In 1973/74 Bennett and (independently) Carl proved that for
1
≤
u
≤
2
1 \le u \le 2
the identity map id:
ℓ
u
↪
ℓ
2
\ell _u \hookrightarrow \ell _2
is absolutely
(
u
,
1
)
(u,1)
-summing, i. e., for every unconditionally summable sequence
(
x
n
)
(x_n)
in
ℓ
u
\ell _u
the scalar sequence
(
‖
x
n
‖
ℓ
2
)
(\|x_n \|_{\ell _2})
is contained in
ℓ
u
\ell _u
, which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a
2
2
-concave symmetric Banach sequence space
E
E
the identity map
id
:
E
↪
ℓ
2
\text {id}: E \hookrightarrow \ell _2
is absolutely
(
E
,
1
)
(E,1)
-summing, i. e., for every unconditionally summable sequence
(
x
n
)
(x_n)
in
E
E
the scalar sequence
(
‖
x
n
‖
ℓ
2
)
(\|x_n \|_{\ell _2})
is contained in
E
E
. Various applications are given, e. g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator
T
T
on
ℓ
2
\ell _2
with values in a
2
2
-concave symmetric Banach sequence space
E
E
is a multiplier from
ℓ
2
\ell _2
into
E
E
. Furthermore, we prove an asymptotic formula for the
k
k
-th approximation number of the identity map
id
:
ℓ
2
n
↪
E
n
\text {id}: \ell _2^n \hookrightarrow E_n
, where
E
n
E_n
denotes the linear span of the first
n
n
standard unit vectors in
E
E
, and apply it to Lorentz and Orlicz sequence spaces.