1. Introduction
A large Schröder number
, which will be called as Schröder number in the remainder of study for short, represents the count of trails from the southwest corner
of an
grid to the northeast corner
, utilizing just single steps north, northeast, or east, which do not rise above the southwest–northeast diagonal, where
. Some terms of the Schröder integer sequence are
and Schröder numbers satisfy the recurrence relation
Furthermore, the limit of the ratio of the consecutive terms of Schröder sequence is
. To access Schröder numbers and related basic and more comprehensive information, readers can consult sources [
1,
2,
3,
4,
5].
There are also fascinating numbers other than Schröder numbers, which have various applications in geometry, probability theory, combinatorics and number theory, such as the Catalan numbers
, whose first few terms are
The Catalan number sequence was discovered by Leonhard Euler in 1751 while investigating the number of different ways to divide a polygon into triangles. There are several equally different ways of expressing Catalan numbers, but the basic relations for them are as follows:
where
,
, and the limit of the ratio of the consecutive terms of Catalan sequence is 4. Researchers wishing to obtain detailed information about Catalan numbers and the necessary basic concepts are advised to consult sources [
6,
7].
Moreover, the relationship between Schröder and Catalan numbers is given by the relation
Each subset of the space
that contains all sequences whose elements are real numbers is called a sequence space. The classical sequence spaces mentioned in the study can be expressed as
for
. From now on, unless otherwise stated,
will represent
for brevity. A Banach space with continuous coordinates is named a BK-space. The spaces
c,
and
are BK-spaces with the norm
, and the space
is a BK-space with the norm
for
. Let
denote the
rth row of an infinite matrix
with real entries. In this case, the expression
is called the
F-transform of
if the series converges for all
. The
-,
- and
-duals of a sequence space
are given the sets
If the
F-transform
of every sequence
y taken from
is in
for an infinite matrix
F, then
F is called a matrix transformation from
to
, and the class of such matrices is represented by
. Furthermore,
if and only if
and
for all
. The matrix domain of an infinite matrix
F on the sequence space
is represented by the set
The normed spaces
and
are called linearly isomorphic spaces if there is a one-to-one, onto, and norm-preserving transformation between them, and they are denoted by
. Researchers who want to deal in detail with the basic topics of summability theory, sequence spaces, and matrix domains can refer to sources [
8,
9,
10,
11].
In recent years, the use of infinite matrices in obtaining new sequence spaces has become quite widespread. However, the idea of using special number sequences in obtaining infinite matrices is a newer approach. After the studies by Kara and Başarır [
12,
13] based on the Fibonacci integer sequence, generating new normed or paranormed sequence spaces by means of integer sequences appears to be an interesting idea. Thus, the integer sequences Lucas [
14,
15], Padovan [
16], Leonardo [
17], Catalan [
18,
19,
20], Bell [
21], Schröder [
22,
23], Mersenne [
24] and Motzkin [
25,
26] were used to construct new infinite matrices and sequence spaces and at the same time some fundamental topics were examined in obtained sequence spaces.
In this study, firstly, we describe a novel matrix with the aid of both Schröder and Catalan numbers, and construct sequence spaces and as the domain of . After that, some properties, such as regularity, completeness and inclusion relations, are examined. In the rest of the paper, the duals of the new spaces are calculated, and the characterization of some classes of matrices and compact operators is presented.
2. A New Regular Schröder–Catalan Matrix and Associated Sequence Spaces
In the current part, the regular Schröder–Catalan matrix is acquired by Schröder and Catalan numbers. Then, with the help of this matrix, BK-spaces and are obtained. After that, it is shown that and and, finally, the Schauder basis of and the inclusion relations of the spaces are presented.
The Schröder–Catalan matrix
is constructed with the help of Schröder and Catalan numbers as follows:
is stated in matrix form as follows:
One point to note is that the Schröder–Catalan matrix is not symmetric, since its transpose is not equal to itself.
The
-transform of
is presented by
Furthermore, from Theorem 2 of Deng and Yan [
27], the inverse
of the Schröder–Catalan matrix
is computed as
After remembering that matrices that transform convergent sequences into convergent sequences by preserving the limit are termed as regular, let us give the relevant lemma and result.
Lemma 1. is regular iff
- (i)
;
- (ii)
;
- (iii)
for all .
Theorem 1. The Schröder–Catalan matrix is regular.
Proof. (
i) and (
ii) is clear by the relation
Now, let us consider the series
, and examine the convergence of this. By the aid of the ratio test and the relation
it is reached that the series
absolutely convergent, and
Thus, Condition (iii) holds.
□
Now, we can introduce the sets
and
by
and
for
. Then, it can be said that
and
are the domains of
on
and
, respectively. It is stated in [
28] that, for a triangle
F and BK-space
,
is a BK-space, too.
Theorem 2. and are BK-spaces withandrespectively. Theorem 3. and .
Proof. The mapping , is linear. From , is incejtive.
Let us take
and
with
It is seen that
is surjective from the equation
Furthermore, is norm keeper from the equality .
In other spaces, the result can be seen in a similar way. □
Definition 1. Assume that is a normed sequence space, and . If, for any , there exists a unique scalars’ sequence asfor , then is Schauder basis for Δ
, and it is written as . Let be a sequence whose other terms are zero, except for the rth term, which is 1. We conclude that the inverse image of the basis of becomes the basis of , since is an isomorphism, given Theorem 3, so it will be given following result without proof.
Theorem 4. The set expressed byis the Schauder basis for . Furthermore, each is represented uniquely by for and . Theorem 5. The inclusion is strict for .
Proof. Let us take , such that . Furthermore, it is known that for . Then, it is seen that and .
If is chosen, it is reached that the inclusion is strict. □
Theorem 6. .
Proof. From the inequality
for
, we see that
. □
4. Matrix Transformations
In this part, necessary and sufficient conditions for some classes of matrices containing new sequence spaces will be stated.
Theorem 10. Assume that, the infinite matrices and are as follows:andfor all . Then, for , , if and only if and . Proof. Assume that
and
. Then,
for all
. It must be
, because
exists. If we take limit for
on (
23), it is reached that
. It is known that
; in that case,
. Thus,
.
Assume that
and
. In that case, it is seen that
, which gives
. Consequently,
exists for all
. Thus, it is reached from (
23) as
that
, and this implies that
. □
Corollary 1. The necessary and sufficient conditions for the classes can be seen in Table 2 for , and the matrices and expressed by (21) and (22). 1. (8),(9) hold with , (6) holds with . | 2. (8),(9) hold with , (7) holds with . |
3. (8),(9) hold with , (8) holds with . | 4. (8),(9) hold with , (8),(9) hold with . |
5. (8),(9) hold with , (8),(10) hold with . | 6. (9),(12) hold with , (11) holds with . |
7. (9),(12) hold with , (12) holds with . | 8. (9),(12) hold with , (9),(12) hold with . |
9. (9),(12) hold with , (10),(12) hold with . | 10. (9),(16) hold with , (13) holds with . |
11. (9),(16) hold with , (14) holds with . | 12. (9),(16) hold with , (15) holds with . |
13. (9),(16) hold with , (9),(16) hold with . | 14. (9),(16) hold with , (17) holds with . |
Table 2.
Conditions of the classes for , and .
Table 2.
Conditions of the classes for , and .
() | | | | c | |
---|
| 1. | 2. | 3. | 4. | 5. |
| 6. | ◃ | 7. | 8. | 9. |
| 10. | 11. | 12. | 13. | 14. |
Theorem 11. Let us assume that the infinite matrices and are expressed by the equality Then, if and only if , where and .
Proof. For a sequence
, it is obtained that
Thus, for all , and iff for all . Consequently, it is reached that if and only if . □
Corollary 2. The necessary and sufficient conditions for the classes can be seen in Table 3 for the matrices and expressed by (24). 5. Characterizations of Compact Operators
Assume that
is a normed space, and
is the unit sphere in
. The acronym
is expressed by
for a BK-space
, and
, provided that the series is finite, where
is space of all finite sequences. In that case,
.
Lemma 2 ([
30]).
The following hold:- (i)
and , for all .
- (ii)
and , for all .
- (iii)
and , for all and .
By is meant the family of all bounded linear operators from to .
Lemma 3 ([
30]).
Let us assume that Δ
and Π
are BK-spaces and . In that case, there is an operator , such that for all . Lemma 4 ([
30]).
Let us assume that is a BK-space. If , in that case, for . Let us assume that
is a bounded set in the metric space
. Then, the Hausdorff measure of non-compactness of
is expressed as
where
is the open ball with center
and radius
for each
. Comprehensive explanations of what has been discussed so far in this section can be obtained from source [
30].
Theorem 12 ([
31]).
Let us assume that is bounded, and an operator stated by for all , and . In that case,for the identity operator on . Let us assume that and are Banach spaces and linear operator from to . In that case, if has a convergent subsequence in for all , it is said that is compact. The acronym denotes the Hausdorff measure of non-compactness of operator . Furthermore, is compact iff .
Comprehensive information about compactness and Hausdorff measure of non-compactness can be obtained from [
32,
33,
34].
Let us suppose the sequences
and
as
for all
.
Lemma 5. Assume that for . In that case, andfor all . Lemma 6. For with (25), the following are satisfied: - (i)
for all .
- (ii)
for all .
- (iii)
for all and .
Proof. It will be proved only for the first part.
(i) It is achieved from Lemma 5 that
and
satisfies for
, and for all
. It is seen that
if and only if
from
with (
3). In that case,
From Lemma 2, we see that
. □
Lemma 7 ([
32]).
Assume that is a BK-space.- (i)
If , in this case, and is compact if .
- (ii)
If , in this case and is compact iff .
- (iii)
If , in this case,and is compact iff . Here, is the collection of all finite subsets of and is the subcollection of including of subsets of with entries which are bigger than k.
For the rest of the study, the existence of the relation between matrices and F will be agreed by taking into account the convergence of the series.
Lemma 8. Assume that the infinite matrix and . If , in that case, and for all and .
Proof. We see that by the aid of Lemma 5. □
Theorem 13. Assume that . Then,
- (i)
If , in this case,and is compact if - (ii)
If , thenand is compact iff - (iii)
If , thenand is compact iffwhere for all .
Proof. (
i) Assume that
and
. It is seen that
, because
converges for each
. By Lemma 6-(iii),
. Then, by Lemma 7-(i), the following expression is reached:
and
is compact if
(ii) Let
. It is known that
, and by Lemma 7-(ii), the following expression is reached:
and
is compact iff
(iii) Let
. By Lemma 6,
. Thus, from Lemma 7-(iii), we see that
and
is compact iff
□
Theorem 14. - (i)
If , in this case,and is compact if - (ii)
If , in this case,and is compact iff - (iii)
If , in this case,and is compact iffwhere .
Proof. It can be proved with similar approach to Theorem 13. □
Theorem 15. - (i)
If , in this case,and is compact if - (ii)
If , in this case,and is compact iff
Proof. It can be proved with similar approach to Theorem 13. □
Lemma 9 ([
32]).
Let . If Δ has AK property or , in this case,and is compact iffwhere and . Theorem 16. Assume that and . In this case,and is compact iffwhere . Proof. Let
. Then,
from Lemma 8. By Lemma 9,
Therefore, from Lemma 6-(iii),
Consequently, by Lemma 9,
is compact iff
□
Theorem 17. Let , thenand is compact iff Proof. It can be proved with similar approach to Theorem 16. □
Theorem 18. Let , thenand is compact iff Proof. It can be also proved with similar approach to Theorem 16. □