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.
In the setting of various complex Banach spaces we consider the questions of when a square of a Hermitian operator is also Hermitian, and when a Hermitian operator has a Hermitian square root. We ...also describe the structure of Hermitian projections and Hermitian square roots of the identity operator.
In this note we generalize the well-known Wigner’s unitary-antiunitary theorem. For smooth normed spaces
X
and
Y
and a surjective mapping
f
:
X
→
Y
such that
|
f
(
x
)
,
f
(
y
)
|
=
|
x
,
y
|
,
x
...,
y
∈
X
, where
·
,
·
is the unique semi-inner product, we show that
f
is phase equivalent to either a linear or an anti-linear surjective isometry. When
X
and
Y
are smooth real normed spaces and
Y
is strictly convex, we show that Wigner’s theorem is equivalent to
{
‖
f
(
x
)
+
f
(
y
)
‖
,
‖
f
(
x
)
-
f
(
y
)
‖
}
=
{
‖
x
+
y
‖
,
‖
x
-
y
‖
}
,
x
,
y
∈
X
.
Phase-isometries between normed spaces Ilišević, Dijana; Omladič, Matjaž; Turnšek, Aleksej
Linear algebra and its applications,
03/2021, Letnik:
612
Journal Article
Recenzirano
Odprti dostop
Let X and Y be real normed spaces and f:X→Y a surjective mapping. Then f satisfies {‖f(x)+f(y)‖,‖f(x)−f(y)‖}={‖x+y‖,‖x−y‖}, x,y∈X, if and only if f is phase equivalent to a surjective linear ...isometry, that is, f=σU, where U:X→Y is a surjective linear isometry and σ:X→{−1,1}. This is a Wigner's type result for real normed spaces.
Generalized circular projections Ilišević, Dijana; Li, Chi-Kwong; Poon, Edward
Journal of mathematical analysis and applications,
11/2022, Letnik:
515, Številka:
1
Journal Article
Recenzirano
We study r-circular projections of matrix norms with some special properties, including the unitarily invariant norms, the unitary congruence invariant norms, and the unitary similarity invariant ...norms. In each case, we determine all values r for the existence of r-circular projections corresponding to isometries of a certain form.
For f:M→N, where M, N are finite or infinite dimensional inner product spaces or Hilbert C⁎-modules over compact operators, we investigate the superstability of the Wigner equation, where the class ...of its approximate solutions is defined by the inequality||〈f(x),f(y)〉|−|〈x,y〉||≤φ(x,y),x,y∈M.
Let
X
and
Y
be normed spaces over
F
∈
{
R
,
C
}
and
f
:
X
→
Y
a surjective mapping. Suppose that
|
ϕ
f
(
y
)
(
f
(
x
)
)
|
=
|
ϕ
y
(
x
)
|
holds for all
x
,
y
∈
X
and all support functionals
ϕ
f
(
y
...)
at
f
(
y
) and
ϕ
y
at
y
, or equivalently, suppose that for all semi-inner products on
X
and
Y
, compatible with given norms,
|
f
(
x
)
,
f
(
y
)
|
=
|
x
,
y
|
holds for all
x
,
y
∈
X
. Then
f
=
σ
U
, where
σ
:
X
→
F
is a phase function, and
U
:
X
→
Y
is a linear or a conjugate linear isometry.
On square roots of isometries Ilišević, Dijana; Kuzma, Bojan
Linear & multilinear algebra,
09/2019, Letnik:
67, Številka:
9
Journal Article
Recenzirano
In various normed spaces we answer the question of when a given isometry is a square of some isometry. In particular, we consider (real and complex) matrix spaces equipped with unitarily invariant ...norms and unitary congruence invariant norms, as well as some infinite dimensional spaces illustrating the difference between finite and infinite dimensions.
Let
A
be a JB∗-triple and let
P
:
A
→
A
be a linear projection. It is proved that
P
+
λ
(
Id
-
P
)
is an isometry for some modulus one complex number
λ
≠
1
if and only if either
λ
=
-
1
,
or
P
is ...hermitian. It is also proved that every rank one bicontractive projection on
A
is hermitian. The particular case when
A
is a C∗-algebra is discussed through several examples.
A nonzero projection P on a complex Banach space X is said to be a generalized tricircular projection if there exist distinct modulus one complex numbers λ and μ, not equal to 1, and nonzero ...projections Q and R on X such that P⊕Q⊕R=I and P+λQ+μR is an isometry. We determine the structure of generalized tricircular projections on minimal norm ideals in B(H), different from the Hilbert–Schmidt class.
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