Let R be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary ⁎-ring case. It is shown that the group, Moore–Penrose, ...core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in R are obtained. It is shown that core and dual core inverse belong to the class of inverses along an element and to the class of (b,c)-inverses.
A positive answer on a question posed in Eshkaftaki et al. DSS-weak majorization and its linear preservers on
spaces. Linear Multilinear Algebra. 2018;66(10):2076-2088 is provided. Precisely, the ...concrete form of bounded linear operators which preserve DSS-weak majorization on discrete Lebesgue space
, when I is an infinite set, is presented. Interestingly, it is shown that the class of DSS-weak majorization preservers is a proper subset of the class of majorization preservers on
, despite the fact that these two classes of preservers coincide Eshkaftaki et al. DSS-weak majorization and its linear preservers on
spaces. Linear Multilinear Algebra. 2018;66(10):2076-2088 when above relations are defined on
when
. In this way, the conjecture given at the end of the paper Eshkaftaki et al. DSS-weak majorization and its linear preservers on
spaces. Linear Multilinear Algebra. 2018;66(10):2076-2088 is confirmed.
The aim of this short note is to indicate that Rao and Mitra's constrained matrix inverse AcrCR (introduced in 1972) is a direct precursor of recently introduced Drazin's (b,c) outer generalized ...inverse. Precisely, if one rewrites definition of AcrCR in purely multiplicative language, then it turns out that two inverses coincide. Recently introduced Mary's inverse along an element and Manjunatha Prasad and Mohana's core-EP inverse are the special cases of Rao and Mitra's constrained inverse.
Let
be a unital ring admitting involution. We introduce an order on
and show that in the case when
is a Rickart
-ring, this order is equivalent to the well-known star partial order. The notion of the ...left-star and the right-star partial orders is extended to Rickart
-rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart
-rings and some known results are generalized. We found matrix forms of elements
and
when
, where
is one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated.
The star, sharp, core and dual core partial orders are already known and investigated in the set of complex matrices. In this paper these orders are extensively studied in the context of arbitrary ...ring with involution. It is shown that the condition a<b, where < is one of the considered partial orders, defines appropriate idempotents and thus the suitable matrix representations of a and b are obtained. Using these representations we proved that a<b if and only if G(b)⊆G(a), where G(a) is appropriate subset of the set of all generalized inverses of a. If <- is the minus partial order, the conditions under which a<-b implies a<b are determined. Also, several well-known results are generalized and some new results are proved.
The core inverse of matrix is generalized inverse which is in some sense in-between the group and Moore–Penrose inverse. In this paper a generalization of core inverse and core partial order to ...Hilbert space operator case is presented. Some properties are generalized and some new ones are proved. Connections with other generalized inverses are obtained. The useful matrix representations of operator and its core inverse are given. It is shown that A is less than B under the core partial order if and only if they have specific kind of simultaneous diagonalization induced by appropriate decompositions of Hilbert space. The relation is also characterized by the inclusion of appropriate sets of generalized inverses. The spectral properties of core inverse are also obtained.
The classes of 1MP-inverses and MP1-inverses are recently introduced classes of generalized inverses of complex matrices. Actually, they coincide with the classes of
{
1
,
2
,
3
}
and
{
1
,
2
,
4
}
...inverses, respectively. We consider these inverses in the context of a ring with involution and prove that their most important characterizations and properties remain true. We show that the binary relations based on these inverses are in fact the well known left-star and right-star partial orders. We extend these relations to the ring case, connect them with the unified theory of partial order relations based on generalized inverses and provide several properties. Finally, we indicate how these results can be applied to bounded Hilbert space operators.
INFINITE TOPOLOGICAL DIRECT SUM OF SUBSPACES DJORDJEVIĆ, DRAGAN S.; RAKIĆ, DRAGAN S.
Bulletin de l'Académie serbe des sciences, Classe des sciences mathématiques et naturelles. Sciences mathématiques,
01/2022
47
Journal Article
Recenzirano
We define and investigate infinite topological direct sum of subspaces of a normed space, as well as corresponding matrix decompositions of bounded linear operators.
We study the star and minus partial orders on the set
$ {\mathcal {B}}(H,K) $
B
(
H
,
K
)
of all bounded operators acting from a Hilbert space H to a Hilbert space K. We extend and strengthen some ...results from matrix case to the case of general operators, which may not possess generalized inverses. By means of some norm inequalities, we give necessary and sufficient condition under which two operators have orthogonal ranges, and thus, we give a characterization of the star partial order. When A = PB for some projection P, we prove the equivalence of
$ A \lt ^*B $
A
<
∗
B
with
$ f(AA^*)A \lt ^*f(BB^*)B $
f
(
A
A
∗
)
A
<
∗
f
(
B
B
∗
)
B
for a wide class of continuous functions f. Also, we prove that
$ A \lt ^*B $
A
<
∗
B
if and only if
$ A \lt ^-B $
A
<
−
B
and
$ A^2 \lt ^{-}B^{2} $
A
2
<
−
B
2
when A is a weak EP operator and B is a self-adjoint operator. Finally, we consider the Moore-Penrose invertibility and the ordinary invertibility of a linear combination of operators when they are related with one of these two orders.
The unified theory for matrix partial orders based on generalized inverses has already been done by Mitra. We consider a special kind of ring R, which generalizes the ring of linear operators on ...finite dimensional vector space, and extend Mitra's approach to it. Thus, we find necessary and sufficient condition for a G-based relation to be a partial order on R. Also, some known results are generalized and some new results are proved.
PDF
