Operators Between Sequence Spaces and Applications

By Bruno de Malafosse, Eberhard Malkowsky and Vladimir Rakočević

This book presents modern methods in functional analysis and operator theory along with their applications in recent research. The book also deals with the solvability of infinite systems of linear equations in various sequence spaces. It uses the classical sequence spaces, generalized Cesaro and difference operators to obtain calculations and simplifications of complicated spaces involving these operators. In order to make it self-contained, comprehensive and of interest to a larger mathematical community, the authors have presented necessary concepts with results for advanced research topics. This book is intended for graduate and postgraduate students, teachers and researchers as a basis for further research, advanced lectures and seminars.

• Language: English
• Publisher: Springer
• Release date: Jan 19, 2021
• ISBN: 9789811597428

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© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021

B. de Malafosse et al.Operators Between Sequence Spaces and Applicationshttps://doi.org/10.1007/978-981-15-9742-8_1

1. Matrix Transformations and Measures of Noncompactness

Bruno de Malafosse¹  , Eberhard Malkowsky² and Vladimir Rakočević³

(1)

University of Le Havre (LMAH), Ore, France

(2)

Faculty of Management, Univerzitet Union Nikola Tesla, Beograd, Serbia

(3)

Department of Mathematics, University of Niš, Niš, Serbia

Bruno de Malafosse

Email: [e-mail]

The major part of this chapter is introductory and included as a reference for the reader’s convenience; it recalls the concepts and results from the theories of sequence spaces, matrix transformations in Sects. 1.1–1.3 and 1.5 and measures of noncompactness in Sects. 1.7–1.10 that are absolutely essential for the book. Although the results of this chapter may be considered as standard in the modern theories of matrix transformations, we included their proofs with the exception of those in Sect. 1.4 on the relations between various kind of duals; these results are mainly included for the sake of completeness and not directly used in the remainder of the book.

We refer the reader interested in matrix transformations to [1, 5, 14, 19, 23, 25, 27, 33–37] and the survey paper [29], and, in measures of noncompactness, to [2–4, 10, 12, 21–23, 30]. Although the concepts and results of this chapter are standard, we decided to cite from [23] in almost all cases, if necessary.

Sections 1.3 and 1.5 contain results that are less standard. They concern the characterizations of matrix transformations from arbitrary FK spaces into the spaces $$c_{0}$$ , c, $$\ell _{\infty }$$ and $$\ell _{1}$$ of all null, convergent and bounded sequences and of all absolutely convergent series, and all known characterizations of classes matrix transformations between the classical sequence spaces $$c_{0}$$ , c, $$\ell _{\infty }$$ and

$$\ell _{p}=\{x=(x_{k}):\sum _{k}|x_{k}|^{p}<\infty \}$$

for

$$1\le p<\infty $$

. For completeness sake, we also prove Crone’s theorem which characterizes the class of all matrix transformations of $$\ell _{2}$$ into itself in Sect. 1.6.

The underlying fundamental concepts for the theories of sequence spaces and matrix transformations are those of linear metric and paranormed spaces of FK, BK and AK spaces, and of various kinds of dual spaces. The general results related to these concepts are used in the characterizations of matrix transformations between the classical sequence spaces. By this, we mean to give necessary and sufficient conditions on the entries of an infinite matrix to map one classical sequence space into another.

We also present an axiomatic introduction of measures of noncompactness on bounded sets of complete metric spaces, recall the definitions and essential properties of the Kuratowski and Hausdorff measures of noncompactness, which are the most prominent measures on noncompactness, and the famous theorem by Goldenštein, Go’hberg and Markus, which gives an estimate for the Hausdorff measure of noncompactness of bounded sets in Banach spaces with a Schauder basis. Finally, we recall the definition of the measure of noncompactness of operators and list some important related properties.

1.1 Linear Metric and Paranormed Spaces

Here, we recall the concepts of linear metric and paranormed spaces, which are fundamental in the theory of sequence spaces and matrix transformations.

The concept of a linear or vector space involves an algebraic structure given by the definition of two operations, namely, the sum of any two of its elements, also called vectors, and the product of any scalar with any vector. It is clear that the set $$\omega $$ of all complex sequences

$$x=(x_{k})_{k=0}^{\infty }$$

 is a linear space with respect to the addition and scalar multiplication defined termwise, that is,

$$\begin{aligned} x+y=(x_{k}+y_{k})_{k=0}^{\infty } \text{ and }&\lambda x=(\lambda x_{k})_{k=0}^{\infty }\\&\text{ for } \text{ all } x=(x_{k})_{k=0}^{\infty }, y=(y_{k})_{k=0}^{\infty }\in \omega \text{ and } \text{ all } \lambda \in \mathbb {C}. \end{aligned}$$

On the other hand, a topological structure of a set may be given by a metric. For instance, $$\omega $$ is a metric space with its metric d defined by

$$\begin{aligned} d(x,y)=\sum \limits _{k=0}^{\infty }\dfrac{1}{2^ {k}}\,\dfrac{|x_{k}-y_{k}|}{1+|x_{k}-y_{k}|} \text{ for } \text{ all } x,y\in \omega . \end{aligned}$$

(1.1)

If a set is both a linear and metric space, then it is natural to require the algebraic operations to be continuous with respect to the metric. The continuity of the algebraic operations in a linear metric space (X, d) means the following: If $$(x_{n})$$ and $$(y_{n})$$ are sequences in X and $$(\lambda _{n})$$ is a sequence of scalars with $$x_{n}\rightarrow x$$ , $$y_{n}\rightarrow y$$ and $$\lambda _{n}\rightarrow \lambda $$ $$(n\rightarrow \infty )$$ , then it follows that

$$x_{n}+y_{n}\rightarrow x+y$$

and

$$\lambda _{n}x_{n}\rightarrow \lambda x$$

$$(n\rightarrow \infty )$$ .

A complete linear metric space is called a Fréchet space. Unfortunately, this terminology is not universally agreed on. Some authors call a complete linear metric space an F space, and a locally convex F space a Fréchet space (e.g. [28, p. 8] or [15, p. 208]) which Wilansky calls an F space. We follow Wilansky’s terminology of a Fréchet space being a complete linear metric space [31–33]. We will see later that $$\omega $$ is a Fréchet space with the metric defined in (1.1).

The concept of a paranorm is closely related to linear metric spaces. It is a generalization of that of absolute value. The paranorm of a vector may be thought of as its distance from the origin. We recall the definition of a paranormed space for the reader’s convenience.

Definition 1.1

Let X be a linear space.

(a) A function

$$p:X\rightarrow \mathbb {R}$$

is called a paranorm, if

$$\begin{aligned}&p(0)=0, \end{aligned}$$

(P.1)

$$\begin{aligned}&p(x)\ge 0 \text{ for } \text{ all } x\in X, \end{aligned}$$

(P.2)

$$\begin{aligned}&p(-x)=p(x) \text{ for } \text{ all } x\in X, \end{aligned}$$

(P.3)

$$\begin{aligned}&p(x+y)\le p(x)+p(y) \text{ for } \text{ all } x,y\in X\,{ (triangle \, inequality)} \end{aligned}$$

(P.4)

$$\begin{aligned}&\text{ if } \,(\lambda _{n})\, \text{ is } \text{ a } \text{ sequence } \text{ of } \text{ scalars } \text{ with } \, \lambda _{n}\rightarrow \lambda \, (n\rightarrow \infty )\, \text{ and }\, (x_{n})\, \text{ is } \text{ a }\\&\text{ sequence } \text{ of } \text{ vectors } \text{ with }\, p(x_{n}-x)\rightarrow 0\, (n\rightarrow \infty )\, \text{ then } \text{ it } \text{ follows } \text{ that }\\&p(\lambda _{n}x_{n}-\lambda x)\rightarrow 0\, (n\rightarrow \infty )\, ( continuity \, of \, multiplication \, by \, scalars). \end{aligned}$$

(P.5)

If p is a paranorm  on X, then (X, p), or X for short, is called a paranormed space. A paranorm p for which $$p(x)=0$$ implies $$x=0$$ is called total.

(b) For any two paranorms p and q, p is called stronger than q if, whenever $$(x_{n})$$ is a sequence with

$$p(x_{n})\rightarrow 0$$

$$(n\rightarrow \infty )$$ , then also

$$q(x_{n})\rightarrow 0$$

$$(n\rightarrow \infty )$$ . If p is stronger than q, then q is said to be weakerthan p. If p is stronger than q and q is stronger than p, then p and q are called equivalent. If p is stronger than q, but p and q are not equivalent, then p is called strictly stronger than q, and q is called strictly weaker than p.

If p is a total paranorm for a linear space X, then it is easy to see that

$$d(x,y)=p(x-y)$$

$$(x,y\in X)$$ defines a metric on X, thus every totally paranormed space is a linear metric space. The converse is also true. The metric of any linear metric space is given by some total paranorm [31, Theorem 10.4.2, p. 183].

The next well-known result shows how a sequence of paranorms may be used to define a paranorm.

Theorem 1.1

Let $$(p_{k})_{k=0}^{\infty }$$ be a sequence of paranorms on a linear space X. We define the so-called Fréchet combination of $$(p_{k})_{k=0}^{\infty }$$ by

$$\begin{aligned} p(x)=\sum \limits _{k=0}^{\infty }\dfrac{1}{2^ {k}}\,\dfrac{p_{k}(x)}{1+p_{k}(x)} \text{ for } \text{ all } x\in X. \end{aligned}$$

(1.2)

Then, we have

(a)

p is a paranorm on X and satisfies

$$\begin{aligned} \begin{array}{c} p(x_{n})\rightarrow 0\,\,(n\rightarrow \infty )\\ \text{ if } \text{ and } \text{ only } \text{ if }\\ p_{k}(x_{n})\rightarrow 0\,\,(n\rightarrow \infty ) \text{ for } \text{ each } k. \end{array} \end{aligned}$$

(1.3)

(b)

p is the weakest paranorm stronger than every $$p_{k}$$ .

(c)

p is total if and only if the set

$$\{p_{k}:k=0,1,\dots \}$$

is total.

(We recall that a set $$\Phi $$ if functions from a linear space X to a linear space is said to be total if given

$$x\in X\setminus \{0\}$$

, there exists $$f\in \Phi $$ with $$f(x)\not =0$$ .)

We obtain as an immediate consequence of Theorem 1.1.

Corollary 1.1

The set $$\omega $$ is a complete, totally paranormed space with its paranorm p defined by

$$\begin{aligned} p(x)=\sum \limits _{k=0}^{\infty }\dfrac{1}{2^{k}}\,\dfrac{|x_{k}|}{1+|x_{k}|} \text{ for } \text{ all } x\in \omega . \end{aligned}$$

(1.4)

Thus, $$\omega $$ is a Fréchet space with its natural metric $$d_{\omega }$$ given by

$$\begin{aligned} d_{\omega }(x,y)=\sum \limits _{k=0}^{\infty }\dfrac{1}{2^{k}}\, \dfrac{|x_{k}-y_{k}|}{1+|x_{k}+y_{k}|} \text{ for } \text{ all } x,y\in \omega . \end{aligned}$$

(1.5)

Furthermore, convergence in $$(\omega ,d_{\omega })$$ and coordinatewise convergence are equivalent, that is,

$$ d_{\omega }(x^{(n)},x)\rightarrow 0\,\,(n\rightarrow \infty ) \text{ if } \text{ and } \text{ only } \text{ if } \lim _{n\rightarrow \infty }x_{k}^{(n)}=x_{k} \text{ for } \text{ each } k. $$

We close this section with an example we prove in detail for the reader’s convenience, since part of it will be used in Part (b) of Examples 1.2, 2.​3, and in Sect. 6,

Example 1.1

Let

$$p=(p_{k})_{k=0}^{\infty }$$

be a sequence of positive reals and

$$ \ell (p)=\left\{ x\in \omega :\sum \limits _{k=0}^{\infty }|x_{k}|^{p_{k}}<\infty \right\} ,\,\, c_{0}(p)=\left\{ x\in \omega :\lim \limits _{k\rightarrow \infty }|x_{k}|^{p_{k}}=0\right\} $$

and

$$ \ell _{\infty }(p)=\left\{ x\in \omega :\sup \limits _{k}|x_{}|^{p_{k}}<\infty \right\} . $$

(a) Then $$\ell (p)$$ , $$c_{0}(p)$$ and $$\ell _{\infty }(p)$$ are linear spaces if and only if the sequence p is bounded.

(b) Let the sequence p be bounded and

$$M=\max \{1,\sup _{k}p_{k}\}$$

. Then $$\ell (p)$$ and $$c_{0}(p)$$ are complete, totally paranormed spaces with their natural paranorms g and $$g_{0}$$ given by

$$\begin{aligned} g(x) =\left( \sum \limits _{k=0}^{\infty }|x_{k}|^{p_{k}}\right) ^{1/M} \text{ for } \text{ all } x\in \ell (p) \end{aligned}$$

and

$$\begin{aligned} g_{0}(x) = \sup \limits _{k}|x_{k}|^{p_{k}/M} \text{ for } \text{ all } x\in c_{0}(p), \end{aligned}$$

and g and $$g_{0}$$ are strictly stronger than the natural paranorm of $$\omega $$ on $$\ell (p)$$ and $$c_{0}(p)$$ , respectively.

But $$g_{0}$$ is a paranorm for $$\ell _{\infty }(p)$$ if and only if

$$\begin{aligned} m=\inf \limits _{k}p_{k}>0, \end{aligned}$$

(1.6)

in which case $$\ell _{\infty }(p)$$ reduces to the classical space $$\ell _{\infty }$$ of bounded complex sequences.

Proof

(a)

First we assume that the sequence p is bounded. We write X(p) for any of the sets $$\ell (p)$$ , $$c_{0}(p)$$ and $$\ell _{\infty }(p)$$ . Let

$$x,y\in X(p)$$

and $$\lambda \in \mathbb {C}$$ be given.

(a.i)

   First we show

$$x+y\in X(p)$$

.

Putting

$$\alpha _{k}=p_{k}/M\le 1$$

for all k, we obtain

$$|x_{k}+y_{k}|^{\alpha _{k}}\le |x_{k}|^{\alpha _{k}}+|y_{k}|^{\alpha _{k}}$$

by inequality (A.1).

If

$$X(p)=c_{0}(p)$$

or

$$X(p)=\ell _{\infty }(p)$$

, then

$$\begin{aligned} \sup \limits _{k}|x_{k}+y_{k}|^{\alpha _{k}}\le \sup \limits _{k}|x_{k}|^{\alpha _{k}}+\sup \limits _{k}|y_{k}|^{\alpha _{k}}, \end{aligned}$$

(1.7)

which implies

$$x+y\in X(p)$$

.

If

$$X(p)=\ell (p)$$

, we get applying Minkowski’s inequality (A.4)

$$\begin{aligned} \left( \sum \limits _{k=0}^{\infty }|x_{k}+y_{k}|^{p_{k}}\right) ^{1/M}&= \left( \sum \limits _{k=0}^{\infty }|x_{k}+y_{k}|^{\alpha _{k}M}\right) ^{1/M}\\&\le \left( \sum \limits _{k=0}^{\infty }\left( |x_{k}|^{\alpha _{k}}+|y_{k}|^{\alpha _{k}}\right) ^{M} \right) ^{1/M}\\&\le \left( \sum \limits _{k=0}^{\infty }|x_{k}|^{\alpha _{k}M}\right) ^{1/M}+ \left( \sum \limits _{k=0}^{\infty }|y_{k}|^{\alpha _{k}M}\right) ^{1/M}\\&= \left( \sum \limits _{k=0}^{\infty }|x_{k}|^{p_{k}}\right) ^{1/M}+ \left( \sum \limits _{k=0}^{\infty }|y_{k}|^{p_{k}}\right) ^{1/M}. \end{aligned}$$

Thus, we have shown

$$\begin{aligned} \left( \sum \limits _{k=0}^{\infty }|x_{k}+y_{k}|^{p_{k}}\right) ^{1/M}\le \left( \sum \limits _{k=0}^{\infty }|x_{k}|^{p_{k}}\right) ^{1/M}+ \left( \sum \limits _{k=0}^{\infty }|y_{k}|^{p_{k}}\right) ^{1/M}, \end{aligned}$$

(1.8)

and so

$$x+y\in \ell (p)$$

. This completes Part (a.i) of the proof.

(a.ii)

   Now we show

$$\lambda x\in X(p)$$

.

We put

$$\Lambda =\max \{1,|\lambda |^{M}\}<\infty $$

. Then $$|\lambda _{k}|^{p_{k}}\le \Lambda $$ for all k and so

$$ \sum \limits _{k=0}^{\infty }|\lambda x_{k}|^{p_{k}}\le \Lambda \sum \limits _{k=0}^{\infty }|x_{k}|^{p_{k}}<\infty $$

and

$$ \sup \limits _{k}|\lambda x_{k}|^{p_{k}}\le \Lambda \sup \limits _{k}|x_{k}|^{p_{k}}<\infty , $$

hence

$$\lambda x\in X(p)$$

. This completes Part (a.ii) of the proof.

Thus, we have shown that if the sequence p is bounded then $$\ell (p)$$ is a linear space.

To show the converse part, we assume that the sequence p is not bounded. Then there exists a subsequence $$(p_{k(j)})_{j=0}^{\infty }$$ of the sequence p such that

$$p_{k(j)}>j+1$$

for all j. We define the sequence

$$x=(x_{k})_{k=0}^{\infty }$$

by

$$ x_{k}= {\left\{ \begin{array}{ll} \dfrac{1}{(j+1)^{2/p_{k(j)}}} &{} \quad (k=k(j))\\ 0 &{} \quad (k\not =k(j)) \end{array}\right. }\quad (j=0,1,\dots ). $$

Then, we have

$$ \sum \limits _{k=0}^{\infty }|x_{k}|^{p_{k}}=\sum \limits _{j=0}^{\infty }\dfrac{1}{(j+1)^{2}}<\infty , $$

hence $$x\in \ell (p)$$ , but

$$ \sup \limits _{k}|2x_{k}|^{p_{k}}>\dfrac{2^{j+1}}{(j+1)^{2}} \text{ for } j=0,1,\dots , $$

that is,

$$2x\not \in \ell _{\infty }(p)$$

. This shows that if the sequence p is not bounded, then the spaces X(p) are not linear spaces. This completes the proof of Part (a).

(b)

Now let the sequence p be bounded.

(b.1)

First we show that g is a total paranorm for

$$X(p)=\ell (p)$$

and $$g_{0}$$ is a total paranorm for

$$X(p)=c_{0}(p)$$

.

We write $$g_{X(p)}=g$$ for

$$X(p)=\ell (p)$$

and $$g_{X(p)}=g_{0}$$ for

$$X(p)=c_{0}(p)$$

. Then, we obviously have

$$g_{X(p)}:X(p)\rightarrow \mathbb {R}$$

,

$$g_{X(p)}(0)=0$$

,

$$g_{X(p)}(x)\ge 0$$

,

$$g_{X(p)}(x)>0$$

implies $$x\not =0$$ and

$$g_{X(p)}(-x)=g_{X(p)}(x)$$

, and (1.8) and (1.7) are the triangle inequalities for $$\ell (p)$$ and $$c_{0}(p)$$ , respectively.

To show the condition in (P.5) of Definition 1.1, let $$(\lambda _{n})_{n=0}^{\infty }$$ be a sequence of scalars with $$\lambda _{n}\rightarrow \lambda $$ $$(n\rightarrow \infty )$$ and $$(x^{(n)})_{n=0}^{\infty }$$ be a sequence of elements

$$x^{(n)}\in X(p)$$

with

$$g_{X(p)}(x^{(n)}-x)\rightarrow 0$$

$$(n\rightarrow \infty )$$ . We observe that by (1.7) or (1.8)

$$\begin{aligned} g_{X(p)}\left( \lambda _{n}x^{(n)}-\lambda x\right)&\le g_{X(p)}\left( (\lambda _{n}-\lambda )(x^{(n)}-x)\right) \nonumber \\&+ g_{X(p)}\left( \lambda (x^{(n)}-x)\right) + g_{X(p)}\left( (\lambda _{n}-\lambda )x\right) . \end{aligned}$$

(1.9)

It follows from $$\lambda _{n}\rightarrow \lambda $$ $$(n\rightarrow \infty )$$ that

$$|\lambda _{n}-\lambda |<1$$

for all sufficiently large n, hence

$$ g_{X(p)}\left( (\lambda _{n}-\lambda )(x^{(n)}-x)\right) \le g_{X(p)}(x^{(n)}-x) \rightarrow 0\,\,(n\rightarrow \infty ). $$

Furthermore, we have

$$ g_{X(p)}\left( \lambda (x^{(n)}-x)\right) \le \Lambda g_{X(p)}(x^{(n)}-x)\rightarrow 0\,\, (n\rightarrow \infty ). $$

Therefore, the first two terms on the right in (1.9) tend to zero as n tends to infinity.

To show that the third term on the right also tend to zero as n tend to infinity, let $$\varepsilon >0$$ be given with $$\varepsilon <1$$ .

$$(\alpha )$$

   First, we consider the case of

$$X(p)=\ell (p)$$

.

Since $$x\in \ell (p)$$ , we can choose a non-negative integer $$k_{0}$$ such that

$$ \left( \sum \limits _{k=k_{0}+1}^{\infty }|x_{k}|^{p_{k}}\right) ^{1/M}&lt; \varepsilon /2. $$

Now we choose a non-negative integer $$n_{0}$$ such that

$$ |\lambda _{n}-\lambda |\le 1 \text{ and } \max \limits _{0\le k\le k_{0}}|\lambda _{n}-\lambda |^{p_{k}}&lt; \left( \dfrac{\varepsilon }{2(g(x)+1)}\right) ^{M} $$

for all $$n\ge n_{0}$$ . Then we obtain for all $$n\ge n_{0}$$

$$\begin{aligned}&amp;g\left( (\lambda _{n}-\lambda )x\right) = \left( \sum \limits _{k=0}^{\infty }|\lambda _{n}-\lambda |^{p_{k}}\,|x_{k}|^{p_{k}}\right) ^{1/M} \le \\&amp;\quad \left( \sum \limits _{k=0}^{k_{0}} |\lambda _{n}-\lambda |^{p_{k}}|x_{k}|^{p_{k}} \right) ^{1/M}+ \left( \sum \limits _{k=k_{0}+1}^{\infty }|\lambda _{n}-\lambda |^{p_{k}} |x_{k}|^{p_{k}} \right) ^{1/M}&lt;\\&amp;\dfrac{\varepsilon }{2(g(x)+1)}\left( \sum \limits _{k=0}^{\infty }|x_{k}|^{p_{k}}\right) ^{1/M}+ \left( \sum \limits _{k=k_{0}+1}^{\infty }|x_{k}|^{p_{k}}\right) ^{1/M} \le \varepsilon /2+\varepsilon /2=\varepsilon . \end{aligned}$$

Hence, the third term in (1.9) also tends to zero as n tends to infinity.

$$(\beta )$$

   Now we consider the case of

$$X(p)=c_{0}(p)$$

.

Since $$x\in c_{0}(p)$$ , we can choose a non-negative integer $$k_{0}$$ such that

$$ |x_{k}|^{\alpha _{k}}\le \varepsilon /2 \text{ for } \text{ all } k\ge k_{0}. $$

Now we choose a non-negative integer $$n_{0}$$ such that

$$ |\lambda _{n}-\lambda |\le 1 \text{ and } \max \limits _{0\le k\le k_{0}}|\lambda _{n}-\lambda |^{p_{k}}&lt; \left( \dfrac{\varepsilon }{2(g_{0}(x)+1)}\right) ^{M}. $$

Then we obtain for all $$n\ge n_{0}$$

$$ \left( |\lambda _{n}-\lambda |\cdot |x_{k}|\right) ^{\alpha _{k}}\le |x_{k}|^{\alpha _{k}} \text{ for } \text{ all } k\ge k_{0}+1 $$

and

$$ \left( |\lambda _{n}-\lambda |\cdot |x_{k}|\right) ^{\alpha _{k}}\le \dfrac{\varepsilon }{2(g_{0}(x)+1)}\cdot |x_{0}|^{\alpha _{k}} \text{ for } 0\le k\le k_{0}. $$

Hence, the third term in (1.9) also tends to zero as n tends to infinity.

Thus, we have shown that $$g_{X(p)}$$ is a total paranorm on X(p). This completes Part (b.1) of the proof.

(b.2)

   Now we show that convergence in X(p) is strictly stronger than coordinatewise convergence, when

$$X(p)=\ell (p)$$

or

$$X(p)=c_{0}(p)$$

.

(b.2.i)

   First, we show that

$$g_{X(p)}(x^{n}-x)\rightarrow 0$$

$$(n\rightarrow \infty )$$ implies $$x_{k}^{(n)}\rightarrow x_{k}$$ $$(n\rightarrow \infty )$$ for each k.

We fix k. Then, we have

$$ |x^ {(n)}-x_{k}|\le \left( g_{X(p)}(x^{(n)}-x)\right) ^{M/p_{k}} \text{ for } \text{ all } n=0,1,\dots $$

and so

$$g_{X(p)}(x^{n}-x)\rightarrow 0$$

$$(n\rightarrow \infty )$$ implies $$x_{k}^{(n)}\rightarrow x_{k}$$ $$(n\rightarrow \infty )$$ , hence convergence in the paranorm of X(p) implies coordinatewise convergence. This completes Part (b.2.i) of the proof.

(b.2.ii)

   Now we show that the converse is not true.

We define the sequence $$(x^{(n)})_{n=0}^{\infty }$$ by $$x^{(n)}=e^{(n)}$$

$$(n=0,1,\dots )$$

. Then obviously $$|x_{k}^{(n)}|=0$$ for all $$n>k$$ , hence

$$\lim _{n\rightarrow \infty }x_{k}^{(n)}=0$$

for each k, but for all distinct values of n and m, we have

$$g(x^{(n)}-x^{(m)})=2^{1/M}$$

and

$$g_{0}(x^{(n)}-x^{(m)})\ge 1$$

, hence $$(x^{(n)})_{n=0}^{\infty }$$ is not a Cauchy sequence, and so not convergent. This shows that coordinatewise convergence does not imply convergence in the paranorm of X(p), in general.

Applying Corollary 1.1, we conclude that the paranorm on X(p) is strictly stronger than that of $$\omega $$ on X(p). This completes Part (b.2.ii) of the proof.

This completes Part (b.2) of the proof.

(b.3)

   Now we show that X(p) is complete, when

$$X(p)=\ell (p)$$

or

$$X(p)=c_{0}(p)$$

.

Let $$(x^{(n)})_{n=0}^{\infty }$$ be a Cauchy sequence in X(p). Then $$(x_{k}^{(n)})_{n=0}^{\infty }$$ is a Cauchy sequence in $$\mathbb {C}$$ for each k, by Part (b.2.i) of the proof, hence convergent,

$$ x=\lim _{n\rightarrow \infty }x_{k}^{(n)}, \text{ say. } $$

(b.3.i)

   First we consider the case of

$$X(p)=\ell (p)$$

.

Let $$\varepsilon >0$$ be given and K be an arbitrary non-negative integer. Then there is a non-negative integer N such that

$$ \left( \sum \limits _{k=0}^{K} |x_{k}^{(n)}-x_{k}^{(m)}|^{p_{k}}\right) ^{1/M}\le g(x^{(n)}-x^{(m)})&lt;\varepsilon \text{ for } \text{ all }\, m,n\ge N $$

which implies

$$\begin{aligned} \left( \sum \limits _{k=0}^{K} |x_{k}^{(n)}-x_{k}|^{p_{k}}\right) ^{1/M}= \lim \limits _{m\rightarrow \infty }\left( \sum \limits _{k=0}^{K} |x_{k}^{(n)}-x_{k}^{(m)}|^{p_{k}}\right) ^{1/M}\le \varepsilon \quad&amp;\\ \text{ for } \text{ all } n\ge&amp;N. \end{aligned}$$

Since K was arbitrary, it follows that

$$\begin{aligned} g(x^{(n)}-x)\le \varepsilon \text{ for } \text{ all } n\ge N, \end{aligned}$$

(1.10)

in particular,

$$x^{(n)}-x\in \ell (p)$$

. Since $$\ell (p)$$ is a linear space by Part (a), we have $$x\in \ell (p)$$ , and we conclude from (1.10) that $$x^{(n)}\rightarrow x$$ $$(n\rightarrow \infty )$$ in $$\ell (p)$$ . Thus, we have shown that $$\ell (p)$$ is complete.

(b.3.ii)

   Now we consider the case of

$$X(p)=c_{0}(p)$$

.

Let $$\varepsilon >0$$ be given. Then there exists a non-negative integer N such that

$$ |x_{k}^{(n)}-x_{k}^{(m)}|^{\alpha _{k}}\le g_{0}(x^{(n)}-x^{(m)})&lt;\varepsilon \text{ for } \text{ all } n,m\ge N \text{ and } \text{ for } \text{ all } k, $$

and so

$$ |x_{k}^{(n)}-x_{k}|^ {\alpha _{k}}= \lim \limits _{m\rightarrow \infty }|x_{k}^{(n)}-x_{k}^{(m)}|^{\alpha _{k}}\le \varepsilon \text{ for } \text{ all } n\ge N \text{ and } \text{ all } k, $$

that is,

$$\begin{aligned} g_{0}(x^{(n)}-x)\le \varepsilon \text{ for } \text{ all } n\ge N. \end{aligned}$$

(1.11)

Since

$$x^{(N)}\in c_{0}(p)$$

, there exists a non-negative integer $$k_{0}$$ such that

$$|x_{k}^{(N)}|^{p_{k}}&lt;\varepsilon $$

for all $$k\ge k_{0}$$ , hence

$$ |x_{k}|^{\alpha _{k}}\le \left| x_{k}^{(N)}-x_{k}\right| ^{\alpha _{k}}+ \left| x_{k}^{(N)}\right| ^{\alpha _{k}}\le \varepsilon ^{1/M}+\varepsilon ^{1/M}\le 2\varepsilon ^{1/M} $$

and so

$$|x_{k}|^{p_{k}}\le 2^{1/M}\varepsilon \le 2\varepsilon $$

for all $$k\ge k_{0}$$ . This and (1.11) together imply $$x^{(n)}\rightarrow x$$ $$(n\rightarrow \infty )$$ in $$c_{0}(p)$$ . Thus, we have shown that $$c_{0}(p)$$ is complete.

(b.4)

   Finally, we show that $$g_{0}$$ is a paranorm for $$\ell _{\infty }(p)$$ if and only if the condition (1.6) is satisfied.

If

$$0&lt;m\le p_{k}\le M$$

for all k, then obviously

$$\ell _{\infty }(p)=\ell _{\infty }$$

, and $$\ell _{\infty }$$ is a Banach space.

Conversely, if $$m=0$$ , then there is a subsequence $$(p_{k(j)})_{j=0}^{\infty }$$ of the sequence $$(p_{k})_{k=0}^{\infty }$$ such that

$$p_{k(j)}&lt;1/(j+1)$$

for

$$j=0,1,\dots $$

. We define the sequence

$$x=(x_{k})_{k=0}^{\infty }$$

by

$$ x_{k}= {\left\{ \begin{array}{ll} 1 &amp;{} \quad (k=k(j))\\ 0 &amp;{} \quad (k\not =k(j)) \end{array}\right. }\quad (j=0,1,\dots ). $$

Then obviously $$x\in \ell _{\infty }(p)$$ . Let $$\lambda _{n}=1/n$$ for all n. Then

$$\lim _{n\rightarrow \infty }\lambda _{n}=0$$

, but

$$ g_{0}(\lambda _{n}x)=\sup \limits _{k}\left| \lambda _{n}x_{k}\right| ^{p_{k}}\ge \lim \limits _{j\rightarrow \infty }\left( \dfrac{1}{n}\right) ^{1/(j+1)}=1 \text{ for } \text{ all } n\in \mathbb {N}, $$

hence

$$g_{0}(\lambda _{n}x)\not \rightarrow 0$$

$$(n\rightarrow \infty )$$ . Thus, $$g_{0}$$ cannot be a paranorm for $$\ell _{\infty }(p)$$ .

This completes the proof.    $$\square $$

1.2 FK and BK Spaces

Here we give a short overview concerning the theory of FK and BK spaces which is the most powerful tool in the characterization of matrix transformations between sequence spaces. The fundamental result of this section is Theorem 1.4 which states that matrix maps between FK spaces are continuous. Most of the results in this section can be found in [21, 23, 31, 33]. We also refer the reader to [14, 34].

We provide the proofs of Theorems 1.2–1.7, and Corollary 1.2 for the reader who may not be too familiar with the theory of BK spaces.

We start with a slightly more general definition.

Definition 1.2

Let H be a linear space and a Hausdorff space. An FH space is a Fréchet space X such that X is a subspace of H and the topology of X is stronger than the restriction of the topology of H on X.

If $$H=\omega $$ with its topology given by the metric $$d_{\omega }$$ of (1.5) in Corollary 1.1, then an FH space is called an FK space.

A BH spaceor a BK spaceis an FH or FK space which is a Banach space.

Remark 1.1

(a)

If X is an FH space, then the inclusion map

$$\iota :X\rightarrow H$$

with $$\iota (x)=x$$ for all $$x\in X$$ is continuous. Therefore, X is continuously embedded in H.

(b)

Since convergence in $$(\omega ,d_{\omega })$$ and coordinatewise convergence are equivalent by Corollary 1.1, convergence in an FK space implies coordinatewise convergence.

(c)

The letters F, H, K and B stand for Fréchet, Hausdorff, Koordinate, the German word for coordinate, and Banach.

(d)

Some authors include local convexity in the definition of an FH space. Since most of the theory presented here can be developed without local convexity, we follow Wilansky [33] and do not include it in our definition. If local convexity, however, is needed then it will explicitly be mentioned.

Example 1.2

(a)

Trivially $$\omega $$ is an FK space with its natural metric of (1.5) in Corollary 1.1.

(b)

If

$$p=(p_{k})_{k=0}^{\infty }$$

is a bounded sequence of positive reals, then $$\ell (p)$$ and $$c_{0}(p)$$ are FK spaces with their metrics d given by the their paranorms, since they are Fréchet subspaces of $$\omega $$ with their metrics stronger than the natural metric of $$\omega $$ on them by Example 1.1.

(c)

We consider the spaces $$\ell _{\infty }$$ , c and $$c_{0}$$ of all bounded, convergent and null sequences, and

$$ \ell _{p}=\left\{ x\in \omega :\sum \limits _{k=0}^{\infty }|x_{k}|^{p}&lt;\infty \right\} \text{ for } 1\le p&lt;\infty . $$

It is well known that $$\ell _{\infty }$$ , c and $$c_{0}$$ are BK spaces with

$$ \Vert x\Vert _{\infty }=\sup \limits _{k}|x_{k}|; $$

since

$$|x_{k}|\le \Vert x\Vert _{\infty }$$

for all x and all k, those spaces are BK spaces.

Since $$\ell _{p}$$ obviously is the special case of $$\ell (p)$$ with the sequence $$p=p\cdot e$$ , and

$$ \Vert x\Vert _{p}=\left( \sum \limits _{k=0}^{\infty }|x_{k}|^{p}\right) ^{1/p} $$

is a norm on $$\ell _{p}$$ for

$$1\le p&lt;\infty $$

, $$\ell _{p}$$ is a BK space by Example 1.1.

We refer to spaces in this part as the classical sequence spaces.

Remark 1.2

We have seen in Example 1.1 (b) that $$g_{0}$$ is only a paranorm for $$\ell _{\infty }(p)$$ when the condition in (1.6) holds, in which case

$$\ell _{\infty }(p)=\ell _{\infty }$$

. Grosse–Erdmann [9] determined a linear topology for $$\ell _{\infty }(p)$$ . He showed in [9, Theorem 2 (ii)] that $$\ell _{\infty }(p)$$ is an IBK space, that is, it can be written as the union of an increasing sequence of BK spaces, and is endowed with the inductive limit topology.

The following results are fundamental. Their proofs use well-known theorems from functional analysis, which are included in Appendix A.2 for the reader’s convenience.

Theorem 1.2

Let X be a Fréchet space, Y be an FH space, $$\mathcal {T}_{Y}$$ , $$\left. \mathcal {T}_{H}\right| _{Y}$$ denote FH topologies on Y and of H on Y, and

$$f:X\rightarrow Y$$

be linear. Then

$$f:X\rightarrow (Y,\left. \mathcal {T}_{H}\right| _{Y})$$

is continuous, if and only if

$$f:X\rightarrow (Y,\mathcal {T}_{Y})$$

is continuous.

Proof

(i)

First, we assume that

$$f:X\rightarrow (Y,\mathcal {T}_{Y})$$

is continuous. Since Y is an FH space, we have

$$\left. \mathcal {T}_{H}\right| _{Y}\subset \mathcal {T}_{Y}$$

, and so

$$f:X\rightarrow (Y,\left. \mathcal {T}_{H}\right| _{Y})$$

is continuous.

(ii)

   Conversely, we assume that

$$f:X\rightarrow (Y,\left. \mathcal {T}_{H}\right| _{Y})$$

is continuous. Then it has closed graph by Theorem A.1 in Appendix A.2. Since Y is an FH space, we again have

$$\left. \mathcal {T}_{H}\right| _{Y}\subset \mathcal {T}_{Y}$$

, and so

$$f:X\rightarrow (Y,\mathcal {T}_{Y})$$

has closed graph. Consequently,

$$f:X\rightarrow (Y,\mathcal {T}_{Y})$$

is continuous by the closed graph theorem, Theorem A.2 in Appendix A.2.    $$\square $$

We obtain as an immediate consequence of Theorem 1.2.

Corollary 1.2

Let X be a Fréchet space, Y be an FK space,

$$f:X\rightarrow Y$$

be linear, and the coordinates

$$P_{n}:X\rightarrow \mathbb {C}$$

for

$$n=0,1,\dots $$

be defined by

$$P_{n}(x)=x_{n}$$

for all $$x\in X$$ . If

$$P_{n}\circ f:X\rightarrow \mathbb {C}$$

is continuous for every n, then

$$f:X\rightarrow Y$$

is continuous.

Proof

Since convergence and coordinatewise convergence are equivalent in $$\omega $$ by Corollary 1.1, the continuity of

$$P_{n}\circ f:X\rightarrow \mathbb {C}$$

for all n implies the continuity of

$$f:X\rightarrow \omega $$

, and so

$$f:X\rightarrow Y$$

is continuous by Theorem 1.2.     $$\square $$

By $$\phi $$ , we denote the set of all finite sequences. Thus,

$$x=(x_{k})_{k=0}^{\infty }\in \phi $$

if and only if there is an integer k such that $$x_{j}=0$$ for all $$j>k$$ .

Let X be a Fréchet spaces. Then we denote by $$X'$$  the set of all continuous linear functionals on X; $$X'$$ is called the continuous dual of X.

Theorem 1.3

Let $$X\supset \phi $$ be an FK space. If the series $$\sum _{k=0}^{\infty }a_{k}x_{k}$$ converge for all $$x\in X$$ , then the linear functional $$f_{a}$$ defined by

$$f_{a}(x)=\sum _{k=0}^{\infty }a_{k}x_{k}$$

for all $$x\in X$$ is continuous.

Proof

We define the functionals $$f_{a}^{[n]}$$ for all $$n\in \mathbb {N}_{0}$$ by

$$f_{a}^{[n]}(x)=\sum _{k=0}^{n}a_{k}x_{k}$$

for all $$x\in X$$ . Since X is an FK space and

$$f_{a}^{[n]}=\sum _{k=0}^{n}a_{k}P_{k}$$

is a finite linear combination of the continuous coordinates $$P_{k}$$

$$(k=0,1,\dots )$$

, we have $$f_{a}^{[n]}\in X'$$ for all n. By hypothesis, the limits

$$f_{a}(x)=\lim _{n\rightarrow \infty }f_{a}^{[n]}(x)$$

exist for all $$x\in X$$ , hence $$f_{a}\in X'$$ by the Banach–Steinhaus theorem, Theorem A.3.     $$\square $$

The next result is one of the most important ones in the theory of matrix transformations between sequence spaces. We need the following notations. Let X and Y be subsets of $$\omega $$ ,

$$A=(a_{nk})_{n,k=0}^{\infty }$$

be an infinite matrix of complex numbers and $$x\in \omega $$ . Then, we write

$$ A_{n}x=\sum \limits _{k=0}^{\infty }a_{nk}x_{k} \text{ for } n=0,1,\dots \text{ and } Ax=(A_{n}x)_{n=0}^{\infty } $$

provided all the series converge; Ax is called the A transform of the sequence x. We write (X, Y) for the class of all infinite matrices that map X into Y, that is, for which the series $$A_{n}x$$ converge for all n and all $$x\in X$$ , and $$Ax\in Y$$ for all $$x\in X$$ .

Theorem 1.4

Any matrix map between FK spaces is continuous.

Proof

Let X and Y be FK spaces,

$$A\in (X,Y)$$

and

$$L_{A}:X\rightarrow Y$$

be defined by

$$L_{A}(x)=Ax$$

for all $$x\in X$$ . Since the maps

$$P_{n}\circ L_{A}:X\rightarrow \mathbb {C}$$

are continuous for all n by Theorem 1.3,

$$L_{A}:X\rightarrow Y$$

is continuous by Corollary 1.2.     $$\square $$

It turns out as a consequence of Theorem 1.2 that the FH topology of an FH space is unique, more precisely, we have the following.

Theorem 1.5

Let $$(X,\mathcal {T}_{X})$$ and $$(Y,\mathcal {T}_{Y})$$ be FH spaces with $$X\subset Y$$ , and $$\left. \mathcal {T}_{Y}\right| _{X}$$ denote the topology of Y on X. Then

$$\begin{aligned} \mathcal {T}_{X}\supset \left. \mathcal {T}_{Y}\right| _{X}; \end{aligned}$$

(1.12)

$$\begin{aligned} \mathcal {T}_{X}=\left. \mathcal {T}_{Y}\right| _{X} \text{ if } \text{ and } \text{ only } \text{ if }\, X\, \text{ is } \text{ a } \text{ closed } \text{ subspace } \text{ of }\, Y. \end{aligned}$$

(1.13)

In particular, the topology of an FH space is unique.

Proof

(i)

First we show the inclusion in (1.12).

Since X is an FH space, the inclusion map

$$\iota :(X,\mathcal {T}_{X})\rightarrow (H,T_{H})$$

is continuous by Remark 1.1 (a). Therefore,

$$\iota :(X,\mathcal {T}_{X})\rightarrow (Y,T_{Y})$$

is continuous by Theorem 1.2. Thus, the inclusion in (1.12) holds.

(ii)

   Now we show the identity in (1.13).

Let $$\mathcal {T}$$ and $$\mathcal {T}'$$ be FH topologies for an FH space. Then it follows by what we have just shown in Part (i) that

$$\mathcal {T}\subset \mathcal {T}'\subset \mathcal {T}$$

.

$$(\alpha )$$

   If X is closed in Y, then X becomes an FH space with $$\left. \mathcal {T}_{Y}\right| _{X}$$ . It follows from the uniqueness that

$$\mathcal {T}_{X}=\left. \mathcal {T}_{Y}\right| _{X}$$

.

$$(\beta )$$

   Conversely, if

$$\mathcal {T}_{X}=\left. \mathcal {T}_{Y}\right| _{X}$$

, then X is a complete subspace of Y, and so closed in Y.    $$\square $$

The next results are also useful.

Theorem 1.6

Let X, Y and Z be FH spaces with

$$X\subset Y\subset Z$$

. If X is closed in Z, then X is closed in Y.

Proof

Since X is closed in $$(Y,\left. \mathcal {T}_{Z}\right| _{Y})$$ , it is closed in $$(Y,\mathcal {T}_{Y})$$ by Theorem 1.5.     $$\square $$

Let Y be a topological space, and $$E\subset Y$$ . Then we write cl $$_{Y}(E)$$ for the closure of E in Y.

Theorem 1.7

Let X and Y be FH spaces with $$X\subset Y$$ , and E be a subset of X. Then, we have

$$ \text{ cl}_{Y}(E)= \text{ cl}_{Y}( \text{ cl}_{X}(E)), { in}\, {particular } \text{ cl}_{X}(E)\subset \text{ cl}_{Y}(E). $$

Proof

Since X is closed in $$(Y,\left. \mathcal {T}_{Z}\right| _{Y})$$ , it is closed in $$(Y,\mathcal {T}_{Y})$$ by Theorem 1.5.     $$\square $$

Example 1.3

(a)

Since $$c_{0}$$ and c are closed in $$\ell _{\infty }$$ , their BK topologies are the same; since $$\ell _{1}$$ is not closed in $$\ell _{\infty }$$ , its BK topology is strictly stronger than that of $$\ell _{\infty }$$ on $$\ell _{1}$$ (Theorem 1.5).

(b)

If c is not closed in an FK space X, then X must contain unbounded sequences (Theorem 1.6).

Now we recall the definition of the AD and AK properties.

Definition 1.3

Let $$X\supset \phi $$ be an FK space. Then X is said  to have

(a) AD if

$$\text{ cl}_{X}(\phi )=X$$

;

(b) AK if every sequence

$$x=(x_{k})_{k=0}^{\infty }\in X$$

has a unique representation

$$ x=\sum \limits _{k=0}^{\infty }x_{k}e^{(k)}, $$

that is, if every sequence x is the limit of its m-sections

$$ x^{[m]}=\sum _{k=0}^{m}x_{k}e^{(k)}. $$

The letters A, D and K stand for abschnittsdicht, the German word for sectionally dense, and Abschnittskonvergenz, the German word for sectional convergence.

Example 1.4

(a) Every FK space with AK obviously has AD.

(b) An Example of an FK space with AD which does not have AK can be found in [33, Example 5.2.14].

(c) The spaces $$\omega $$ , $$c_{0}(p)$$ , $$\ell (p)$$ for

$$p=(p_{k})_{k=0}^{\infty }\in \ell _{\infty }$$

, in particular, $$c_{0}$$ and $$\ell _{p}$$

$$(1\le p&lt;\infty )$$

have AK.

(d) The space c does not have AK, since $$e\in c$$ .

Now we recall the concept of a Schauder basis. We refer the reader to [18, 24] for further studies.

Definition 1.4

A Schauder basisof a linear metric space X is a sequence $$(b_{n})$$ of vectors such that for every vector $$x\in X$$ there is a unique sequence $$(\lambda _{n})$$ of scalars with

$$ \sum \limits _{n=0}^{\infty }\lambda _{n} b_{n}=x, \text{ that } \text{ is, } \lim \limits _{m\rightarrow \infty }\sum \limits _{n=0}^{m}\lambda _{n}b_{n}=x. $$

For finite-dimensional spaces, the concepts of Schauder and algebraic bases coincide. In most cases of interest, however, the concepts differ. Every linear space has an algebraic basis, but there are linear spaces without a Schauder basis which we will soon see.

We recall that a metric space (X, d) is said to be separableif it has a countable dense subset; this means there is a countable set $$A \subset X$$ such that for all $$x\in X$$ and all $$\varepsilon >0$$ there is an element $$a\in A$$ with

$$d(x,a)&lt;\varepsilon $$

.

The next result is well known from elementary functional analysis.

Theorem 1.8

Every complex linear metric space with a Schauder basis is separable.

We close this section with two well-known important examples.

Example 1.5

The space $$\ell _{\infty }$$ has no Schauder basis, since it is not separable.

Example 1.6

We put $$b^{(-1)}=e$$ and $$b^{(k)}=e^{(k)}$$ for

$$k=0,1,\dots $$

. Then the sequence $$(b^{(n)})_{n=-1}^{\infty }$$ is a Schauder basis of c; more precisely every sequence

$$x=(x_{k})_{k=0}^{\infty }\in c$$

has a unique representation

$$\begin{aligned} x=\xi \,e+\sum \limits _{k=0}^{\infty }(x_{k}-\xi )e^{(k)} \text{ where } \xi =\xi (x)= \lim \limits _{k\rightarrow \infty }x_{k}. \end{aligned}$$

(1.14)

1.3 Matrix Transformations into the Classical Sequence Spaces

Now we apply the results of the previous section to characterize the classes (X, Y) where X is an arbitrary FK space and Y is any of the spaces $$\ell _{\infty }$$ , c, $$c_{0}$$ and $$\ell _{1}$$ .

Let (X, d) be a metric space, $$\delta >0$$ and $$x_{0}\in X$$ . Then, we write

$$ \overline{B}_{\delta }(x_{0})=\overline{B}_{X,\delta }(x_{0})= \{x\in X:d(x,x_{0})\le \delta \} $$

for the closed ball of radius $$\delta $$ with its centre in $$x_{0}$$ .

If $$X\subset \omega $$ is a linear metric space and $$a\in \omega $$ , then we write

$$\begin{aligned} \Vert a\Vert _{\delta }^{*}=\Vert a\Vert _{X,\delta }^{*}= \sup \limits _{x\in \overline{B}_{\delta }[0]}\left| \sum \limits _{k=0}^{\infty }a_{k}x_{k}\right| , \end{aligned}$$

(1.15)

provided the expression on the right hand exists and is finite which is the case whenever the series $$\sum _{k=0}^{\infty }a_{k}x_{k}$$ converge for all $$x\in X$$ (Theorem 1.3).

Let us recall that a subset S of a linear space X is said to be absorbing if, for every $$x\in X$$ , there is $$\varepsilon >0$$ such that $$\lambda x\in S$$ for all scalars $$\lambda $$ with $$|\lambda |\le \varepsilon $$ .

The next statement is well known.

Remark 1.3

([23, Remark 3.3.4 (j)]) Let (X, p) be a paranormed space. Then the open and closed neighbourhoods

$$N_{r}(0)=\{x\in X:p(x)&lt;r\}$$

and

$$\overline{N}_{r}(0)=\{x\in X:p(x)\le r\}$$

of 0 are absorbing for all $$r>0$$ .

The first result characterizes the class $$(X,\ell _{\infty })$$ for arbitrary FK spaces X.

Theorem 1.9

Let X be an FK space. Then we have

$$A\in (X,\ell _{\infty })$$

if and only if

$$\begin{aligned} \Vert A\Vert _{\delta }^{*}=\sup \limits _{n}\Vert A_{n}\Vert _{\delta }^{*}&lt;\infty \text{ for } \text{ some } \delta &gt;0. \end{aligned}$$

(1.16)

Proof

First, we assume that (1.16) is satisfied. Then the series $$A_{n}x$$ converge for all $$x\in \overline{B}_{\delta }(0)$$ and for all n, and $$Ax\in \ell _{\infty }$$ for all $$x\in \overline{B}_{\delta }(0)$$ . Since the set $$\overline{B}_{\delta }(0)$$ is absorbing by Remark 1.3, we conclude that the series $$A_{n}x$$ converge for all n and all $$x\in X$$ , and $$Ax\in \ell _{\infty }$$ for all $$x \in X$$ .

Conversely, we assume

$$A\in (X,\ell _{\infty })$$

. Then the map

$$L_{A}:X\rightarrow \ell _{\infty }$$

defined by

$$\begin{aligned} L_{A}(x)=Ax \text{ for } \text{ all } x\in X \end{aligned}$$

(1.17)

is continuous by Theorem 1.4. Hence, there exist a neighbourhood N of 0 in X and a real $$\delta >0$$ such that

$$\overline{B}_{\delta }(0)\subset N$$

and

$$\Vert L_{A}(x)\Vert _{\infty }&lt;1$$

for all $$x\in N$$ . This implies (1.16).     $$\square $$

Let X and Y be Banach spaces. Then we use the standard notation $$\mathcal {B}(X,Y)$$ for the set of all bounded linear operators from X to Y. It is well known that $$\mathcal {B}(X,Y)$$ is a Banach space with the operator normdefined by

$$ \Vert L\Vert =\sup \{\Vert L(x)\Vert :\Vert x\Vert =1\} \text{ for } \text{ all } L\in \mathcal {B}(X,Y); $$

we write $$X^{*}$$ for $$X'$$ with the norm defined by

$$ \Vert f\Vert =\sup \{|f(x)|:\Vert x\Vert =1\} \text{ for } \text{ all } f\in X'. $$

Theorem 1.10

Let X and Y be BK spaces.

(a)

Then

$$(X,Y)\subset \mathcal {B}(X,Y)$$

, that is, every

$$A\in (X,Y)$$

defines an operator

$$L_{A}\in \mathcal {B}(X,Y)$$

by (1.17).

(b)

If X has AK then

$$\mathcal {B}(X,Y)\subset (X,Y)$$

, that is, for each

$$L\in \mathcal {B}(X,Y)$$

there exists

$$A\in (X,Y)$$

such that (1.17) holds.

(c)

We have

$$A\in (X,\ell _{\infty })$$

if and only if

$$\begin{aligned} \Vert A\Vert _{(X,\ell _{\infty })}=\sup \limits _{n}\Vert A_{n}\Vert _{X}^{*}= \sup \limits _{n}\left( \sup \left\{ |A_{n}x|:\Vert x\Vert =1\right\} \right) &lt;\infty ; \end{aligned}$$

(1.18)

if

$$A\in (X,\ell _{\infty })$$

, then

$$\begin{aligned} \Vert L_{A}\Vert =\Vert A\Vert _{(X,\ell _{\infty })}. \end{aligned}$$

(1.19)

Proof

(a)   This is Theorem 1.4.

(b)   Let

$$L\in \mathcal {B}(X,Y)$$

be given. We write

$$L_{n}=P_{n}\circ L$$

for all n, and put

$$a_{nk}=L_{n}(e^{(k)})$$

for all n and k. Let

$$x=(x_{k})_{k=0}^{\infty }\in X$$

be given. Since X has AK, we have

$$x=\sum _{k=0}^{\infty }x_{k}e^{(k)}$$

, and since Y is a BK space, it follows that $$L_{n}\in X^{*}$$ for all n. Hence, we obtain

$$L_{n}(x)=\sum _{k=0}^{\infty }x_{k}L_{n}(e^{(k)})=\sum _{k=0}^{\infty }a_{nk}x_{k}=A_{n}x$$

for all n, and so

$$L(x)=Ax$$

.

(c)    This follows from Theorem 1.9 and the definition of $$\Vert A\Vert _{(X,\ell _{\infty })}$$ .    $$\square $$

In many cases, the characterizations of the classes (X, c) and $$(X,c_{0})$$ can easily be obtained from the characterization of the class $$(X,\ell _{\infty })$$ .

Theorem 1.11

Let X be an FK space with AD, and Y and $$Y_{1}$$ be FK spaces with $$Y_{1}$$ a closed subspace of Y. Then

$$A\in (X,Y_{1})$$

if and only if

$$A\in (X,Y)$$

and $$Ae^{(k)}\in Y_{1}$$ for all k.

Proof

First, we assume

$$A\in (X,Y_{1})$$

. Then $$Y_{1}\subset Y$$ implies

$$A\in (X,Y)$$

, and $$e^{(k)}\in X$$ for all k implies $$Ae^{(k)}\in Y_{1}$$ for all k.

Conversely, we assume

$$A\in (X,Y)$$

and $$Ae^{(k)}\in Y_{1}$$ for all k. We define the map

$$L_{A}:X\rightarrow Y$$

by (1.17). Then $$Ae^{(k)}\in Y_{1}$$ implies

$$L_{A}(\phi )\subset Y_{1}$$

. By Theorem 1.4, $$L_{A}$$ is continuous, hence

$$L_{A}(\text{ cl}_{X}(\phi ))\subset \text{ cl}_{Y}(L_{A}(\phi ))$$

. Since $$Y_{1}$$ is closed in Y, and $$\phi $$ is dense in the AD space X, we have

$$L_{A}(X)=L_{A}(\text{ cl}_{X}(\phi ))\subset \text{ cl}_{Y}(L_{A}(\phi )) \subset \text{ cl}_{Y}(Y_{1})=\text{ cl}_{Y_{1}}(Y_{1})=Y_{1}$$

by Theorem 1.5.    $$\square $$

The following result is sometimes referred to as improvement of mapping.

Theorem 1.12

(Improvement of mapping) Let X be an FK space,

$$X_{1}=X\oplus e=\{x_{1}=x+\lambda e: x\in X,\,\,\lambda \in \mathbb {C}\}$$

, and Y be a linear subspace of $$\omega $$ . Then

$$A\in (X_{1},Y)$$

if and only if

$$A\in (X,Y)$$

and $$Ae\in Y$$ .

Proof

First, we assume

$$A\in (X_{1},Y)$$

. Then $$X\subset X_{1}$$ implies

$$A\in (X,Y)$$

, and $$e\in X_{1}$$ implies $$Ae\in Y$$ .

Conversely, we assume

$$A\in (X,Y)$$

and $$Ae\in Y$$ . Let $$x_{1}\in X_{1}$$ be given. Then there are $$x\in X$$ and $$\lambda \in \mathbb {C}$$ such that

$$x_{1}=x+\lambda e$$

, and it follows that

$$Ax_{1}=A(x+\lambda e)=Ax+\lambda Ae\in Y$$

.     $$\square $$

We need Lemma [26] for the characterization of the class $$(X,\ell _{1})$$ . Although it is elementary, we provide its proof here.

Lemma 1.1

([26]) Let

$$a_{0},a_{1},\dots ,a_{n}\in \mathbb {C}$$

. Then the following inequality holds:

$$\begin{aligned} \sum \limits _{k=0}^{n}|a_{k}|\le 4\cdot \max \limits _{N\subset \{0,\dots ,n\}} \left| \sum \limits _{k\in N}a_{k}\right| . \end{aligned}$$

(1.20)

Proof

First we consider the case when

$$a_{0},a_{1},\dots ,a_{n}\in \mathbb {R}$$

.

We put

$$N^{+}=\{k\in \{0,\dots ,n\}:a_{k}\ge 0\}$$

and

$$N^{-}=\{k\in \{0,\dots ,n\}:a_{k}&lt;0\}$$

, and obtain

$$ \sum \limits _{k=0}^{n}|a_{k}|= \left| \sum \limits _{k\in N^{+}}a_{k}\right| + \left| \sum \limits _{k\in N^{-}}a_{k}\right| \le 2\cdot \max \limits _{N\subset \{0,\dots ,n\}}\left| \sum \limits _{k\in N}a_{k}\right| . $$

Now we assume

$$a_{0},a_{1},\dots ,a_{n}\in \mathbb {C}$$

.

We write

$$a_{k}=\alpha _{k}+i\beta _{k}$$

for

$$k=0,1,\dots ,n$$

and, for any subset N of $$\{0,\dots ,n\}$$ , we put

$$ x_{N}=\sum \limits _{k\in N}\alpha _{k},\,\, y_{N}=\sum \limits _{k\in N}\beta _{k} \text{ and } z_{N}=x_{N}+iy_{N}= \sum \limits _{k\in N}a_{k}. $$

Now we choose subsets $$N_{r}$$ , $$N_{i}$$ and $$N_{*}$$ of $$\{0,\dots ,n\}$$ such that

$$ \left| x_{N_{r}}\right| =\max \limits _{N\subset \{0,\dots ,n\}}|x_{N}|,\,\, \left| y_{N_{i}}\right| =\max \limits _{N\subset \{0,\dots ,n\}}|y_{N}| \text{ and } \left| z_{N_{*}}\right| =\max \limits _{N\subset \{0,\dots ,n\}}|z_{N}|. $$

Then, we have for all subsets N of $$\{0,\dots ,n\}$$

$$ |x_{N}|,|y_{N}|\le \left| z_{N_{*}}\right| \text{ and } \left| x_{N_{r}}\right| +\left| y_{N_{i}}\right| \le 2\cdot \left| z_{N_{*}}\right| . $$

Finally, it follows by the first part of the proof that

$$\begin{aligned} \sum \limits _{k=0}^{n}|a_{k}|&amp;\le \sum \limits _{k=0}^{n}|\alpha _{k}|+\sum \limits _{k=0}^{n}|\beta _{k}|\le \left( \left| x_{N_{r}}\right| +\left| x_{N_{i}}\right| \right) \\&amp;\le 4\cdot \left| z_{N_{*}}\right| = 4\cdot \max _{N\subset \{0,\dots ,n\}}\left| \sum \limits _{k\in N}a_{k}\right| .\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \square \end{aligned}$$

Now we give the characterization of the class $$(X,\ell _{1})$$ for arbitrary FK spaces. The proof is similar to that of Theorem 1.9.

Theorem 1.13

([20, Satz 1]) Let X be an FK space. Then

$$A\in (X,\ell _{1})$$

if and only if

../images/500557_1_En_1_Chapter/500557_1_En_1_Equ26_HTML.png

(1.21)

Proof

(i)

First we show the sufficiency of the condition in (1.21).

We assume that (1.21) is satisfied. Then the series $$A_{n}x$$ converge for all $$x\in B_{\delta }{[}0{]}$$ and for all n. Let $$m\in \mathbb {N}_{0}$$ be given. Then we have by Lemma 1.1

$$\begin{aligned} \sum \limits _{n=0}^{m}|A_{n}x|&amp;\le 4\cdot \max \limits _{N_{m}\subset \{0,\ldots ,m\}} \left| \sum \limits _{n\in N_{m}}\sum \limits _{k=0}^{\infty }a_{nk}x_{k}\right| \\&amp;= 4\cdot \max \limits _{N_{m}\subset \{0,\ldots ,m\}} \left| \sum \limits _{k=0}^{\infty }\left( \sum \limits _{n\in N_{m}}a_{nk} \right) x_{k}\right| \\&amp;\le 4\cdot \max \limits _{N_{m}\subset \{0,\ldots ,m\}} \left\| \left( \sum \limits _{n\in N_{m}}a_{nk} \right) _{k=0}^{\infty }\right\| _{X,\delta }^{*}\\&amp;\le 4 \cdot \Vert A\Vert _{(X,\ell _{1})}&lt;\infty , \end{aligned}$$

for all $$x\in \overline{B}_{\delta }(0)$$ . Since $$m\in \mathbb {N}_{0}$$ was arbitrary, $$Ax\in \ell _{1}$$ for all $$x\in \overline{B}_{\delta }(0)$$ . Since the set $$\overline{B}_{\delta }(0)$$ is absorbing by Remark 1.3, the series $$A_{n}x$$ converge for all n and all $$x\in X$$ , and $$Ax\in \ell _{1}$$ for all $$x\in X$$ . Therefore, we have

$$A\in (X,\ell _{1})$$

. Thus, we have shown