Let X and Y be smooth normed spaces which are either real and dimX≥3, or infinite dimensional complex, and one of them is reflexive. Then a surjective mapping from X to Y preserves Birkhoff–James ...orthogonality in both directions if and only if it has the form x↦τ(x)Ux for some surjective linear or conjugate linear isometry U:X→Y and some scalar–valued mapping τ on X. In particular, there exists a surjective mapping from X to Y preserving Birkhoff–James orthogonality in both directions if and only if X and Y are isometrically isomorphic or conjugate isometrically isomorphic. Several illustrative examples and relations with Wigner's theorem are also given.
We introduce the notion of geometric structure spaces of Banach spaces equipped with closure space structure. It is shown that the nonlinear equivalence of Banach spaces based on Birkhoff-James ...orthogonality induces homeomorphisms between geometric structure spaces. As applications, we prove that spaces of continuous functions are isomorphic with respect to the structure of Birkhoff-James orthogonality if and only if they are isometrically isomorphic, and that classical sequence spaces c0 and ℓp are completely classified from the viewpoint of Birkhoff-James orthogonality.
Nonlinear commuting maps on ℒ (X) Costara, Constantin
Linear & multilinear algebra,
02/2021, Letnik:
69, Številka:
3
Journal Article
Recenzirano
Let X be a Banach space over the real or the complex field, and denote by
the algebra of all bounded linear operators on it. We characterize maps
such that
Given two different complex numbers
a
and
b
, a bounded linear operator
T
acting on an infinite-dimensional complex Hilbert space
H
is said to be {
a, b
}-quadratic if (
T
−
a
)(
T
−
b
) = 0. We ...provide in this paper a complete description of all surjective maps Φ (not necessarily additive) on the algebra
ℬ
(
H
)
of all bounded linear operators on
H
that satisfy
S
−
λT
is {
a, b
}-quadratic if and only if Φ(
S
) −
λΦ
(
T
) is {
a, b
}-quadratic for every
S
,
T
∈
ℬ
(
H
)
and
λ
∈ ℂ.
Let
M
n
be the algebra of all
n×
n complex matrices and
P
n
the set of all idempotents in
M
n
. Suppose
φ:
M
n
→
M
n
is a surjective map satisfying
A−
λB∈
P
n
if and only if
φ(
A)−
λφ(
B)∈
P
n
,
A,
...B∈
M
n
,
λ∈
C
. Then either
φ is of the form
φ(
A)=
TAT
−1,
A∈
M
n
, or
φ is of the form
φ(
A)=
TA
t
T
−1,
A∈
M
n
, where
T∈
M
n
is a nonsingular matrix.
Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H and
the set of all idempotents in B(H). Suppose
is a surjective map satisfying
if and only ...if
. Then ϕ is either of the form
and T is a continuous invertible linear operator on H, or of the form
and T is a continuous invertible conjugate linear operator on H.
Given a conjugation C on a separable complex Hilbert space H, a bounded
linear operator T on H is said to be C-skew symmetric if CTC = -T*. This
paper describes the maps, on the algebra of all ...bounded linear operators
acting on H, that preserve the difference of C-skew symmetric operators for
every conjugation C on H.
Let
X
be an infinite-dimensional Banach space, and
B
(
X
)
be the algebra of all bounded linear operators on
X
. A map
Δ
, from
B
(
X
)
into a closed subsets of
C
is said to be
∂
-spectrum if
∂
(
σ
(
...T
)
)
⊆
Δ
(
T
)
⊆
σ
(
T
)
for all
T
∈
B
(
X
)
. Here,
σ
(
T
)
is spectrum of
T
and
∂
(
σ
(
T
)
)
the boundary of
σ
(
T
)
. In this paper, we determine the forms of all surjective maps
ϕ
from
B
(
X
)
into itself that satisfy either
Δ
(
ϕ
(
T
)
ϕ
(
S
)
)
=
Δ
(
T
S
)
for all
T
,
S
∈
B
(
X
)
or
Δ
(
ϕ
(
T
)
ϕ
(
S
)
ϕ
(
T
)
)
=
Δ
(
T
S
T
)
for all
T
,
S
∈
B
(
X
)
.
Nonlinear maps preserving certain subspaces Benbouziane, H.; Bouramdane, Y.; Kettani, M. Ech-Chérif El
Proyecciones (Antofagasta),
03/2019, Letnik:
38, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Abstract Let X be a Banach space and let B(X) be the Banach algebra of all bounded linear operators on X. We characterise surjective (not necessarily linear or additive) maps ϕ : B(X) → B(X) such ...that F(ϕ (A)◇ ϕ (B)) = F(A ◇ B) for all A,B ∈ B(X) where F(A) denotes any of R(A) or N(A), anda ◇ B denotes any binary operations A−B, AB and ABA for all A,B ∈B(X).
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