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# Operators constructed by means of basic sequences and nuclear matrices

*Advances in Difference Equations*
**volume 2019**, Article number: 504 (2019)

## Abstract

In this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space \(\ell _{1}\) of all absolutely summable sequences. Examples of nuclear operators over the space \(\ell _{1}\) are given and used to construct operators over general Banach spaces with specific approximation numbers.

## Introduction and basic definitions

Banach spaces, which are separable and reflexive, can exist without a Schauder basis as proved by Enflo in 1973 [11]. However, in 1972, Morrell and Retherford [8] showed that in each infinite-dimensional Banach space and for any sequence of positive numbers, that is, monotonically convergent to zero \((\lambda _{i})_{i\in N}\), where \(N=\{1,2,3,\ldots \}\), one can construct a weakly square-summable basic sequence whose norms equal to \((\lambda _{i})_{i \in N}\).

In 1977, Makarov and Faried [7] showed how to construct compact operators of the form \(\sum_{i\in N} \mu _{i}f_{i}\otimes x_{i}\) between arbitrary infinite-dimensional Banach spaces such that its sequence of approximation numbers has a specific rate of convergence to zero. It was also proved that the operator ideal, whose sequence of approximation numbers are *p*-summable, is a small ideal; see [4, 10, 11].

In this work, we show how to construct compact operators between arbitrary infinite-dimensional Banach spaces using a countable number of basic sequences and nuclear operators, represented in the form of an infinite-dimensional matrix \((\mu _{ij})_{i,j\in N}\) defined over the space \(\ell _{1}\) of all absolutely summable sequences, which verifies

for every \(i\in N\). For such double-summation operators, a choice of matrix elements is more convenient than choosing sequence elements in the case of single-summation operators. Such a construction will help give counterexamples of operators between Banach spaces without a Schauder basis. An upper estimate of the sequence of approximation numbers is given for such double-summation operators. For basic notions and some related results, one can see [1, 6, 9, 13].

The following notations are used throughout this study. The normed space of bounded linear operators from a normed space *X* into a normed space *Y* is denoted by \(L(X, Y)\), while the dual space of the normed space *X* is denoted by \(X^{*}=L(X, R)\), where *R* is the set of real numbers.

Also as mentioned before, the space \(\{x=(x_{i})_{i=1}^{\infty }:\sum_{i}|x_{i}|^{p} <\infty \}\) of all sequences of real numbers that are *p*-absolutely summable, is denoted by \(\ell _{p}\), which is equipped with the norm \(\|x\|=(\sum_{i\in N}|x_{i}|^{p})^{\frac{1}{p}}\). The space \(\{x=(x_{i})_{i=1}^{\infty }: \lim x_{i}=0\}\) of all sequences of real numbers that are convergent to zero, is denoted by \(c_{o}\), which is equipped with the norm \(\|x\|=\sup_{i\in N}|x_{i}|\).

### Definition 1.1

([12])

A map *s*, which assigns a unique sequence \(\{s_{r}(T)\}_{r=0}^{ \infty }\) of real numbers to every operator \(T\in {L(X,Y)}\), is called an *s*-number sequence if the following conditions are verified:

- 1.
\(\|T\|=s_{0}(T)\geq s_{1}(T)\geq \cdots \geq 0\) for \(T\in L(X,Y)\).

- 2.
\(s_{r+m}(U+V)\leq s_{r}(U)+s_{m}(V)\) for \(U,V\in L(X,Y)\).

- 3.
\(s_{r}(UTV)\leq \|U\|s_{r}(T)\|V\|\) for \(V\in L(X_{0},X), T \in L(X,Y)\) and

\(U\in L(Y,Y_{0})\).

- 4.
\(s_{r}(T)=0\) if and only if \(\operatorname{rank}(T)\leq r\) for \(T\in L(X,Y)\).

- 5.
\(s_{r}(I_{k})=\bigl\{ \begin{array}{l@{\quad}l} 1, & \text{for }r< k; \\ 0, & \text{for }r\geq k, \end{array} \)

where \(I_{k}\) is the identity operator on Euclidean space \(\ell _{2} ^{k}\).

As an examples of *s*-numbers, we mention the approximation numbers \(\alpha _{r}(T)\), Gelfand numbers \(c_{r}(T)\), Kolmogorov numbers \(d_{r}(T)\), and Tikhomirov numbers \(d_{r}^{*}(T)\), defined by

- 1.
\(\alpha _{r}(T)=\inf \{\|T-A\|: A\in L(X,Y)\) and \(\operatorname{rank}(A)\leq r\}\). Clearly, we always have \(\|T\|=\alpha _{0}(T)\geq \alpha _{1}(T)\geq \alpha _{2}(T)\geq \cdots \geq 0\).

- 2.
\(c_{r}(T)=\alpha _{r}(J_{Y}T)\), where \(J_{Y}\) is a metric injection from the space

*Y*into a higher space \(\ell ^{\infty }( \varLambda )\) of all bounded-real functions for a suitable index set*Λ*. - 3.
$$\begin{aligned} d_{r}(T)=\inf_{\operatorname{dim}K\leq r} \sup_{ \Vert x \Vert \leq 1} \inf_{y\in K} \Vert Tx-y \Vert , \end{aligned}$$
where \(K\subseteq Y\).

- 4.
\(d_{r}^{*}(T)=d_{r}(J_{Y}T)\).

### Definition 1.2

([11])

An operator \(T\in L(X,Y)\) is nuclear if and only if it can be represented in the form

with \(a_{1}, a_{2},\ldots \in X^{*}\) and \(y_{1}, y_{2}, \ldots \in Y\), such that

On the class \(N(X,Y)\) of all nuclear operators from *X* into *Y*, a norm \(\nu (T)\) is defined by

where the inf is taken over all possible representations of the operator *T*.

## Basic theorems and technical lemmas

It is well known that an infinite matrix defines a linear continuous operator from the space \(\ell _{1}\) into itself if its columns are absolutely uniformly-summable; see [3, 4, 10].

### Lemma 2.1

([11], 6.3.6)

*An operator*
\(T\in L(\ell _{1},\ell _{1})\)
*is nuclear if and only if there is an infinite matrix*
\((\sigma _{ik})_{i,k\in N}\)
*such that*

*and*

*In this case*

### Lemma 2.2

([3])

*If*
\((T_{i})_{i=1}^{\infty }\)*is an absolutely summable sequence of bounded linear operators then*

*where the* inf *is taken over all possible representations for*

The following is a consequence of Lemma 2 in [2].

### Theorem 2.3

*Let*
\((x_{i})_{i=1}^{\infty }\)*be a sequence in a Banach space**X**such that*

*then the series*
\(\sum_{i=1}^{\infty }\lambda _{i}x_{i}\)*converges unconditionally in**X**for every sequence*
\((\lambda _{i})_{i=1}^{ \infty }\in c_{o}\).

### Theorem 2.4

(Morrell and Retherford [8])

*Let**X**be an infinite*-*dimensional Banach space and let*
\((\lambda _{i})_{i=1} ^{\infty }\in c_{o}\)*with*
\(0<\lambda _{i}<1\), *then there is a basic sequence*
\((x_{i})_{i=1}^{\infty }\)*in**X**such that*
\(\|x_{i}\|=\lambda _{i}\)*for all*
\(i=1,2,\ldots \)*that verifies*

### Remark 2.5

Theorem 2.4 is valuable in the case of sequences that are slowly convergent to zero \((\lambda _{i})_{i=1}^{\infty }\). Indeed, if \((\lambda _{i})_{i=1}^{\infty }\) converges rapidly to zero then \(\sum_{i=1}^{\infty }\|x_{i}\|<\infty \) and hence, one can write

### Theorem 2.6

(Dini’s theorem [5])

*For a convergent series*
\(\sum_{i=1}^{\infty }a_{i}\)*of positive real numbers*, *the series*

*where*
\(R_{i}=\sum_{j=i}^{\infty }a_{j}\)*is the remainder of the series*
\(\sum_{i=1}^{\infty }a_{i}\).

### Theorem 2.7

([7])

*Let**X**and**Y**be infinite*-*dimensional Banach spaces and let*
\((\lambda _{r})_{r=1}^{\infty }\)*be a monotonically decreasing sequence of positive real numbers*, *then there is a completely continuous operator*
\(A\in L(X,Y)\)*verifying*

### Lemma 2.8

([3])

*Let*
\(\{\xi _{i}\}_{i\in N}\)*be a bounded family of real numbers and let*
\(K\subseteq N\)*be an arbitrary subset of indices*, *such that* card*K**is the number of elements in**K*. *Then*

## Main results

### Proposition 3.1

*Let**X**and**Y**be infinite*-*dimensional Banach spaces and let*
\(M=(\mu _{ij})_{i,j\in N}\)*be an infinite matrix verifying that*:

- 1.
\(\lim_{j}\mu _{ij}=0 \)

*for every*\(i\in N\). - 2
\(\sum_{i=1}^{\infty }\sup_{j=1}^{\infty } \vert \mu _{ij} \vert <\infty\).

*Let*
\((f_{ij})_{i,j\in N}\)*be a matrix of functionals in*
\(X^{*}\)*and*
\((z_{ij})_{i,j\in N}\)*be a matrix of elements in**Y**that verifies*

*for every**F**in*
\(Y^{*}\)*and every**x**in**X*. *Then the expression*

*defines a linear continuous operator from**X**into**Y*.

### Proof

Let

then from Dini’s theorem 2.6 we get

From condition (1) and Theorem 2.3, the formula

defines a linear continuous operator \(T_{i}\in L(X,Y)\) for every \(i=1,2,\ldots \) .

Now we need to prove the unconditional convergence of the series

In order to do so, it is enough to apply again Theorem 2.3, noting that \(\lambda _{n}\rightarrow 0\) and we only have to verify that

In fact,

Then the expression

defines a linear continuous operator from *X* into *Y*. □

### Remark 3.2

From Theorem 2.4 and for every \(i=1,2,\ldots \) , there exist a basic sequence of functionals \(\{f_{ij}\}_{j=1}^{\infty }\) in \(X^{*}\) and a basic sequence of elements \(\{z_{ij}\}_{j=1}^{\infty }\) in *Y* such that

and

Basic sequences can be found by choosing different convergent to zero sequences \((\lambda _{i})_{i=1}^{\infty }\in c_{o}\), as mentioned in Theorem 2.4, according to their rate of convergence.

As a consequence of Proposition 3.1 and Remark 3.2 we get the following result.

### Theorem 3.3

*Let**X**and**Y**be Banach spaces and let*
\(\{f_{ij}\}_{j=1}^{\infty }\)*and*
\(\{z_{ij}\}_{j=1}^{\infty }\), *where*
\(i\in N\), *be basic sequences in*
\(X^{*}\)*and**Y*, *respectively*. *Verifying the following*,

- 1.
\(\sum_{j=1}^{\infty } \vert f_{ij}(x) \vert ^{2}< \Vert x \Vert ^{2}\)

*for every*\(x\in X\),*and*\(i\in N\). - 2.
\(\sum_{j=1}^{\infty } \vert F(z_{ij}) \vert ^{2}< \Vert F \Vert ^{2}\)

*for every*\(F\in Y^{*}\)*and*\(i\in N\),*then every nuclear operator*$$\begin{aligned} M=\{\mu _{ij}\}:\ell _{1}\rightarrow \ell _{1}, \quad\textit{with } \lim_{j}\mu _{ij}=0, \end{aligned}$$*defines an operator*\(T:X\rightarrow Y\)*of the form*$$\begin{aligned} T(x)=\sum_{i=1}^{\infty }\sum _{j=1}^{\infty }\mu _{ij} f_{ij}(x) z _{ij}. \end{aligned}$$

### Proof

The proof follows directly from Proposition 3.1 and Remark 3.2. □

### Theorem 3.4

*Let**X**and**Y**be infinite*-*dimensional Banach spaces and let*
\(\{\mu _{i}\}_{i=1}^{\infty }\)*be a sequence of real numbers that is convergent to zero and*
\(\{f_{i}\}_{i=1}^{\infty }\), \(\{z_{i}\}_{i=1} ^{\infty }\)*be sequences in*
\(X^{*}\)*and**Y*, *respectively*. *Verifying that*

*and*

*Then for the operator*

*we have*

*where**K**is any subset of the index set**N**with*
\(\operatorname{card} K \leq n\).

### Proof

For every operator \(T\in L(X,Y)\) and every subset of indices \(K\subset N\) with \(\operatorname{card} K\leq n\), we define a finite rank operator

with \(\operatorname{rank}(A_{K})\leq n\). From the definition of approximation numbers we get

Since this relation is true for every index subset *K* with \(\operatorname{card} K\leq n\),

□

### Remark 3.5

As a consequence of Theorem 3.4 and by using Lemma 2.8, we can get the following similar result:

### Theorem 3.6

*Let**X**and**Y**be infinite*-*dimensional Banach spaces and let*
\((\mu _{ij})_{i,j\in N}\)*be an infinite matrix with linearly independent rows such that conditions of Proposition *3.1*are verified*, *and let*
\(\{f_{ij}\}_{j=1}^{\infty }\), \(\{z_{ij}\}_{j=1}^{\infty }\)*for*
\(i=1,2,\ldots \) , *be sequences in*
\(X^{*}\)*and**Y*, *respectively*, *such that conditions of Theorem *3.4*are fulfilled for all*
\(i=1,2,\ldots \) . *Then for the operator*

*we have*

*where**K**is a subset of the index set**N**with*
\(\operatorname{card} K \leq n_{i}\).

### Proof

From Lemma 2.2, Theorem 3.4 and by using the same operator \(T_{i}\) defined by Eq. (2) throughout the proof of Proposition 3.1, we get

This relation is true for every \(\varSigma n_{i}=n\), then we get the proof.

In the following, we are going to give two examples of nuclear operators over \(\ell _{1}\) and use them to construct operators over general Banach spaces with specific approximation numbers. □

### Example 3.7

Consider the operator \(A\in L(c_{0},\ell _{1})\) such that \(A=(a_{ij})_{i,j=1} ^{\infty }\), where

Also, consider \(B\in L(\ell _{1},c_{0})\), such that

where

Thus we have \(D=AB\in L(\ell _{1},\ell _{1})\), such that

where

Let \(D=(\mu _{ij})_{i,j=1}^{\infty }\), then this operator has the following properties:

- 1.
$$\begin{aligned} \sum_{i=1}^{\infty } \vert \mu _{ii} \vert &=1+\biggl(\frac{1}{8}+\frac{1}{8} \biggr)+\biggl( \frac{1}{36}+\frac{1}{36}+\frac{1}{36}+ \frac{1}{36}\biggr)+\biggl(\frac{1}{128}+ \frac{1}{128}+ \cdots \biggr)+\cdots \\ &=\sum_{i=1}^{\infty }\frac{1}{i^{2}}= \frac{\pi ^{2}}{6}. \end{aligned}$$
- 2.
$$\begin{aligned} \nu (D)=\sum_{i=1}^{\infty }\sup _{j} \vert \mu _{ij} \vert = \frac{\pi ^{2}}{6}< \infty, \end{aligned}$$
then by using Lemma 2.1

*D*is a nuclear operator. - 3.
\(\operatorname{Trac}(D)=1+(\frac{1}{8}-\frac{1}{8})+( \frac{1}{36}-\frac{1}{36}+\frac{1}{36}-\frac{1}{36})+(\frac{1}{128}- \frac{1}{128}+\cdots )+\cdots =1\).

- 4.
\(D=(\mu _{ij})_{i,j=1}^{\infty }\) is having linearly independent rows.

Now, for \(D=(\mu _{ij})_{i,j=1}^{\infty }\) and by using Proposition 3.1 and Theorem 3.6 one can construct an operator \(T\in L(X,Y)\) for any Banach spaces \(X,Y\) of the form

where \(\{f_{ij}\}_{i,j=1}^{\infty }\), \(\{z_{ij}\}_{i,j=1}^{\infty }\), are basic sequences in \(X^{*}\) and *Y*, respectively, such that conditions of Theorem 3.4 are fulfilled for all \(i=1,2,\ldots \) .

Now by applying Eq. (3), one can get

Hence, we have

which is consistent with the properties of the approximation numbers.

By applying Eq. (3) in the case of \(n=0\), we get

### Example 3.8

Consider the operator \(J\in L(\ell _{1},\ell _{1})\) such that \(J=(\lambda _{ij})_{i,j=1}^{\infty }\) where \(\lambda _{ij}= \frac{ij}{2^{i+j}}\), then this operator has the following properties:

- 1.
\(\nu (J)=\sum_{i=1}^{\infty }\sup _{j} \vert \lambda _{ij} \vert =\sum _{i=1}^{\infty }\frac{i}{2^{i}}\sup _{j}(\frac{j}{2^{j}})=1<\infty,\) then by using Lemma 2.1

*J*is a nuclear operator. - 2.
\(J=(\lambda _{ij})_{i,j=1}^{\infty }\) has linearly independent rows.

Now for \(J=(\lambda _{ij})_{i,j=1}^{\infty }\) and by using Proposition 3.1 and Theorem 3.6, one can construct an operator \(T\in L(X,Y)\) for any Banach spaces \(X,Y\) on the form,

where \(\{f_{ij}\}_{i,j=1}^{\infty }\) and \(\{z_{ij}\}_{i,j=1}^{\infty }\) are basic sequences in \(X^{*}\) and *Y*, respectively, such that conditions of Theorem 3.4 are fulfilled for all \(i=1,2,\ldots \) .

Applying Eq. (3) yields

Thus, we have \((\alpha _{n}(T))_{n=1}^{\infty }\in \ell _{1}\) because

Applying Eq. (3) in the case of \(n=0\) yields

noting that this is independent of the selection of \(\{f_{ij}\}_{i,j=1} ^{\infty }\) and \(\{z_{ij}\}_{i,j=1}^{\infty }\).

If we choose \(\{f_{ij}\}_{i,j=1}^{\infty }\) and \(\{z_{ij}\}_{i,j=1} ^{\infty }\) such that

then we get

which means that *T*, in this case, is a nuclear operator.

## Conclusion

By using nuclear operators defined over \(\ell _{1}\) with particular representation, one can construct compact operators over general Banach spaces with specific approximation numbers. Such compact operators are been constructed using a countable number of basic sequences and nuclear operators. For such nuclear operators, its construction in a matrix form will yield to double-summation operators. This double-summation gives more freedom rather than choosing sequence elements in the case of single-summation operators. Such a construction will help give counterexamples of operators between Banach spaces without a Schauder basis.

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### Acknowledgements

The authors would like to thank the reviewers for valuable comments and suggestions which helped improving this work.

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## Funding

This project was supported by the Deanship of scientific research at Prince Sattam Bin Abdulaziz University under the research project 2017/01/7606.

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### Cite this article

Morsy, A., Faried, N., Harisa, S.A. *et al.* Operators constructed by means of basic sequences and nuclear matrices.
*Adv Differ Equ* **2019**, 504 (2019). https://doi.org/10.1186/s13662-019-2445-1

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DOI: https://doi.org/10.1186/s13662-019-2445-1

### MSC

- 47B06
- 47B38
- 47B99
- 39B42
- 47A56

### Keywords

- Nuclear operators
*s*-numbers- Schauder basis
- Basic sequence