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Let [Formula: see text] be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all Moore–Penrose invertible elements in [Formula: see text]. We study partial orders on [Formula: see text] that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart ∗-rings the concept of another partial order, called the plus order, which has been recently introduced on the set of all bounded linear operators between Hilbert spaces. Properties of these relations are investigated and some known results are thus generalized.

As solutions of some operator equations, we present two new outer inverses for bounded linear operators between two Hilbert spaces. Precisely, we introduce and characterize the weighted weak core i...

Let M n ( F ) be the set of all n × n matrices over the field F , where F = R or F = C . We characterize maps on M n ( F ) which are surjective and preserve either the left-star or the right-star order in both directions, but are not necessarily additive.

We investigate properties of the minus partial order in regular modules, present new characterizations of this order, and generalize some known results. We introduce a new relation in the general module theoretic setting that is analogous to the space pre-order on complex matrices and study how it is related to the minus partial order.

Skripta »Opcije, terminske pogodbe in model binomskega drevesa« je namenjena slušateljem predmeta Temelji finančnega inženiringa, ki se izvaja na Faklteti za naravoslovje in matematiko. V njej so predstavljeni temeljni pojmi finančnega inženiringa. Najprej so podane predpostavke, na katerih temelji večina matematičnih modelov, ki simulirajo dogajanja na finančnih trgih. Dokazi trditev in izrekov so utemeljeni na načelu nearbitražnosti. Predstavljeni so nekateri izvedeni finančni instrumenti. Tako so v osrednjem delu skripte najprej analizirane standandizirane in nestandardizirane terminske pogodbe. Sledi predstavitev modela binomskega drevesa in njegova uporaba pri vrednotenju evropskih in ameriških nakupnih ter prodajnih opcij. Skripta je opremljena s številnimi rešenimi zgledi in primeri.

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all core-EP invertible elements in an arbitrary unital ring [Formula: see text] with involution. A new partial order is introduced on [Formula: see text] by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

Let (mathcal{R}) be a ring with identity and let $mathcal{J}_{mathcal{R}}$ be a collection of subsets of $mathcal{R}$ such that their left and right annihilators are generated by the same idempotent. We extend the notion of the sharp, the left-sharp, and the right-sharp partial orders to $mathcal{J}_{mathcal{R}}$, present equivalent definitions of these orders, and study their properties. We also extend the concept of the core and the dual core orders to $mathcal{J}_{mathcal{R}}$, show that they are indeed partial orders when $mathcal{R}$ is a Baer $ast$-ring, and connect them with one-sided sharp and star partial orders.

Let H n + $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$ (ℝ) be the cone of all positive semidefinite (symmetric) n × n real matrices. Matrices from H n + $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$ (ℝ) play an important role in many areas of engineering, applied mathematics, and statistics, e.g. every variance-covariance matrix is known to be positive semidefinite and every real positive semidefinite matrix is a variance-covariance matrix of some multivariate distribution. Three of the best known partial orders that were mostly studied on various sets of matrices are the Löwner, the minus, and the star partial orders. Motivated by applications in statistics authors have recently investigated the form of maps on H n + $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$ (ℝ) that preserve either the Löwner or the minus partial order in both directions. In this paper we continue with the study of preservers of partial orders on H n + $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$ (ℝ). We characterize surjective, additive maps on H n + $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$ (ℝ), n ≥ 3, that preserve the star partial order in both directions. We also investigate the form of surjective maps on the set of all symmetric real n × n matrices that preserve the Löwner partial order in both directions.

Let H be a Hilbert space and B(H) the set of all bounded linear operators on H. In the paper we consider the generalized concept of order relations on B(H) which was proposed by Šemrl and which covers the star partial order, the left-star partial order, the right-star partial order, and the minus partial order. We also connect this concept with the sharp partial order.

Let A be a Rickart ring and let A ( 1 ) be the set of all regular elements in A . The set of all a ∈ A such that a ≤ b are characterized, where b ∈ A ( 1 ) is given and ≤ is the minus partial order. In case when A is a Rickart *-ring, such sets are characterized for the diamond, the left-star, the right-star, the left-sharp, and the right-sharp partial orders. Some recent results of Mosic et al. on partial orders in B ( H ) , the algebra of all bounded linear operators on a Hilbert space H , are thus generalized.

We study orders in proper $*$-rings that are derived from the core-nilpotent decomposition. The notion of the C-N-star partial order and the S-star partial order is extended from $M_ {n} ( \mathbb{C)}$, the set of all $n \times n$ complex matrices, to the set of all Drazin invertible elements in proper $*$-rings with identity. Properties of these orders are investigated and their characterizations are presented. For a proper $*$-ring $\mathcal{A}$ with identity, it is shown that on the set of all Drazin invertible elements $a \in \mathcal{A}$ where the core part of $a$ is an EP element, the C-N-star partial order implies the star partial order.

We study orders in unital rings that are derived from the core-nilpotent decomposition. The notion of the Drazin order, the C-N partial order and the S-minus partial order is extended from , the set of all matrices over a field , to the set of all Drazin invertible elements in rings with identity. Properties of these orders are investigated, their characterizations are presented, and some known results are thus generalized.

Bijective maps on matrices over arbitrary fields with sufficiently many elements which preserve Lie product are characterized.

The minus partial order is already known for sets of matrices over a field and bounded linear operators on arbitrary Hilbert spaces. Recently, this partial order has been studied on Rickart rings. In this paper, we extend the concept of the minus relation to the module theoretic setting and prove that this relation is a partial order when the module is regular. Moreover, various characterizations of the minus partial order in regular modules are presented and some well-known results are also generalized., 15 pages

Let $H_{n}^{+}(\mathbb{R})$ be the cone of all positive semidefinite $n\times n$ real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L��wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on $H_{n}^{+}(\mathbb{R})$. We describe the form of all surjective maps on $H_{n}^{+}(\mathbb{R})$, $n>1$, that preserve the L��wner partial order in both directions. We present an equivalent definition of the minus partial order on $H_{n}^{+}(\mathbb{R})$ and also characterize all surjective, additive maps on $H_{n}^{+}(\mathbb{R})$, $n\geq3$, that preserve the minus partial order in both directions., 17 pages

We study properties of a relation in *-rings, called the core-EP (pre)order which was introduced by H. Wang on the set of all n × n complex matrices [Linear Algebra Appl., 2016, 508: 289–300] and has been recently generalized by Y. Gao, J. Chen, and Y. Ke to *-rings [Filomat, 2018, 32: 3073–3085]. We present new characterizations of the core-EP order in *-rings with identity and introduce the notions of the dual core-EP decomposition and the dual core-EP order in *-rings.

Let ϕ be an injective, continuous, Lie product preserving map on Mn (ℝ), n > 3. In the paper we show that then there exist an invertible matrix T ∈ Mn (ℝ) and a continuous function ψ:Mn (ℝ)→ ℝ, where ψ(A) = 0 for all matrices of trace zero, such that either ϕ(A) = TAT −1 + ψ(A)I for all A ∈ Mn (ℝ), or ϕ(A) = −TAtT−1 + ψ(A)I for all A ∈ Mn (ℝ). We determine that a similar proposition holds true for the set Mn (ℂ), n > 3.

We classify spectrum preserving additive bijections which map the set of linear (possibly unbounded) maps on a complex Banach space X to the set of linear (possibly unbounded) maps on a complex Banach space Y .

We consider the generalized concept of order relations in Rickart rings and Rickart -rings which was proposed by Semrl and which covers the star partial order, the left-star partial order, the right-star partial order and the minus partial order. We show that on Rickart rings the definitions of orders introduced by Jones and Semrl are equivalent. We also connect the generalized concept of order relations with the sharp order and prove that the sharp order is a partial order on the subset of elements in a ring with identity which have the group inverse. Properties of the sharp partial order in are studied and some known results are generalized.

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.

Let H be an infinite dimensional complex Hilbert space, and let B(H) be the set of all bounded linear operators on H . In the paper equivalent definitions for the left-star and the right-star partial orders on B(H) are given and bijective additive maps on B(H) which preserve the left-star or the right-star partial order in both directions are characterized. Mathematics subject classification (2010): 06A06, 15A03, 15A04, 15A86.

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

Let $\mathcal{A}$ be a Rickart $\ast$ -ring and let $\leq^{\ast},\leq^{\sharp },\leq^{\oplus}$ , and $\leq_{\oplus}$ denote the star, the sharp, the core, and the dual core partial orders in $\mathcal{A}$ , respectively. The sets of all $b\in\mathcal{A}$ such that $a\leq b$ , along with the sets of all $b\in\mathcal{A}$ such that $b\leq a$ , are characterized, where $a\in\mathcal{A}$ is given and where $\leq $ is one of the partial orders: $\leq^{\ast}$ , or $\leq^{\sharp}$ , or $\leq^{\oplus}$ , or $\leq_{\oplus}$ . Such sets of elements that are above or below a given element under the minus partial order $\leq^{-}$ in a Rickart ring $\mathcal{A}$ are also studied. Some recent results of Cvetković-Ilić et al. on partial orders in $\mathcal{B}(H)$ , the algebra of all bounded linear operators on a Hilbert space $H$ , are thus generalized.

Let 𝔻 be an arbitrary division ring and Mn(𝔻) the multiplicative semigroup of all n × n matrices over 𝔻. We describe the general form of non-degenerate homomorphisms from M2(𝔻) to M3(𝔻).

Let H be an infinite-dimensional complex Hilbert space and let B ( H ) be the algebra of all bounded linear operators on ( H ). In the paper the equivalent definition of the star partial order on B ( H ), using selfadjoint idempotent operators, is introduced. Also some properties of the generalized concept of order relations on B ( H ), defined with the help of idempotent operators, are investigated.

Let D be an arbitrary division ring and M n ( D ) the multiplicative semigroup of all n × n matrices over D . We study non-degenerate, injective homomorphisms from M 2 ( D ) to M 4 ( D ) . In particular, we present a structural result for the case when D is the ring of quaternions.

Let X be a compact Hausdorff space which satisfies the first axiom of countability, I = [ 0 , 1 ] and C ( X , I ) the set of all continuous functions from X to I. In the paper we will give a description of bijective maps on C ( X , I ) which preserve the order in both directions.

We generalize the notion of the left-sharp and the right-sharp partial orders to [Formula: see text] where [Formula: see text] is a ring with identity and [Formula: see text] the subset of elements in [Formula: see text] which have the group inverse. We connect these orders to well-known sharp and minus partial orders. Properties of one-sided sharp partial orders in [Formula: see text] are studied and some known results are generalized.

Let $X$ be a compact Hausdorff space which satisfies the first axiom of countability, and $C_+(X)$ the set of all continuous functions from $X$ to $% [0,\infty)$. If $\varphi :C_{+}(X)\rightarrow C_{+}(X)$ is a bijective map which preserves the order in both directions, then there exists a homeomorphism $\omega :X\mathcal{\rightarrow }X$ and for each $x\in X$ a bijective, increasing map $m_{x}:\left[ 0,\infty \right) \mathcal{% \rightarrow }\left[ 0,\infty \right) $ such that $\varphi (f)(x)=m_{x}(f(\omega (x))),$ for all $x\in X$ and $f\in C_{+}(X).$