Bounded Operators to $\ell$-K\"othe Spaces

TL;DR

This paper characterizes bounded operators between certain Fréchet and l-K"othe spaces, extending previous results to tensor products and nuclear spaces, with implications for operator boundedness and factorization.

Contribution

It provides a necessary and sufficient condition for bounded operators between specific Fréchet and l-K"othe spaces, extending prior work to tensor products and nuclear spaces.

Findings

Characterization of bounded operators between Fréchet and l-K"othe spaces.

Extension of bounded factorization results to tensor products of nuclear spaces.

Sufficient conditions for bounded factorization involving nuclear l-K"othe spaces.

Abstract

For Fr{\'e}chet spaces E and F we write (E,F) \in {B} if every continuous linear operator from E to F is bounded. Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We obtain a necessary and sufficient condition for (E,F) \in {B} when F = \lambda^{l}(B). We say that a triple (E,F,G) has the bounded factorization property and write (E,F,G) \in {BF} if each continuous linear operator T : E \longrightarrow G that factors over F is bounded. We extend some results in \cite{Ter03} to l-K\"{o}the spaces and obtain a sufficient condition for (E,\lambda^{l_1}(A) \hat{otimes}_{pi} \lambda^{l_2}(B), \lambda^{l_3}(C)) \in {BF} when \lambda^{l_1}(A) and \lambda^{l_2}(B) are nuclear.

Equations

$$\lambda (A)=\left\{x=(x_n):{\Vert x\Vert }_k=\sum _n\left|x_n\right|a_n^k<\infty ,\forall k\in \mathbb{N}\right\}$$

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

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

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

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

$${\Vert T\Vert }_{k,N(k)}<\infty ,\forall k\in \mathbb{N};$$

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

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

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

$$\underset{x\ne 0,x\in \lambda (A)}{\sup }\frac{{\Vert Tx\Vert }_m}{{\Vert x\Vert }_k}<+\infty$$

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

$${\Vert Tx\Vert }_m$$

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

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

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

$${\Vert T\Vert }_{r,N}$$

$$a_v^r{\Vert u\Vert }_N^{\ast }\le C\underset{1\le k\le k_0}{\max }{a}_v^k{\Vert u\Vert }_{N(k)}^{\ast }$$

$$Tx=\sum _{v=1}^{\infty }{e}_v{}^{\prime }(Tx)e_v=(e_v{}^{\prime }T(x))=(u_v(x)),x\in E$$

$$\underset{{\Vert x\Vert }_{N(k)}\le 1}{\sup }\left(\underset{|{\beta }_v|\le 1}{\sup }\left|\sum _v{\beta }_v{u}_v(x)a_v^k{e}_v\right|\right)=M(k)$$

$${\Vert Tx\Vert }_r$$

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

$$c_j^r{\Vert u\Vert }_N^{\ast }\le C\underset{1\le k\le k_0}{\max }\left\{{\Vert u\Vert }_{N(k)}^{\ast }{b}_i^k\right\}\underset{1\le k\le k_0}{\max }\left\{\frac{c_j^k}{b_i^{N(k)}}\right\}$$

$$\sum _i\frac{b_i^{N(k)}}{b_i^{S(k)}}=\theta (k)<\infty ,k\in \mathbb{N}.$$

$$Tx=\sum _i\sum _j{u}_i(x)r_{ji}{e}_j$$

$$c_j^r{\Vert u_i\Vert }_N^{\ast }\le C\underset{1\le k\le k_0}{\max }\left\{{\Vert u_i\Vert }_{S(k)}^{\ast }{b}_i^k\right\}\underset{1\le k\le k_0}{\max }\left\{\frac{c_j^k}{b_i^{S(k)}}\right\}$$

$${\Vert S\Vert }_{k,N(k)}=\underset{{\Vert x\Vert }_{N(k)}\le 1}{\sup }{\Vert Sx\Vert }_k=\underset{{\Vert x\Vert }_{N(k)}\le 1}{\sup }\sum _i\left|u_i(x)\right|b_i^k=\sum _i{\Vert u_i\Vert }_{N(k)}^{\ast }{b}_i^k$$

$${\Vert R\Vert }_{k,N(k)}=\underset{i}{\sup }\frac{{\Vert R{e}_i\Vert }_k}{{\Vert e_i\Vert }_{N(k)}}=\underset{i}{\sup }\underset{|{\beta }_j|\le 1}{\sup }\left|\sum _j\frac{r_{ji}{\beta }_j{c}_j^k{e}_j}{b_i^{N(k)}}\right|$$

$$\sum _i{\Vert u_i\Vert }_{S(k)}^{\ast }{b}_i^k\le \sum _i{\Vert u_i\Vert }_{N(k)}^{\ast }{b}_i^k={\Vert S\Vert }_{k,N(k)}$$

$$\sum _j\frac{\left|r_{ji}\right|c_j^k}{b_i^{S(k)}}\le \theta (k)\underset{j}{\sup }\frac{\left|r_{ji}\right|c_j^k}{b_i^{N(k)}}\le \theta (k){\Vert R\Vert }_{k,N(k)}$$

$${\Vert Tx\Vert }_r$$

Full text

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

1. Introduction

For an infinite matrix $A=(a_{n}^{k})$ with $0\leq a_{n}^{k}\leq a_{n}^{k+1}$ and $\displaystyle\sup_{k}a_{n}^{k}>0$ for every $n$ and $k$ we denote by $\lambda(A)$ the corresponding $l_{1}$-Köthe space, that is,

[TABLE]

Equipped with the system of seminorms $\left\{\left\|.\right\|_{k},k\in\mathbb{N}\right\}$, $\lambda(A)$ is a Fréchet space.

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 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]

Let us remind that $\left\|(e_{n})\right\|_{k}=a_{n}^{k}.$ We denote by $\mathcal{L}(E,F)$ the space of all continuous linear operators between Fréchet spaces $E$ and $F.$ For $T\in\mathcal{L}(E,F)$ we consider the following operator seminorms

[TABLE]

which may take the value $+\infty$. In particular, for any one dimensional operator $T=u\otimes x$ which sends each $z\in E$ to $u(z)x\in F$, we have

[TABLE]

where ${\left\|u\right\|}_{q}^{*}=\sup\left\{\left|u(x)\right|:\left\|x\right\|_{q}\leq 1\right\}$.

We recall that $T:E\longrightarrow F$ is continuous if there is a map $N:\mathbb{N}\longrightarrow\mathbb{N}$ such that

[TABLE]

$T$ is bounded if $\exists N\in\mathbb{N}$ such that

[TABLE]

We write $(E,F)\in\mathcal{B}$ if every continuous linear operator from $E$ to $F$ is bounded. For Fréchet spaces $E$ and $F$, in [4], Vogt proved that $(E,F)\in\mathcal{B}$ 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 $T\in\mathcal{L}(E,F)$.

We say that a triple $(E,F,G)$ has the bounded factorization property and write $(E,F,G)\in\mathcal{BF}$ if each continuous linear operator $T:E\longrightarrow G$ that factors over $F$ is bounded. In [3], the property $\mathcal{BF}$ is characterized not only for triples of Köthe spaces but also for the general case of Fréchet spaces. Our aim here to extend some results in [3] to $\ell$-Köthe space case.

2. BOUNDED OPERATORS TO $\ell$-KOTHE SPACES

If we follow the steps of Crone and Robinson Theorem [1], we obtain the following.

Lemma 2.1**.**

$T\in\mathcal{L}(\lambda(A),{\lambda}^{\ell}(B))$* iff $\forall m$, $\exists k$ such that*

[TABLE]

Proof.

$T\in\mathcal{L}(\lambda(A),\lambda^{\ell}(B))$ iff $\forall m$, $\exists k$ such that

[TABLE]

For $x=e_{n}$, we obtain the result.

Conversely, suppose that $\forall m$, $\exists k$ such that

[TABLE]

Let $x\in\lambda(A)$.

[TABLE]

So, $T\in\mathcal{L}(\lambda(A),\lambda^{\ell}(B))$. ∎

Notice that when domain is $\ell$-Köthe space, we can not use this argument.

Our first result is the following.

Theorem 2.2**.**

*The following are equivalent:

i) $(\lambda(A),{\lambda}^{\ell}(B))\in\mathcal{B}$

ii) for every sequence $N(k)$, there is $N\in\mathbb{N}$ such that for each $r\in\mathbb{N}$ we have $k_{0}\in\mathbb{N}$ and $C>0$ with*

[TABLE]

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

Proof.

Suppose that $(\lambda(A),{\lambda}^{\ell}(B))\in\mathcal{B}$. Consider $S:\lambda(A)\longrightarrow\lambda^{\ell}(B)$ with $S=e_{i}{{}^{\prime}}\otimes e_{v}$ where $e_{i}{{}^{\prime}}(x)=x_{i}$ for all $x\in\lambda(A).$ Since $S$ is the operator of rank one, we note that

[TABLE]

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

For the converse, let $\displaystyle Te_{i}=\sum_{v=1}^{\infty}u_{vi}e_{v}$. Since $T$ is continuous, by Lemma 2.1, there is $N(k)$ such that

[TABLE]

So we find $N\in\mathbb{N}$ such that

[TABLE]

Therefore, $T$ is bounded. ∎

Now, consider the $\ell$-Köthe space $\lambda^{\ell}(A)$ and any Fréchet space $E$. Then, we obtain the following.

Theorem 2.3**.**

*The following are equivalent:

i) $(E,\lambda^{\ell}(A))\in\mathcal{B}$

ii) for every sequence $N(k)$, there is $N\in\mathbb{N}$ such that for each $r\in\mathbb{N}$ we have $k_{0}\in\mathbb{N}$ and $C>0$ with*

[TABLE]

for all $v\in\mathbb{N}$, $u\in E{{}^{\prime}}$.

Proof.

Suppose that $(E,\lambda^{\ell}(A))\in\mathcal{B}$. Similar to the proof of Theorem 2.2, consider the operator of rank one $A=y\otimes e_{v}$ where $y\in E{{}^{\prime}}$. The result follows from (1.1).

For the converse, let $T:E\longrightarrow\lambda^{\ell}(A)$ be continuous linear operator. Let

[TABLE]

where $u_{v}=e_{v}{{}^{\prime}}\circ T.$

Then, by continuity we find $N(k)$ such that

[TABLE]

Let $\left|u_{v}(x)\right|=\theta_{v}u_{v}(x)$ where $\theta_{v}=\pm 1$ and $\beta_{v}\theta_{v}=\alpha_{v}$, note that

[TABLE]

Hence $T$ is bounded. ∎

3. BOUNDED FACTORIZATION PROPERTY FOR $\ell$-KOTHE SPACES

We need the following theorem [3, Theorem 2.2].

Theorem 3.1**.**

For Fréchet spaces $E,F$ and $G$ we have $(E,F,G)\in\mathcal{BF}$ if and only if for every sequence $N(k)$ there is $N\in\mathbb{N}$ such that for each $r\in\mathbb{N}$ we have $k_{0}=k_{0}(r)\in\mathbb{N}$ and $C=C(r)>0$ so that the following inequality

[TABLE]

is satisfied for every $R\in L(F,G)$, $S\in L(E,F)$ where $T=RS$.

The next result is obtained by following the lines of [3, Corollary 3.1].

Theorem 3.2**.**

Let $E$ be a Fréchet space, $\lambda^{\ell}(B),\lambda^{\tilde{\ell}}(C)$ be ${\ell}$-Köthe spaces and $\lambda^{\ell}(B)$ be nuclear. Then $(E,\lambda^{\ell}(B),\lambda^{\tilde{\ell}}(C))\in\mathcal{BF}$ if and only if for every sequence $N(k)$ there is $N\in\mathbb{N}$ such that for each $r\in\mathbb{N}$ we have $k_{0}\in\mathbb{N}$ and $C>0$ with

[TABLE]

for all $i\in\mathbb{N},j\in\mathbb{N}$ and $u\in E{{}^{\prime}}.$

Proof.

Let $S=u\otimes e_{i}$ and $R=e_{i}{{}^{\prime}}\otimes e_{j}$ where $u\in E{{}^{\prime}}$. Then $RS:E\longrightarrow G$ is the operator of rank one which sends each $x\in E$ to $u(x)e_{j}$. If we apply Theorem 3.1 we obtain the result.

For sufficiency, we take $S\in L(E,\lambda^{\ell}(B))$, $R\in L(\lambda^{\ell}(B),\lambda^{\tilde{\ell}}(C))$ and $T=RS$. Since $\lambda^{\ell}(B)$ is nuclear, $\exists S(k)$ such that $S(k)>N(k)$ and

[TABLE]

We can write $\displaystyle Sx=\sum_{i}u_{i}(x)e_{i}$ where $u_{i}=e_{i}{{}^{\prime}}\circ S\in E{{}^{\prime}}$ and $\displaystyle Re_{i}=\sum_{j}r_{ji}e_{j}.$ Therefore

[TABLE]

For this $S(k)$ we choose $N\in\mathbb{N}$ such that for each $r\in\mathbb{N}$ we obtain $k\in\mathbb{N}$ and $C>0$ with

[TABLE]

for all $i\in\mathbb{N},j\in\mathbb{N},u_{i}\in E{{}^{\prime}}.$

Since all types nuclear Köthe spaces determined by one and the same matrix $B$ coincide [2, Corollary 2, p.22]$,\lambda(B)=\lambda^{\ell}(B)$ and we have

[TABLE]

[TABLE]

Therefore, we have

[TABLE]

and

[TABLE]

(see [2, proof of Corollary 2, p.22]) Hence, by (3.2) we obtain that

[TABLE]

Therefore, $T$ is bounded. ∎

Recall that projective tensor product of two $l_{1}$-Köthe spaces $\lambda(A)$ and $\lambda(B)$ is isomorphic to $\lambda(D)$ where $d_{vz}^{k}=a_{v}^{k}b_{z}^{k}.$

Theorem 3.2 enables us to get:

Theorem 3.3**.**

Suppose $(E,\lambda^{\ell_{1}}(A))\in\mathcal{B}$ and $(\lambda^{\ell_{2}}(B),\lambda^{\ell_{3}}(C))\in\mathcal{B}$ where $E$ is a Fréchet space, $\lambda^{\ell_{3}}(C)$ is an $\ell$-Köthe space, $\lambda^{\ell_{1}}(A)$ and $\lambda^{\ell_{2}}(B)$ are nuclear $\ell$-Köthe spaces. Then $(E,\lambda^{\ell_{1}}(A)\hat{\otimes}_{\pi}\lambda^{\ell_{2}}(B),\lambda^{\ell_{3}}(C))\in\mathcal{BF}$.

Proof.

Given $N(k)$ which is assumed to be non-decreasing. Since $\lambda^{\ell_{2}}(B)$ is nuclear and $(\lambda^{\ell_{2}}(B),\lambda^{\ell_{3}}(C))\in\mathcal{B}$, we obtain that $\lambda^{\ell_{2}}(B)=\lambda(B)$ and by Theorem 2.2, $\exists n\in\mathbb{N}$ such that $\forall r\in\mathbb{N}$ we have $k_{0}=k_{0}(r)\in\mathbb{N}$ and $C_{1}=C_{1}(r)>0$ with

[TABLE]

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

We then determine $S(k)$ such that $S(k)=N(n)$ if $1\leq k\leq n$ and $S(k)>N(k)$ if $n+1\leq k\leq s_{0}$. Since $(E,\lambda^{\ell_{1}}(A))\in\mathcal{B}$ by Theorem 2.3, for this $S(k)$, we find $m\in\mathbb{N}$ such that $\forall q\in\mathbb{N}$ we have $s_{0}=s_{0}(q)$ and $C_{2}=C_{2}(q)$

[TABLE]

Therefore, for this $N(k)$ we have $\tilde{s_{0}}=s_{0}(N(k))$ and $\tilde{C_{2}}=C_{2}(N(k))$ with

[TABLE]

Let $s=\max\left\{k_{0},\tilde{s_{0}}\right\}$ and $C=C(r)=C_{1}\tilde{C_{2}}$. We have proved that $\exists m\in\mathbb{N}$ such that $\forall r\in\mathbb{N}$ we have $s\in\mathbb{N}$ and $C>0$ with

[TABLE]

for all $j\in\mathbb{N},v\in\mathbb{N},i\in\mathbb{N}$ and $u\in E{{}^{\prime}}.$

If $\lambda(A)^{\ell_{1}}$ and $\lambda^{\ell_{2}}(B)$ are nuclear $\ell$-Köthe spaces, then

[TABLE]

is nuclear where $d_{vi}^{k}=a_{v}^{k}b_{i}^{k}$ [2, Corollary 2, p.22].

By Theorem 3.2 we obtain that $(E,\lambda^{\ell_{1}}(A)\hat{\otimes}_{\pi}\lambda^{\ell_{2}}(B),\lambda^{\ell_{3}}(C))\in\mathcal{BF}$. ∎