A note on triangular operators on Smooth Sequence Spaces

TL;DR

This paper investigates conditions under which triangular operators, defined by specific matrices, are linear and continuous between certain nuclear K"othe spaces, extending understanding of their properties and transpose behavior.

Contribution

It introduces necessary and sufficient conditions for the linearity and continuity of Cauchy Product maps between nuclear K"othe spaces and their transposes.

Findings

Established criteria for linearity of Cauchy Product maps

Determined conditions for continuity of these operators

Analyzed the transpose of the operators in this context

Abstract

For a scalar sequence {(\theta_n)}_{n \in \mathbb{N}}, let C be the matrix defined by c_n^k = \theta_{n-k+1} if n > k, c_n^k = 0 if n < k. The map between K\"{o}the spaces \lambda(A) and \lambda(B) is called a Cauchy Product map if it is determined by the triangular matrix C. In this note we introduced some necessary and sufficient conditions for a Cauchy Product map on a nuclear K\"{o}the space \lambda(A) to nuclear G_1-space \lambda(B) to be linear and continuous. Its transpose is also considered.

Equations

$$\|x\|_{k}=\sum_{n}|x_{n}|a_{n}^{k}<\infty,\quad\forall k\in\mathbb{N}\$$

$$\sum _n\frac{a_n^k}{a_n^m}<+\infty$$

$$\Vert x{\Vert }_k=\underset{n}{\sup }|x_n|a_n^k,k\in \mathbb{N}.$$

$$\underset{n}{\sup }\frac{{\Vert T{e}_n\Vert }_k}{{\Vert e_n\Vert }_m}<+\infty$$

$$C=\left(\begin{array}{ccccc}{\theta }_1& 0& 0& 0& \cdots \\ {\theta }_2& {\theta }_1& 0& 0& \cdots \\ {\theta }_3& {\theta }_2& {\theta }_1& 0& \cdots \\ ⋮& & & \ddots & \end{array}\right)$$

$$\underset{j\ge n}{\sup }|{\theta }_{j-n+1}|b_j^k\le \rho a_n^m,\forall n\in \mathbb{N}.$$

$$b_n^k\le C{a}_n^m$$

$$\Vert T_{\theta }{e}_n{\Vert }_k$$

$$\underset{n}{\sup }\frac{{\Vert T_{\theta }{e}_n\Vert }_k}{{\Vert e_n\Vert }_{m_2}}<\infty$$

$$C^t=\left(\begin{array}{ccccc}{\theta }_1& {\theta }_2& {\theta }_3& {\theta }_4& \cdots \\ 0& {\theta }_1& {\theta }_2& {\theta }_3& \cdots \\ 0& 0& {\theta }_1& {\theta }_2& \cdots \\ ⋮& & & \ddots & \end{array}\right)$$

$$\underset{1\le i\le n}{\sup }|{\theta }_{n+1-i}|b_i^k\le \mu a_n^m,\forall n\in \mathbb{N}.$$

$$|{\theta }_n|\le C{a}_n^m,\forall n$$

$$b_n^k\le \frac{\mu }{|{\theta }_1|}{a}_n^m$$

$$\lambda (A)\subset \lambda (B)$$

$$\Vert T_{\theta }{}^{\prime }{e}_n{\Vert }_k$$

$$\underset{n}{\sup }\frac{{(a_n^m)}^2}{a_n^j}<\infty$$

$$\underset{n}{\sup }\frac{{\Vert T_{\theta }{}^{\prime }{e}_n\Vert }_k}{{\Vert e_n\Vert }_j}<\infty$$

$$\underset{n}{\sup }\left\{\underset{j\ge n}{\sup }\left|{\theta }_{j-n+1}\right|\frac{b_j^k}{a_n^m}\right\}\le \underset{n}{\sup }\left\{\underset{j\ge n}{\sup }\left|{\eta }_{j-n+1}\right|\frac{b_j^k}{a_n^m}\right\}<\infty$$

Full text

This research was partially supported by the Turkish Scientific and Technological Research Council

1. Introduction

We refer the reader to [3], [4] and [5] for the terminology used but not defined here. Let $A=(a_{n}^{k})_{n,k\in\mathbb{N}}$ be a matrix of real numbers such that $0\leq a_{n}^{k}\leq a_{n}^{k+1}$ for all $n,k$ and $\displaystyle\sup_{k}a_{n}^{k}>0$. The $\ell^{1}$ - Köthe space $\lambda(A)$ defined by the matrix $A$ is the space of all sequences of scalars $x=(x_{n})$ such that

[TABLE]

With the topology generated by the system of seminorms $\left\{\left\|.\right\|_{k},k\in\mathbb{N}\right\}$, it is a Fréchet space.

The topological dual of $\lambda(A)$ is isomorphic to the space of all sequences $u$ for which $\left|u_{n}\right|\leq Ca_{n}^{k}$ for some $k$ and $C>0.$

It is well known that a Köthe space $\lambda(A)$ associated with the matrix $A$ is nuclear if and only if for each $k$ there exists $m$ such that

[TABLE]

and in this case the fundamental system of norms $\displaystyle\|x\|_{k}=\sum_{n}|x_{n}|a_{n}^{k}$ can be replaced by the equivalent system of norms

[TABLE]

The infinite and finite type power series spaces are well known examples of Köthe spaces given by the matrices $(e^{k\alpha_{n}})$ respectively $(e^{-\frac{\alpha_{n}}{k}})$ where $(\alpha_{n})$ is a monotonically increasing sequence going to infinity. The space $A(\mathbb{C})$ of all entire functions on $\mathbb{C}$ and the space $A(\mathbb{D})$ of all holomorphic functions on the unit disc can be represented as an infinite respectively finite type power series spaces.

Smooth sequence spaces were introduced in [6] as a generalization of power series spaces. A Köthe set $A=\left\{(a_{n}^{k})\right\}$ is called a $G_{\infty}$-set and the corresponding Köthe space $\lambda(A)$ a $G_{\infty}$-space if $A$ satisfies the followings :

(1) $a_{n}^{1}=1$, $a_{n}^{k}\leq a_{n+1}^{k}$ for each k and n;

(2) $\forall k$ $\exists j$ with $(a_{n}^{k})^{2}=O(a_{n}^{j})$

A Köthe set $B=\left\{(b_{n}^{k})\right\}$ is called a $G_{1}$-set and the corresponding Köthe space $\lambda(B)$ a $G_{1}$-space if $B$ satisfies the followings :

(1) $0<b_{n+1}^{k}\leq b_{n}^{k}<1$ for each $k$ and $n$;

(2) $\forall k$ $\exists j$ with $b_{n}^{k}=O((b_{n}^{j})^{2})$

We need the following result [1].

Lemma 1.1**.**

Let $\lambda(A)$ and $\lambda(B)$ be Köthe spaces. A map $T:\lambda(A)\longrightarrow\lambda(B)$ is continuous linear map if and only if for each k there exists m such that

[TABLE]

If $(a_{n})$, $(b_{n})$ are two sequences of scalars, then the Cauchy product $(c_{n})=(a_{n})\ast(b_{n})$ of $(a_{n})$ and $(b_{n})$ is defined by $\displaystyle{c_{n}=\sum_{k=1}^{n}a_{n+1-k}b_{k}}$.

Now let $\theta=(\theta_{n})$ be a fixed sequence of scalars and let $\lambda(A)$, $\lambda(B)$ be two nuclear $\ell^{1}$-Köthe spaces. We define the Cauchy Product mapping $T_{\theta}$ from $\lambda(A)$ into $\lambda(B)$ by $T_{\theta}x=\theta\ast x,\quad x=(x_{n})\in\lambda(A).$ So, $T_{\theta}:\lambda(A)\longrightarrow\lambda(B)$ can be determined by the lower triangular matrix

[TABLE]

2. Cauchy Product Map on Köthe spaces

In this section we introduce some necessary and sufficient conditions for the map $T_{\theta}$ to be linear and continuous.

Theorem 2.1**.**

Let $\lambda(A)$ be a nuclear Köthe space, $\lambda(B)$ be a nuclear $G_{1}$-space. Then the Cauchy product map $T_{\theta}:\lambda(A)\longrightarrow\lambda(B)$ is linear continuous operator if and only if the following hold:

i) $\theta\in\lambda(B)$

ii) $\lambda(A)\subset\lambda(B)$

Proof.

Let $T_{\theta}:\lambda(A)\longrightarrow\lambda(B)$ be a continuous linear operator.

Note that ${\|T_{\theta}e_{n}\|}_{k}={\|(0,0,...,0,\theta_{1},\theta_{2},\cdots)\|}_{k}=\displaystyle{\sup_{j\geq n}|\theta_{j-n+1}|b_{j}^{k}}\quad$ for $n\in\mathbb{N}.$ Clearly ${\|e_{n}\|}_{m}=a_{n}^{m}.$ So, by Lemma 1.1 $\forall k$, $\exists m$, $\exists\rho>0$ such that

[TABLE]

Choose $j=n$. Then $\forall k$, $\exists m$, $\exists C>0$ such that

[TABLE]

i.e, $\lambda(A)\subset\lambda(B)$. Since $T_{\theta}e_{1}\in\lambda(B),$ it follows that $\theta\in\lambda(B)$.

Conversely, since $B$ is a $G_{1}$-set and by $ii$ and $i$ we have for a given k, there are $m_{1}(k)$ and $m_{2}(m_{1})$ such that

[TABLE]

Therefore, $\forall k$, $\exists m_{2}$ such that

[TABLE]

that is, $T_{\theta}$ is continuous. ∎∎

We consider the map $T_{\theta}{{}^{\prime}}:\lambda(A)\longrightarrow\lambda(B)$ which is determined by the matrix $C^{t}$ (the transpose of C) and try to find necessary and sufficient conditions for the continuity of $T_{\theta}{{}^{\prime}}$.

Theorem 2.2**.**

Let $\lambda(A)$ be a nuclear $G_{\infty}$-space, $\lambda(B)$ be a nuclear Köthe space. Then, $T_{\theta}{{}^{\prime}}:\lambda(A)\longrightarrow\lambda(B)$ which is given above is linear continuous operator if and only if the following hold:

i) $\theta\in\lambda(A)^{\prime}$

ii) $\lambda(A)\subset\lambda(B)$

Proof.

The matrix $C^{t}$ of the operator $T_{\theta}{{}^{\prime}}:\lambda(A)\longrightarrow\lambda(B)$ is the following upper triangular matrix:

[TABLE]

Let $T_{\theta}{{}^{\prime}}:\lambda(A)\longrightarrow\lambda(B)$ be a continuous linear operator.

Note that ${\|T_{\theta}{{}^{\prime}}e_{n}\|}_{k}={\|(\theta_{n},\theta_{n-1},\cdots,\theta_{1},0,0,\cdots)\|}_{k}=\displaystyle{\sup_{1\leq i\leq n}|\theta_{n+1-i}|b_{i}^{k}}\quad$ for $n\in\mathbb{N}.$ So, by Lemma 1.1 $\forall k$, $\exists m$, $\exists\mu>0$ such that

[TABLE]

Let $i=1$. Hence $\exists m$, $\exists C=\frac{\mu}{b_{1}^{k}}>0$ such that

[TABLE]

i.e, $\theta\in\lambda(A)^{\prime}$.

Let $i=n$. Then $\forall k$, $\exists m$ such that

[TABLE]

i.e,

[TABLE]

On the other hand, since $A$ is a $G_{\infty}$-set and by $(i)$ and $(ii)$ for a given $k$, there are $m_{1}$ and $m_{2}(k)$ and $m=max\left\{m_{1},m_{2}\right\}$ such that

[TABLE]

Since $\lambda(A)$ is $G_{\infty}$ - space, for this $m$, $\exists j$ such that

[TABLE]

Therefore, $\forall k$, $\exists j$ such that

[TABLE]

that is, $T_{\theta}{{}^{\prime}}$ is continuous. ∎∎

It is known that $\mathcal{S}$ is a normal sequence space if whenever $\left|x_{i}\right|<\left|y_{i}\right|$ and $y=(y_{i})\in\mathcal{S}$, then $x=(x_{i})\in\mathcal{S}$ [2].

Remark 2.1**.**

Now we write $\theta\in\mathcal{S}$ when the Cauchy product map $T_{\theta}:\lambda(A)\longrightarrow\lambda(B)$ above is continuous. If $\theta,\eta\in\mathcal{S}$, $\lambda\in\mathcal{K}$, then clearly $T_{\theta+\eta}$ and $T_{\lambda\theta}$ will be continuous since $T_{\theta}$ and $T_{\eta}$ are continuous. Hence $\mathcal{S}$ is a vector space.

Now, let $\left|\theta_{i}\right|<\left|\eta_{i}\right|,\forall i$, $\eta\in\mathcal{S}.$ Since $T_{\eta}$ is continuous, for all $k$ we find $m$ so that

[TABLE]

i.e. $T_{\theta}$ is continuous.

Therefore $\theta\in\mathcal{S}$. Hence we obtain that $\mathcal{S}$ is a normal sequence space.