Abstract
Define an infinite matrix
D
α
=
(
d
n
,
v
α
)
\mathfrak{D}^{\alpha}=(d^{\alpha}_{n,v})
by
d
n
,
v
α
=
{
v
α
σ
(
α
)
(
n
)
,
v
∣
n
,
0
,
v
∤
n
,
d^{\alpha}_{n,v}=\begin{cases}\dfrac{v^{\alpha}}{\sigma^{(\alpha)}(n)},&v\mid n,\\
0,&v\nmid n,\end{cases}
where
σ
(
α
)
(
n
)
\sigma^{(\alpha)}(n)
is defined to be the sum of the 𝛼-th power of the positive divisors of
n
∈
N
n\in\mathbb{N}
, and construct the matrix domains
ℓ
p
(
D
α
)
\ell_{p}(\mathfrak{D}^{\alpha})
(
0
<
p
<
∞
0<p<\infty
),
c
0
(
D
α
)
c_{0}(\mathfrak{D}^{\alpha})
,
c
(
D
α
)
c(\mathfrak{D}^{\alpha})
and
ℓ
∞
(
D
α
)
\ell_{\infty}(\mathfrak{D}^{\alpha})
defined by the matrix
D
α
\mathfrak{D}^{\alpha}
.
We develop Schauder bases and determine 𝛼-, 𝛽- and 𝛾-duals of these new spaces.
We characterize some matrix transformation from
ℓ
p
(
D
α
)
\ell_{p}(\mathfrak{D}^{\alpha})
,
c
0
(
D
α
)
c_{0}(\mathfrak{D}^{\alpha})
,
c
(
D
α
)
c(\mathfrak{D}^{\alpha})
and
ℓ
∞
(
D
α
)
\ell_{\infty}(\mathfrak{D}^{\alpha})
to
ℓ
∞
\ell_{\infty}
, 𝑐,
c
0
c_{0}
and
ℓ
1
\ell_{1}
.
Furthermore, we determine some criteria for compactness of an operator (or matrix) from
X
∈
{
ℓ
p
(
D
α
)
,
c
0
(
D
α
)
,
c
(
D
α
)
,
ℓ
∞
(
D
α
)
}
X\in\{\ell_{p}(\mathfrak{D}^{\alpha}),c_{0}(\mathfrak{D}^{\alpha}),c(\mathfrak{D}^{\alpha}),\ell_{\infty}(\mathfrak{D}^{\alpha})\}
to
ℓ
∞
\ell_{\infty}
, 𝑐,
c
0
c_{0}
or
ℓ
1
\ell_{1}
.