n}=\frac{1}{n!}
\end{array}$, Jλ = J is the integration operator and if λn = 1, Jλ = U, is the multiplication one (shift operator), see [11].
We assume that the operator Jλ, where the sequence (λn) are positive real numbers without lost of generality, is continuous on λp(A), that is the following condition is fulfilled
is called the generalized derivation operator, being the usual derivation when $\begin{array}{}
\displaystyle
d_{n}=\frac{1}{n!}
\end{array}$.
Isomorphisms commuting with Jλ. Bases in Köthe spaces
We characterize the isomorphisms between Köthe spaces that commute with the generalized integration operator Jλ determining some bases, related with it, on λ1(A).
Theorem 1
Let T : λ1(A) → λ1(A) be a continuous linear operator.$\begin{array}{}
\displaystyle
\left\{ \frac{1}{\lambda_{n}}T^{n}x\right\}_{n=0}^{\infty}
\end{array}$, x ∈ λ1(A) is a basis in λ1(A) if and only if there exists an isomorphism S : λ1(A) → λ1(A) such that T ∘ S = S ∘ Jλ and x = Sδ0.
Proof. If $\begin{array}{}
\displaystyle
\left\{ \frac{1}{\lambda_{n}}T^{n}x\right\}
\end{array}$, n ≥ 0, is a basis in λ1(A), then there exists an isomorphism S such that $\begin{array}{}
\displaystyle
S\delta_{n}=\frac{1}{\lambda_{n}}T^{n}x
\end{array}$, n = 0, 1, 2,... It follows that Sδ0 = x and for n ∈ ℕ
$\begin{array}{}
\displaystyle
\left\{ \frac{1}{\lambda_{n}}J_{\lambda}^{n} x\right\}
\end{array}$, x ∈ λ1(A) is a basis in λ1(A) if and only if there exists an isomorphism T : λ1(A) → λ1(A) that commutes with Jλ and x = T δ0.
Proposition 3
[13] A linear operator T : λ1(A) → λ1(A) is continuous and commutes with Jλ if and only if
(c.f. [13]). Let T be a linear operator from λ1(A) onto itself commuting with Jλ and b = (bn) = T (δ0) (b0 ≠ 0). If T−1is the formal operator given by the inverse matrix of T , c = (cn) = T−1(δ0) and k,
then T is an isomorphism if and only if b, c ∈ λ1(A).
Remark 1
Recall that the matrix (ti,j) of a continuous linear operator T commuting with Jλ is lower triangular so, formally, (ti,j) has an inverse of the same type if T δ0 = (bn) with b0 ≠ 0. The operator T−1 given by this inverse matrix is always linear and commutes with Jλ . Then a continuous operator T is an isomorphism if and only if T−1 is continuous and T−1 can be written
$\begin{array}{}
\displaystyle
a_{m+n}^{k}\leq C_{k}a_{m}^{k}a_{n}^{k}
\end{array}$, ∀k, that is, the spaces ℓ1(ak) are Banach algebras.
λm+n ≤ Cλmλn, ∀m, n.
Let$\begin{array}{}
\displaystyle
T=\sum\limits_{n=0}^{\infty}\frac{b_{n}}{\lambda_{n}}J_{\lambda}^{n}
\end{array}$be a linear operator on λ1(A) commuting with Jλ.
Then T is an isomorphism if and only if any of the following equivalent conditions are satisfied:
The sequence$\begin{array}{}
\displaystyle
(\frac{b_{n}}{\lambda_{n}})
\end{array}$is an exponential (invertible) element of all the Banach algebras ℓ1(bk), $\begin{array}{}
\displaystyle
b_{n}^{k}=\lambda_{n}a_{n}^{k}
\end{array}$, for all k.
If$\begin{array}{}
\displaystyle
\left( \frac{b_{n}}{\lambda_{n}}\right)
\end{array}$is an exponential (invertible) element of λ1(B), $\begin{array}{}
\displaystyle
B=\left( b_{n}^{k}\right)
=(\lambda_{n}a_{n}^{k})
\end{array}$, then the system
[13] Let T be a linear operator on λ1(A) commuting with Jλ, $\begin{array}{}
\displaystyle
T=\sum\limits_{n=0}^{\infty}\,\frac{b_{n}}{\lambda_{n}}
J_{\lambda}^{n}
\end{array}$, b0 ≠ 0.
If there exists$\begin{array}{}
\displaystyle
M_{k}=\lim\limits_{n\to\infty}\frac{\lambda_{n+1}}{\lambda_{n}}
\frac{a_{n+1}^{k}}{a_{n}^{k}}
\end{array}$, Mk ≠ 0, for a suitable k, then the function$\begin{array}{}
\displaystyle
\phi (z)=\sum\limits_{n=0}^{\infty} \frac{b_{n}}{\lambda_{n}}z^{n}
\end{array}$is an holomorphic one with no zeros in a disc D(0, ρ), with ρ ≥ Mk.
If$\begin{array}{}
\displaystyle
\lim\limits_{n\to\infty}\frac{\lambda_{n+1}}{\lambda_{n}}\frac{a_{n+1}^{k}}{a_{n}^{k}}=\infty
\end{array}$for a suitable k, then the function$\begin{array}{}
\displaystyle
\phi (z)=\sum\limits_{n=0}^{\infty}\frac{b_{n}}{\lambda_{n}}z^{n}
\end{array}$is an entire function without zeros.
Proposition 8
(c.f. [13]). Let T be a linear operator on λ1(A) commuting with Jλ
If the function$\begin{array}{}
\displaystyle
\phi (z)=\sum\limits_{n=0}^{\infty}\frac{b_{n}}{\lambda_{n}}z^{n}
\end{array}$is holomorphic without zeros in a disc$\begin{array}{}
\displaystyle
\mathbb{D}_{\rho}
\end{array}$, ρ > supk {Mk} or ρ = ∞, then T is an isomorphism from λ1(A) onto itself.
then the only entire functions without zeros that give continuous linear operators on λ1(A) are the constants.
Example 10
The space of holomorphic functions, $\begin{array}{}
\displaystyle
\mathscr{H}(\mathbb{D}_R)
\end{array}$, on the disc$\begin{array}{}
\displaystyle
\mathbb{D}_R=\mathbb{D}(0,R)
\end{array}$, 0 < R ≤ ∞ is a Köthe space λ1(A), with$\begin{array}{}
\displaystyle
A=(a_{n}^{k})=(t_{k}^{n})
\end{array}$, where (tk) is an increasing sequence of real positive numbers converging to R.
If λn = 1, ∀n then a continuous linear operator$\begin{array}{}
\displaystyle
T=\sum \limits_{n=0}^{\infty}{b_n}U^{n}
\end{array}$on$\begin{array}{}
\displaystyle
\mathscr{H}(\mathbb{D}_{R})
\end{array}$, commuting with the multiplication operator U, is an isomorphism if and only if the function$\begin{array}{}
\displaystyle
\phi (z)=\sum\limits_{n=0}^{\infty}{b_{n}}z^{n}\in\mathscr{H}(\mathbb{D}_{R})
\end{array}$and has no zeros in the disc$\begin{array}{}
\displaystyle
\mathbb{D}_{R}
\end{array}$, see [11].
If$\begin{array}{}
\displaystyle
\lambda_{n}=\frac{1}{n!}
\end{array}$, ∀n then Jλ = J and a linear continuous operator T on$\begin{array}{}
\displaystyle
\mathscr{H}(\mathbb{D}_{R})
\end{array}$, commuting with J, is an isomorphism if and only if the function$\begin{array}{}
\displaystyle
\phi(z)=\sum\limits_{n=0}^\infty b_{n}z^{n} \in \mathscr{H}(\mathbb{D}_{R})
\end{array}$and b0 ≠ 0, see [11].
Example 11
The space λ1(A) = Λ∞(α), A = (ekαn) with (αn) an increasing sequence of positive numbers going to infinity, is an infinite power series space.
If λn = 1, ∀n, and αm+n ≤ C + αn + αm, ∀m, n, then a continuous linear operator T on Λ∞(α), commuting with U, is an isomorphismif and only if the sequence T δ0 = (bn) ∈ Λ∞(α) and the function$\begin{array}{}
\displaystyle
\phi (z)=\sum\limits_{n=0}^{\infty}b_nz^{n}
\end{array}$has no zeros in the closed disk D(0,1) (if$\begin{array}{}
\displaystyle
\lim\limits_{n\to\infty} \frac{\alpha_{n}}{n}=0
\end{array}$or has no zeros in the complex plane (if$\begin{array}{}
\displaystyle
\lim\limits_{n\to\infty} \frac{\alpha _{n}}{n}>0
\end{array}$[20].
If$\begin{array}{}
\displaystyle
\lambda_{n}=\frac{1}{n!}
\end{array}$, ∀n, αm+n ≤ C + αn + αm, ∀m,n, and$\begin{array}{}
\displaystyle
\lim\limits_{n\to\infty}\frac{\alpha _{n}}{n}<\infty
\end{array}$, then a continuous linear operator T is an isomorphism on Λ∞(α) commuting with J if and only if T δ0 = (bn) ∈ Λ∞(α) and b0 ≠ 0.
Example 12
The conditions of the proposition 9 are fulfilled, for instance, if λln = 1 or$\begin{array}{}
\displaystyle
\lambda_{n}=\frac{1}{n!}
\end{array}$and$\begin{array}{}
\displaystyle
a_{n}^{k}=e^{n^{\alpha}k}
\end{array}$, α > 0.
Two continuous operators commuting with Jλ commute with each other [2] but the converse is not true. For example, take an operator given by an infinite two-block matrix
and the operator $\begin{array}{}
\displaystyle
J_{\lambda}^{2}
\end{array}$. We show that for certain spaces the result is true.
Theorem 13
Let T be a linear operator from λp(A) to λp(A), p = 0, p ∈ [1, +∞) commuting with Jλ , $\begin{array}{}
\displaystyle
T=\sum\limits_{n=0}^\infty \frac{b_{n}}{\lambda_{n}}J_{\lambda}^{n}
\end{array}$and$\begin{array}{}
\displaystyle
\left\{ \lambda_{n}U^{n}\left( \frac{b_{j+1}}{\lambda_{j+1}}\right) _{j=0}^{\infty }\right\} _{n=0}^{\infty }
\end{array}$is a basis of λ1(A). Then any continuous linear operator S on λp(A),commuting with T , commutes with Jλ.
Proof. It is similar to the proof of theorem 3.5 in [20].
Gončarov polynomials in a nuclear Köthe space
Conditions for the generalized Gončarov polynomials to be a basis in the nuclear space spaces λ1(A) are given.
Given a sequence of complex numbers $\begin{array}{}
\displaystyle
(z_n)_{n=0}^{\infty}
\end{array}$, the Gončarov polynomials Gn(z;z0,...,zn−1) are recursively defined by
where (dn) is a sequence of positive real numbers.
Recall that if X is a locally convex space, a biorthogonal system {ei, fi}, ei ∈ X, fi ∈ X′, fi(ej) = δij, is complete, if the finite linear combinations of (ei) are dense in X, see [14].
If we define the functionals Dm, Lm, m ≥ 0 on $\begin{array}{}
\displaystyle
\mathscr{H}(\mathbb{D}_{R})
\end{array}$ by
then $\begin{array}{}
\displaystyle
\left\{ G_{m}(z;z_{0},z_{1},\dots,z_{m-1});D_{m}\right\} _{m=0}^{\infty }
\end{array}$ and $\begin{array}{}
\displaystyle
\left\{ Q_{m}(z;z_{0},z_{1},\dots,z_{m-1});L_{m}\right\} _{m=0}^{\infty }
\end{array}$ are biorthogonal systems for $\begin{array}{}
\displaystyle
\mathscr{H(}\mathbb{D}_{R})
\end{array}$.
Theorem 14
(c.f. [8]). If λ1(A) is nuclear, a complete biorthogonal system, (ei, fi), fi = ( fi,j), is a Schauder basis for λ1(A) if and only if ∀k ∈ ℕ there exists r = r(k) ∈ ℕ such that:
(c.f. [9]). Let (tk) be a sequence such that tk < tk+1and$\begin{array}{}
\displaystyle
\lim\limits_{k\to\infty}t_{k}=R
\end{array}$, 0 < R ≤ ∞. The Gončarov polynomials Gn(z; z0,...,zn−1) are a Schauder basis in$\begin{array}{}
\displaystyle
\mathscr{H}(\mathbb{D}_{R})
\end{array}$, if and only if ∀k ∈ ℕ, there exists r = r(k) such that
(c.f. [10]). The generalized Gončarov polynomials Qn(z; z0,...,zn−1) are a basis in$\begin{array}{}
\displaystyle
\mathscr{H}(\mathbb{D}_{R})
\end{array}$, 0 < R ≤ ∞, if and only if ∀k ∈ ℕ, ∃r = r(k) such that
The generalized Gončarov polynomials $\begin{array}{}
\displaystyle
\left\{Q_{n}(z;z_{0},\dots,z_{n-1})\right\}_{n=0}^{\infty }
\end{array}$ are a complete system in a nuclear space λ1(A) and Ln ∈ (λ1(A))′ if and only if
If λ1(A) is a nuclear space, the generalized Gončarov polynomials$\begin{array}{}
\displaystyle
\left\{ Q_{n}(z;z_{0},\dots,z_{n-1})\right\}_{n=0}^{\infty }
\end{array}$are a basis in λ1(A) if and only if ∀k ∈ ℕ, ∃r = r(k) ∈ ℕ such that:
