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.
This paper deals with left star, star, and core partial orders for complex matrices. For each partial order, we present an order-isomorphism between the down-set of a fixed matrix B and a certain set ...(depending on the partial order) of orthogonal projectors whose matrix sizes can be considerably smaller than that of the matrix B. We study the lattice structure and we give properties of the down-sets. We prove that the down-set of B ordered by the core partial order and by the star partial order are sublattices of the down-set ordered by the left star partial order. We analize the existence of supremum and infimum of two given matrices and we give characterizations of these operations (whenever they exist). Some of the results given in the paper are already known in the literature but we present a different proof based on the previously established order-isomorphism.
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.
We give a simple necessary and sufficient condition for the existence of star supremum for two arbitrary operators on a Hilbert space. It is shown that the results of Hartwig (1979) 15, and Janowitz ...(1983) 18, when applied to the ring of bounded operators on a Hilbert space, require much simpler conditions. We also give a complete answer to a question stated by Hartwig and Drazin (1982) in 16, regarding the extremal values of range and null-space of the star infimum.
Partial orders on B(H) Cvetković-Ilić, Dragana S.; Mosić, Dijana; Wei, Yimin
Linear algebra and its applications,
09/2015, Letnik:
481
Journal Article
Recenzirano
Odprti dostop
In this paper we characterize the sets of all B∈B(H) such that AρB and the sets of all B∈B(H) such that BρA, where A∈B(H) is given and ρ∈{≤−,≤⁎,≤#,≤⊕}.
Let H be an infinite-dimensional complex Hilbert space and let B(H) be the algebra of all bounded linear operators on H. The equivalent definition of the DMP relation on B(H), using operator block ...matrices, is introduced. We present a new interpretation of this relation which allows to generalize many known results for matrices to general Hilbert spaces regarding the DMP relation and its relationship with the star order and minus partial order. Also some properties of DMP relation, defined with the help of idempotent operators, are investigated.
We define and investigate a class of partial orders, called one-sided
w
-core partial orders, extending the recently introduced
w
-core partial order. We give several characterizations. In ...particular, it is shown that
a
is below
b
under the left
w
-core partial order (resp., right
w
-core partial order) if and only if
a
is below
b
under the left star partial order (resp.,
wa
is below
wb
under the right sharp partial order, provided that
w
is invertible). Also, the relationship with several well-known other partial orders is considered.
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.
We introduce the notions of coherent and precoherent elements in a Rickart *-ring, generalizing this concept from the ring of bounded operators on a Hilbert space. Some interesting properties of such ...elements are demonstrated, resembling those of bounded operators, e.g. the range additivity and the parallel summation. As an application, we solve some problems regarding the star partial order on Rickart *-rings.
PDF
