Last Updated：2025/11/30
(mathematics) A theorem in operator theory, stating that, if T is a contraction on a Hilbert space H, and U is its minimal unitary dilation acting on some Hilbert space K, and R is an operator on H commuting with T, then there is an operator S on K commuting with U such that RTⁿ=P_HSUⁿ|_H;∀n≥0, and ‖S‖=‖R‖. In other words, an operator from the commutant of T can be lifted
to an operator in the commutant of the unitary dilation of T.
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commutant lifting theorem
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Source Word
commutant lifting theorem
Proper noun
(mathematics)
A
theorem
in
operator
theory,
stating
that,
if
T
is
a
contraction
on
a
Hilbert
space
H,
and
U
is
its
minimal
unitary
dilation
acting
on
some
Hilbert
space
K,
and
R
is
an
operator
on
H
commuting
with
T,
then
there
is
an
operator
S
on
K
commuting
with
U
such
that
RTⁿ=P_HSUⁿ|_H;∀n≥0,
and
‖S‖=‖R‖.
In
other
words,
an
operator
from
the
commutant
of
T
can
be
"lifted"
to
an
operator
in
the
commutant
of
the
unitary
dilation
of
T.
Japanese Meaning
(数学/演算子論) ヒルベルト空間上の縮小 T に対して、T と可換な演算子 R が存在すると、その可換性を保ったまま、T の最小ユニタリ拡大 U に作用する対応する演算子 S（‖S‖ = ‖R‖）が存在することを示す定理。
Sense（1）
(mathematics)
A
theorem
in
operator
theory,
stating
that,
if
T
is
a
contraction
on
a
Hilbert
space
H,
and
U
is
its
minimal
unitary
dilation
acting
on
some
Hilbert
space
K,
and
R
is
an
operator
on
H
commuting
with
T,
then
there
is
an
operator
S
on
K
commuting
with
U
such
that
RTⁿ=P_HSUⁿ|_H;∀n≥0,
and
‖S‖=‖R‖.
In
other
words,
an
operator
from
the
commutant
of
T
can
be
"lifted"
to
an
operator
in
the
commutant
of
the
unitary
dilation
of
T.
