We present several distance formulae in the space of compact operators in terms of extreme points and semi-inner-products. We characterize best approximation to an element out of a subspace and ...obtain a sufficient condition for unique best approximation. The usefulness of the results is demonstrated with examples. Finally, using the distance formulae we characterize approximate Birkhoff-James orthogonality of an element to a subspace in the space of compact operators.
We characterize the left-symmetric points as well as the right-symmetric points in the sense of Birkhoff-James orthogonality, of the Banach spaces
. As an application of our study, we produce an ...elementary proof of the well-known result: T is an isometry on
if and only if T is a signed permutation. This illustrates the pivotal role played by the set of left-symmetric points in determining the isometry group of a given Banach space.
Let
be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace
we partially solve the operator version of Birkhoff-James orthogonality problem, if
is orthogonal ...to
when does there exist a unit vector x
0
such that
and
is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for
in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for
is a L
1
-predual space as well as a M-ideal in Y, we show that if
is orthogonal to
then there is a unit vector x
0
with
and
is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved.
In this paper, we discuss a version of Birkhoff-James orthogonality in the
-algebra of all bounded linear operators on a finite-dimensional Hilbert space. We obtain a characterization of this ...relation and use it to describe some classes of operators. We also characterize linear preservers of this type of orthogonality.
Very recently, motivated by the result of Bhatia and Šemrl which characterizes the Birkhoff-James orthogonality of operators on a finite dimensional Hilbert space in terms of norm attaining points, ...the Bhatia-Šemrl property was introduced. The main purpose of this article is to study the denseness of the set of multilinear maps with the Bhatia-Šemrl property which is contained in the set of norm attaining ones. Contrary to the most of previous results which were shown for operators on real Banach spaces, we prove the denseness for multilinear maps on some complex Banach spaces. We also show that the denseness of operators does not hold when the domain space is c0 for arbitrary range. Moreover, we find plenty of Banach spaces Y such that only the zero operator has the Bhatia-Šemrl property in the space of operators from c0 to Y.
We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let T,A∈B(X,Y), where X is a real Banach space and Y is a real normed linear space. ...We find sufficient condition for T⊥BA⇔Tx⊥BAx for some x∈SX with ‖Tx‖=‖T‖, and use it to show that T is a smooth point in B(X,Y) if T attains its norm at unique (upto multiplication by scalar) vector x∈SX, Tx is a smooth point of Y and supy∈C‖Ty‖<‖T‖ for all closed subsets C of SX with d(±x,C)>0. For operators on a Hilbert space H we show that T⊥BA⇔Tx⊥BAx for some x∈SH with ‖Tx‖=‖T‖ if and only if the norm attaining set MT={x∈SH:‖Tx‖=‖T‖}=SH0 for some finite dimensional subspace H0 and ‖T‖Ho⊥<‖T‖. We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.
We study the best coapproximation problem in the Banach space
$ \ell _1^n, $
ℓ
1
n
,
by using Birkhoff-James orthogonality techniques. Given a subspace
$ \mathbb {{Y}} $
Y
of
$ \ell _1^n $
ℓ
1
n
, we ...completely identify the elements x in
$ \ell _1^n, $
ℓ
1
n
,
for which best coapproximations to x out of
$ \mathbb {{Y}} $
Y
exist. The methods developed in this article are computationally effective and it allows us to present an algorithmic approach to the concerned problem. We also identify the coproximinal subspaces and co-Chebyshev subspaces of
$ \ell _1^n $
ℓ
1
n
.
We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of ...the operator in terms of the geometry of the concerned spaces. In particular, it is shown that such a study is related to the preservation of k-smoothness and the extremal properties of the unit ball of a Banach space. As an application of the results obtained in this direction, we obtain a refinement of the well-known Blanco-Koldobsky-Turnsek characterization of isometries on some polyhedral Banach spaces, including ℓ∞n,ℓ1n.
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