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On Motzkin sequence spaces via q -analog and compact operators
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Abstract
We aim to develop a
q
q
-analog of recently introduced Motzkin sequence spaces by Erdem et al. [
Motzkin sequence spaces and Motzkin core
, Numer. Funct. Anal. Optim.
45
(2024), no. 4–6, 283–303] by using
q
q
-Motzkin numbers and introduce sequence spaces
c
(
M
(
q
)
)
c\left({\mathfrak{M}}\left(q))
and
c
0
(
M
(
q
)
)
.
{c}_{0}\left({\mathfrak{M}}\left(q)).
We investigate some topological properties, compute bases, and obtain their duals. For
X
∈
{
c
(
M
(
q
)
)
,
c
0
(
M
(
q
)
)
}
X\in \{c\left({\mathfrak{M}}\left(q)),{c}_{0}\left({\mathfrak{M}}\left(q))\}
and
Y
∈
{
ℓ
∞
,
c
,
c
0
,
ℓ
1
}
,
Y\in \{{\ell }_{\infty },c,{c}_{0},{\ell }_{1}\},
some results pertaining to characterization of matrix class
(
X
,
Y
)
\left(X,Y)
is given. We devote the final section to obtain necessary and sufficient conditions for a matrix operator to be compact on the space
c
0
(
M
(
q
)
)
{c}_{0}\left({\mathfrak{M}}\left(q))
via Hausdorff measure of noncompactness.
Title: On Motzkin sequence spaces via
q
-analog and compact operators
Description:
Abstract
We aim to develop a
q
q
-analog of recently introduced Motzkin sequence spaces by Erdem et al.
[
Motzkin sequence spaces and Motzkin core
, Numer.
Funct.
Anal.
Optim.
45
(2024), no.
4–6, 283–303] by using
q
q
-Motzkin numbers and introduce sequence spaces
c
(
M
(
q
)
)
c\left({\mathfrak{M}}\left(q))
and
c
0
(
M
(
q
)
)
.
{c}_{0}\left({\mathfrak{M}}\left(q)).
We investigate some topological properties, compute bases, and obtain their duals.
For
X
∈
{
c
(
M
(
q
)
)
,
c
0
(
M
(
q
)
)
}
X\in \{c\left({\mathfrak{M}}\left(q)),{c}_{0}\left({\mathfrak{M}}\left(q))\}
and
Y
∈
{
ℓ
∞
,
c
,
c
0
,
ℓ
1
}
,
Y\in \{{\ell }_{\infty },c,{c}_{0},{\ell }_{1}\},
some results pertaining to characterization of matrix class
(
X
,
Y
)
\left(X,Y)
is given.
We devote the final section to obtain necessary and sufficient conditions for a matrix operator to be compact on the space
c
0
(
M
(
q
)
)
{c}_{0}\left({\mathfrak{M}}\left(q))
via Hausdorff measure of noncompactness.
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