Abstract
In this paper, we study the following extremal problem and its relevance to the sum of the so-called superoptimal singular values of a matrix function: Given an
m
×
n
matrix function
Φ, when is there a matrix function
Ψ
∗
in the set
A
k
n
,
m
such that
∫
T
trace
(
Φ
(
ζ
)
Ψ
∗
(
ζ
)
)
d
m
(
ζ
)
=
sup
Ψ
∈
A
k
n
,
m
|
∫
T
trace
(
Φ
(
ζ
)
Ψ
(
ζ
)
)
d
m
(
ζ
)
|
?
The set
A
k
n
,
m
is defined by
A
k
n
,
m
=
def
{
Ψ
∈
H
0
1
(
M
n
,
m
)
:
‖
Ψ
‖
L
1
(
M
n
,
m
)
⩽
1
,
rank
Ψ
(
ζ
)
⩽
k
a.e.
ζ
∈
T
}
.
To address this extremal problem, we introduce Hankel-type operators on spaces of matrix functions and prove that this problem has a solution if and only if the corresponding Hankel-type operator has a maximizing vector. The main result of this paper is a characterization of the smallest number
k for which
∫
T
trace
(
Φ
(
ζ
)
Ψ
(
ζ
)
)
d
m
(
ζ
)
equals the sum of all the superoptimal singular values of an admissible matrix function
Φ (e.g. a continuous matrix function) for some function
Ψ
∈
A
k
n
,
m
. Moreover, we provide a representation of any such function
Ψ when
Φ is an admissible very badly approximable unitary-valued
n
×
n
matrix function.