Dilation and Birkhoff-James orthogonality Pal, Sourav; Roy, Saikat
Journal of mathematical analysis and applications,
09/2024, Volume:
537, Issue:
2
Journal Article
Peer reviewed
For any two elements x and y in a normed space X, x is said to be Birkhoff-James orthogonal to y, denoted by x⊥By, if ‖x+λy‖≥‖x‖ for every scalar λ. Also, for any ε∈0,1), x is said to be ...ε-approximate Birkhoff-James orthogonal to y, denoted by x⊥Bεy, if‖x+λy‖2≥‖x‖2−2ε‖x‖‖λy‖,for all scalarsλ. For any ρ>0, a unitary operator U acting on a Hilbert space K is said to be a unitary ρ-dilation of an operator T on a Hilbert space H if H⊆K and Tn=ρPHUn|H for every nonnegative integer n, where PH:K→H is the orthogonal projection. Also, when ρ=1 and T is a contraction, U is called a unitary dilation of T. We obtain the following main results.(1)We find necessary and sufficient conditions such that for any two contractions T,A on H, their Schäffer unitary dilations UT˜ and UA˜ on the space ⊕−∞∞H are Birkhoff-James orthogonal. Also, counter example shows that in general UT˜⊥̸BUA˜ even if T⊥BA.(2)For any ρ>0 and for two Hilbert space operators T,A with T⊥BA, we show that if ‖T‖=ρ then UT⊥BUA for any unitary ρ-dilations UT of T and UA of A acting on a common Hilbert space. Also, we show by an example that the condition that ‖T‖=ρ cannot be ignored.(3)For any ρ>0, we explicitly construct examples of Hilbert space operators T,A such that T⊥̸BA but any of their unitary ρ-dilations UT,UA acting on a common Hilbert space are Birkhoff-James orthogonal.(4)We find a characterization for the ε-approximate Birkhoff-James orthogonality of operators on complex Hilbert spaces.(5)For any ρ>0 and for any Hilbert space operators T,A, we find a sharp bound on ε such that T⊥BA implies UT⊥BεUA for any unitary ρ-dilations UT of T and UA of A acting on a common space. Also, we show by an example that in general the bound on ε cannot be improved.(6)We construct families of generalized Schäffer-type unitary dilations for a Hilbert space contraction in two different ways. Then we show that one of them preserves Birkhoff-James orthogonality while any two members UT,UA from the other family are always Birkhoff-James orthogonal irrespective of the orthogonality of T and A.(7)We show that Andô dilation of a pair of commuting contractions of the form (T,ST), where S is a unitary that commutes with T, are orthogonal. Also, we explore orthogonality of regular unitary dilation of a pair of commuting contractions. However, Birkhoff-James orthogonality is independent of commutativity of operators.
We characterize the class of surjective (conjugate) linear mappings Φ:B(H)→B(H) that preserve the strong Birkhoff–James orthogonality in both directions. We also give characterizations of rank-one ...operators as well as coisometries in terms of the strong Birkhoff–James orthogonality.
Let
A
be a unital
C
∗
-algebra with unit
1
A
and let
a
∈
A
be a positive and invertible element. Suppose that
S
(
A
)
is the set of all states on
A
and let
S
a
(
A
)
=
f
f
(
a
)
:
f
∈
S
(
A
)
,
f
(
a
...)
≠
0
.
The norm
‖
x
‖
a
for every
x
∈
A
is defined by
‖
x
‖
a
=
sup
φ
∈
S
a
(
A
)
φ
(
x
∗
a
x
)
.
In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm
‖
·
‖
a
in
A
,
namely
a
-Birkhoff–James orthogonality. The characterization of
a
-Birkhoff–James orthogonality in
A
by means of the elements of generalized state space
S
a
(
A
)
is provided. As an application, a characterization for the best approximation to elements of
A
in a subspace
B
with respect to
‖
·
‖
a
is obtained. Moreover, a formula for the distance of an element of
A
to the subspace
B
=
C
1
A
is given. We also study the strong version of
a
-Birkhoff–James orthogonality in
A
.
The classes of
C
∗
-algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong
a
-Birkhoff–James orthogonality and
A
-valued inner product orthogonality in
A
implies that the center of
A
is trivial. Finally, we show that if the (strong)
a
-Birkhoff–James orthogonality is right-additive (left-additive) in
A
,
then the center of
A
is trivial.
In this paper we characterize Birkhoff–James orthogonality of linear operators defined on a finite dimensional real Banach space X. We also explore the left symmetry of Birkhoff–James orthogonality ...of linear operators defined on X. Using some of the related results proved in this paper, we finally prove that T∈L(lp2) (p≥2,p≠∞) is left symmetric with respect to Birkhoff–James orthogonality if and only if T is the zero operator.
We present a complete characterization of the right-symmetric points in the one sum of two Banach spaces. We also obtain some basic properties of the left-symmetric (right-symmetric) points in the p ...sum,
$ 1\leq p\not =2 \lt \infty $
1
≤
p
≠
2
<
∞
(
$ 1 \lt p\not =2 \lt \infty $
1
<
p
≠
2
<
∞
), of two Banach spaces. Using these properties we (a) give examples of Banach spaces which do not have any non-zero left-symmetric points and (b) prove a complete characterization of those left-symmetric and right-symmetric points in the p sum,
$ 1 \lt p\not =2 \lt \infty $
1
<
p
≠
2
<
∞
, of two Banach spaces, whose components satisfy an additional norm assumption. We give examples of Banach spaces where all non-zero left-symmetric or right-symmetric points satisfy this additional norm assumption. We also present an alternative proof of the recently obtained characterization of the left-symmetric and the right-symmetric points in
$ \ell _p^n $
ℓ
p
n
,
$ n\geq 3 $
n
≥
3
, and
$ \ell _p $
ℓ
p
,
$ 1\leq p\not =2 \lt \infty $
1
≤
p
≠
2
<
∞
.
Let
be a Hilbert
-module over a
-algebra
and let
be the set of states on
. In this paper, we first compute the norm derivative for nonzero elements x and y of
as follows:
We then apply it to ...characterize different concepts of orthogonality in
. In particular, we present a simpler proof of the classical characterization of Birkhoff-James orthogonality in Hilbert
-modules. Moreover, some generalized Daugavet equation in the
-algebra
of all bounded linear operators acting on a Hilbert space
is solved.
For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, ...characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.
In this paper we explore the connection between strict convexity of a real normed linear space X and orthogonality of operators in the sense of Birkhoff–James in K(X), the space of all compact linear ...operators on X. We prove that a real reflexive Banach space X is strictly convex iff for any T,A∈K(X), T⊥BA⇒T⊥SBA or Ax=0 for some x∈SX with ‖Tx‖=‖T‖. We prove that if H is an infinite dimensional real Hilbert space and T∈K(H), then for all A∈B(H), A⊥BT⇒T⊥BA if and only if T is the zero operator. We also prove that for a real Hilbert space H, T⊥BA⇒A⊥BT for all A∈B(H) if and only if T is the zero operator.
A Banach space X is said to be isomorphic to another Y with respect to the structure of Birkhoff-James orthogonality, denoted by X∼BJY, if there exists a (possibly nonlinear) bijection between X and ...Y that preserves Birkhoff-James orthogonality in both directions. It is shown that X≅Y if either X or Y is finite dimensional and X∼BJY, and that ℓp≁BJℓq if 1<p<q<∞. Moreover, if H is a Hilbert space with dimH≥3 and H∼BJX, then H=X. In the two-dimensional case, it turns out that ℓp,q2∼BJℓ22, which indicates that nonlinear Birkhoff-James orthogonality preservers between Banach spaces are not necessarily scalar multiples of isometric isomorphisms.
We give necessary and sufficient condition that an element of an arbitrary C⁎-algebra is an isolated vertex of the orthograph related to the mutual strong Birkhoff-James orthogonality. Also, we prove ...that for all C⁎-algebras except C,C⊕C and M2(C) all non isolated points make a single connected component of the orthograph which diameter is less than or equal to 4, i.e. any two non isolated points can be connected by a path with at most 4 edges. Some related results are given.
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