A Remark on a paper of P. B. Djakov and M. S. Ramanujan

TL;DR

This paper investigates the conditions under which unbounded operators exist between l-K"othe spaces, showing that such operators imply the existence of unbounded quasi-diagonal operators and exploring their implications for the structure of these spaces.

Contribution

It establishes a link between unbounded operators and quasi-diagonal operators in l-K"othe spaces, providing new insights into their structure and operator boundedness.

Findings

Existence of unbounded operators implies existence of unbounded quasi-diagonal operators.

Conditions under which all continuous operators are bounded are characterized.

Unbounded operators lead to the presence of a common basic subspace under certain conditions.

Abstract

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\"{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\"{o}the matrices when every continuous linear operator between l-K\"{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\"{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

Equations

$$\Vert x\Vert =\underset{|{\beta }_n|\le 1}{\sup }\left|\sum _n{e}_n{}^{\prime }(x){\beta }_n{e}_n\right|$$

$${\Vert (x_n)\Vert }_k=\Vert (x_n{a}_n^k)\Vert$$

$${\Vert T\Vert }_{p,q}=\sup \left\{{\Vert Tx\Vert }_p:{\Vert x\Vert }_q\le 1\right\},p,q\in \mathbb{N}$$

$${\Vert T\Vert }_{p,q}={\Vert u\Vert }_q^{\ast }{\Vert x\Vert }_p$$

$${\Vert T\Vert }_{k,N(k)}<\infty ,$$

$${\Vert T\Vert }_{r,N}<\infty .$$

$${\Vert T\Vert }_{r,N}\le C\underset{1\le k\le k_0}{\max }{\Vert T\Vert }_{k,N(k)}$$

$$T{e}_n=m_n{\tilde{e}}_{k(n)},n\in \mathbb{N}$$

$$\forall p\exists q,k\forall s,l\exists r,C:\frac{b_m^s}{a_n^k}\le C\max \left\{\frac{b_m^q}{a_n^p},\frac{b_m^r}{a_n^l}\right\}$$

$${\Vert Tx\Vert }_k\le \frac{1}{2^k}{\Vert x\Vert }_k,\forall x\in {\lambda }^{\ell }(A)$$

$$\underset{n}{\sup }\frac{{\Vert T{e}_n\Vert }_{k+1}}{{\Vert e_n\Vert }_k}=\infty ,k\in \mathbb{N}.$$

$$\frac{\Vert T{e}_{n_j}{\Vert }_{k_j+1}}{\Vert e_{n_j}{\Vert }_{k_j}}\ge 2^j,\forall j$$

$$\underset{|{\alpha }_v|\le 1}{\sup }\left|\sum _v{\theta }_{nv}{\alpha }_v\left(\underset{k}{\sup }\frac{b_v^k}{a_n^k}\right)\widetilde{e_v}\right|$$

$$\le \sum _k\frac{1}{a_n^k}\underset{|{\alpha }_v|\le 1}{\sup }\left|\sum _v{\theta }_{nv}{\alpha }_v{b}_v^k\widetilde{e_v}\right|$$

$$\le \sum _k\frac{{\Vert T{e}_n\Vert }_k}{{\Vert e_n\Vert }_k}\le \sum _k\frac{1}{2^k}\le 1$$

$$\underset{|{\alpha }_v|\le 1}{\sup }\left|\sum _v{\theta }_{n_{j}v}{\alpha }_v\left(\underset{k}{\sup }\frac{b_v^k}{a_{n_j}^k}\right)\widetilde{e_v}\right|\le 1\le \frac{1}{2^j}\underset{|{\alpha }_v|\le 1}{\sup }\left|\sum _v{\theta }_{n_{j}v}{\alpha }_v\frac{b_v^{k_j+1}}{a_{n_j}^{k_j}}\widetilde{e_v}\right|$$

$$t_j:=\underset{k}{\sup }\frac{b_{v_j}^k}{a_{n_j}^k}\le \frac{1}{2^j}\frac{b_v^{k_j+1}}{a_{n_j}^{k_j}}$$

$$D{e}_{n_j}=t_j^{-1}\widetilde{e_{v_j}},j\in \mathbb{N}$$

$$D{e}_n=0\text{if}n\ne n_j$$

$${\Vert Dx\Vert }_k\le {\Vert x\Vert }_k\forall k$$

$$\frac{\Vert D{e}_{n_{j_m}}{\Vert }_{k+1}}{\Vert e_{n_{j_m}}{\Vert }_k}\ge 2^{j_m}\to \infty$$

$$\frac{b_v^r}{a_i^N}\le C\underset{1\le k\le k_0}{\max }\frac{b_v^k}{a_i^{N(k)}}$$

$${\Vert T\Vert }_{k,N(k)}={\Vert e_i{}^{\prime }\Vert }_{N(k)}{\Vert e_v\Vert }_k=\frac{b_v^k}{a_j^{N(k)}}$$

$$\underset{i}{\sup }\frac{{\Vert T{e}_i\Vert }_k}{{\Vert e_i\Vert }_{N(k)}}=\underset{i}{\sup }\frac{\left|t_i\right|b_{z(i)}^k}{a_i^{N(k)}}=C(k)<\infty .$$

$$\frac{\left|t_i\right|b_{z(i)}^r}{a_i^N}\le C\underset{1\le k\le k_0}{\max }\frac{\left|t_i\right|b_{z(i)}^k}{a_i^{N(k)}}\le C\underset{1\le k\le k_0}{\max }C(k).$$

Full text

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

1. Introduction

Following [2], we denote by $\ell$ a Banach sequence space in which the canonical system $(e_{n})$ is an unconditional basis. The norm $\left\|.\right\|$ is called monotone if $\left\|x\right\|\leq\left\|y\right\|$ whenever $\left|x_{n}\right|\leq\left|y_{n}\right|$, $x=(x_{n})$, $y=(y_{n})\in\ell$, $n\in\mathbb{N}$. Let $\Lambda$ be the class of such spaces with monotone norm. In particular, $l_{p}\in\Lambda$ and $c_{0}\in\Lambda$. It is known that every Banach space with an unconditional basis $(e_{n})$ has a monotone norm which is equivalent to its original norm. Indeed, it is enough to put

[TABLE]

where $\left|.\right|$ denotes the original norm, $(e_{n}{{}^{\prime}})$ denote the sequence of coefficient functionals.

Let $\ell\in\Lambda$ and $\left\|.\right\|$ be a monotone norm in $\ell$. If $A=(a_{n}^{k})$ is a Köthe matrix, the $\ell$-Köthe space $\lambda^{\ell}(A)$ is the space of all sequences of scalars $(x_{n})$ such that $(x_{n}a_{n}^{k})\in\ell$ with the topology generated by the seminorms

[TABLE]

For any linear operator $T:X\longrightarrow Y$ between Fréchet spaces we consider the following operator seminorms

[TABLE]

which may take the value $+\infty$. In particular, for any one dimensional operator $T=u\otimes x$, we have

[TABLE]

The operator $T$ is continuous if and only if for all $k$ there is $N(k)$ such that

[TABLE]

$T$ is bounded if and only if there is $N\in\mathbb{N}$ such that for all $r\in\mathbb{N}$,

[TABLE]

We write $(X,Y)\in\mathcal{B}$ if every continuous linear operator on $X$ to $Y$ is bounded. Zahariuta [7] obtained that if the matrices $A$ and $B$ satisfy the conditions $d_{2}$ and $d_{1}$, repectively, then $(\lambda^{l_{1}}(A),\lambda^{l_{1}}(B))\in\mathcal{B}$. This phenomenon was studied extensively by Vogt [6] not only for Köthe spaces but also for the general case of Fréchet spaces. In case of $\ell$-Köthe spaces, there is no characterization of pairs $(X,Y)$ with the property $\mathcal{B}.$

For Fréchet spaces $X$ and $Y$, in [6], Vogt proved that $(X,Y)\in\mathcal{B}$ if and only if for every sequence $N(k)$, $\exists N\in\mathbb{N}$ such that $\forall r\in\mathbb{N}$ we have $k_{0}\in\mathbb{N}$ and $C>0$ with

[TABLE]

for all $T\in\mathcal{L}(X,Y)$.

An operator $T:\lambda^{\ell}(A)\longrightarrow\lambda^{\ell}(B)$ is called quasi-diagonal if there exists $k:\mathbb{N}\longrightarrow\mathbb{N}$ and constants $m_{n}$ such that

[TABLE]

Following [4], a pair of Köthe spaces $(\lambda^{\ell}(B),\lambda^{\ell}(A))$ satisfies the condition $\mathcal{S}$ if,

[TABLE]

In [3] it was proved that the existence of an unbounded continuous linear operator from nuclear $l_{1}$-Köthe space to another implies the existence of a continuous unbounded quasi-diagonal operator. Also, if the both Köthe spaces are nuclear, in [5], Nurlu and Terzioğlu proved that the existence of an unbounded continuous linear operator on $\lambda^{l_{1}}(A)$ to $\lambda^{l_{1}}(B)$ implies, under some conditions, the existence of a common basic subspaces of $\lambda^{l_{1}}(A)$ and $\lambda^{l_{1}}(B)$. Djakov and Ramanujan generalized these results by omitting nuclearity condition [1].

Let $X=\lambda^{\ell}(A)$ and $Y=\lambda^{\ell}(B)$ be the $\ell$- Köthe spaces. Here, we modify Proposition $1$ in [1] for $\ell$- Köthe spaces and using it we obtain a necessary and sufficient condition in terms of corresponding Köthe matrices when $(X,Y)\in\mathcal{B}$. Also we observe a common basic subspace between $\ell$- Köthe spaces $X$ and $Y$ when $(X,Y)\notin\mathcal{B}$ and $(Y,X)\in\mathcal{S}$ following the same lines in [1].

2. Bounded and unbounded operators in $\ell$-Köthe spaces

Let $\lambda^{\ell}(A),\lambda^{\ell}(B)$ be $\ell$-Köthe spaces. As in [1] we obtain the following.

Proposition 2.1**.**

Let $\lambda^{\ell}(A)$ and $\lambda^{\ell}(B)$ be $\ell$- Köthe spaces. If there exists a continuous linear unbounded operator $T:\lambda^{\ell}(A)\longrightarrow\lambda^{\ell}(B)$, then there exists a continuous unbounded quasi-diagonal operator on $\lambda^{\ell}(A)$ to $\lambda^{\ell}(B).$

Proof.

Let $T:\lambda^{\ell}(A)\longrightarrow\lambda^{\ell}(B)$ be continuous and unbounded. We may assume without loss of generality that

[TABLE]

[TABLE]

Indeed, one may obtain these by using appropriate multipliers and passing to a subsequence of seminorms, if necessary. Let $(k_{j})$ be a sequence of integers such that each $k\in\mathbb{N}$ appears in it infinitely many times and choose an increasing subsequence $(n_{j})$ such that

[TABLE]

Let us remind that $\left\|\tilde{e_{v}}\right\|_{k}=b_{v}^{k}$ and $\left\|e_{n}\right\|_{k}=a_{n}^{k}$ and let $\displaystyle Te_{n}=\sum_{v}\theta_{nv}\tilde{e_{v}}.$ Note that,

[TABLE]

Therefore we obtain that

[TABLE]

So there is a $v_{j}$ such that

[TABLE]

Otherwise we obtain a contradiction to (2.1) by monotonicity of $\left\|.\right\|.$

Now, consider the quasi-diagonal operator $D:\lambda^{\ell}(A)\longrightarrow\lambda^{\ell}(B)$ defined by

[TABLE]

[TABLE]

Let $\displaystyle x=\sum_{j}x_{n_{j}}e_{n_{j}}\in\lambda^{\ell}(A).$ So, $\displaystyle Dx=\sum_{j}x_{n_{j}}t_{j}^{-1}\tilde{e}_{v_{j}}$. Since $\left|x_{n_{j}}t_{j}^{-1}b_{v_{j}}^{k}\right|\leq\left|x_{n_{j}}a_{n_{j}}^{k}\right|$, by monotonicity we obtain that $\left\|\left(x_{n_{j}}t_{j}^{-1}b_{v_{j}}^{k}\right)\right\|\leq\left\|\left(x_{n_{j}}a_{n_{j}}^{k}\right)\right\|,$ i.e.,

[TABLE]

Hence, $D$ is continuous.

Similarly, it is easy to see that $D$ is unbounded since for a fixed $k$, there is a subsequence $(j_{m})$ such that $k_{j_{m}}=k$, $m\in\mathbb{N}$ and

[TABLE]

as $m\rightarrow\infty.$ This completes the proof. ∎

Proposition 2.1 enables us to prove the sufficiency part of the following theorem. Notice that sufficiency can not be obtained directly for a general linear map.

Theorem 2.2**.**

Let $\lambda^{\ell}(A)$ and $\lambda^{\ell}(B)$ be $\ell$-Köthe spaces. $(\lambda^{\ell}(A),\lambda^{\ell}(B))\in\mathcal{B}$ if and only if for every sequence $N(k)\uparrow\infty$ there exists $N\in\mathbb{N}$ such that for each $r\in\mathbb{N}$ we have $k_{o}\in\mathbb{N}$ and $C>0$ with

[TABLE]

for all $v\in\mathbb{N},i\in\mathbb{N}$.

Proof.

Suppose $(\lambda^{\ell}(A),\lambda^{\ell}(B))\in\mathcal{B}$. Consider $T:\lambda^{\ell}(A)\longrightarrow\lambda^{\ell}(B)$ with $T=e_{i}^{\prime}\otimes e_{v}$ where $e_{i}{{}^{\prime}}(x)=x_{i}$ for all $x\in\lambda^{\ell}(A).$

Since $T$ is the operator of rank one, we note that

[TABLE]

Similarly $\left\|T\right\|_{r,N}=\frac{b_{v}^{r}}{a_{i}^{N}}.$ The result follows from (1.1).

Conversely we want to show that every continuous linear quasi-diagonal operator is bounded. Let $T:\lambda^{\ell}(A)\longrightarrow\lambda^{\ell}(B)$ be a continuous quasi-diagonal operator defined by $T(e_{i})=t_{i}\tilde{e}_{z(i)}$. By continuity, $\exists N(k)$ such that

[TABLE]

Thus for this $N(k)$, $\exists N\in\mathbb{N}$ such that $\forall r\in\mathbb{N}$ we have $k_{o}\in\mathbb{N}$ and $C>0$ with

[TABLE]

Hence $\left\|T\right\|_{r,N}<\infty$, i.e., $T$ is bounded. In view of Proposition 2.1, we obtain the result. ∎

$\lambda^{\ell}(A)$ and $\lambda^{\ell}(B)$ have a common basic subspace if there is a quasi-diagonal operator $T:X\rightarrow Y$ such that the restriction of $T$ to some infinite dimensional basic subspace of $X$ is an isomorphism. We observe the following extension of Proposition $3$ in [1] to the $\ell$-Köthe space case. The proof is the same as in [1].

Corollary 2.3**.**

If $(\lambda^{\ell}(B),\lambda^{\ell}(A))\in\mathcal{S}$ and there exists a continuous unbounded operator $T:\lambda^{\ell}(A)\longrightarrow\lambda^{\ell}(B)$, then $\lambda^{\ell}(A)$ and $\lambda^{\ell}(B)$ have a common basic subspace.