The minus order and range additivity

Djikić, Marko S.; Fongi, Guillermina; Maestripieri, Alejandra

doi:10.1016/j.laa.2017.05.045

Cited by 14 publications

(18 citation statements)

References 31 publications

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“…Also A * W A = A * W B, so that W 1/2 A * ≤ W 1/2 B. In the same way W 1/2 B * ≤ W 1/2 C, then since * ≤ is transitive, W 1/2 A * ≤ W 1/2 C and therefore A * W A = A * W C. On the other hand, if A * W ≤ B and B * W ≤ C, then A − ≤ B and B − ≤ C, and since − ≤ is transitive (see [18,Proposition 3.11]), we also have that A − ≤ C. Then, by Proposition 5.1, A * W ≤ C and the relation * W ≤ is a partial order. In a similar way, it can be proven that * W ≤ , ≤ * W are transitive.…”

Section: Definition Givenmentioning

confidence: 92%

“…The left minus order was defined in [18] for operators in L(H). Using Theorem 3.3, it is easy to see that this notion is weaker than the minus order, in the infinite dimensional setting.…”

Section: Minus Order and Compatibilitymentioning

confidence: 99%

“…In [20], Drazin introduced the star order on semigroups with involutions and in [8] More generally, given a positive operator W, we now introduce the weighted star order, which is the minus order with an additional condition on the angle between the ranges of the operators involved, imposed by the weight W. Proof. Observe that * W ≤ , ≤ * W and * W ≤ are clearly reflexive and also antisymmetric, because − ≤, ≤ − and − ≤ are antisymmetric (see [18,Proposition 3.11]).…”

Section: Weighted Star Order and Applicationsmentioning

confidence: 99%

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Shorted operators and minus order

Contino

Giribet

Maestripieri

2018

Linear and Multilinear Algebra

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Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W ∈ L(H) a positive operator. Given a closed subspace S of H, we characterize the shorted operator W /S of W to S as the maximum and as the infimum of certain sets, for the minus order − ≤. Also, given A ∈ L(H) with closed range, we study the following operator approximation problem considering the minus order:We show that, under certain conditions, the shorted operator of W /R(A) is the minimum of this problem and we characterize the set of solutions. KEYWORDS Shorted operators; minus order; oblique projections ≤ B (where the symbol − ≤ stands for the minus order of operators) if the Dixmier angle between R(A) and R(B − A) and the Dixmier angle between R(A * ) and R(B * − A * ) are less than 1, where R(T ) stands for the range of the operator T. Independently, in [31]Šemrl gave another characterization of the minus order, he showed that A − ≤ B if and only if there are bounded (oblique) projections, i.e. idempotents, P and Q such that A = P B and A * = QB * . In [18], a new characterization of the minus order for operators acting on Hilbert spaces was given in terms of the so called range additivity property. Namely, it was proved that A − ≤ B is equivalent to the range of B being the direct sum of ranges of A and B − A and the range of B * being the direct sum of ranges of A * and B * − A * , which generalizes previous results presented in [31]. This plays an equivalent role to the rank additivity characterization when A and B are matrices [21, 26]. In [5] the notion of shorted operator appears in relation with the minus order. Given a closed subspace S of H and W ∈ L(H) a positive operator, in 1947, Krein [23], proved the existence of a maximum (with respect to the order induced by the cone of positive operators) of the set M(W, S) = {X ∈ L(H) : 0 ≤ X ≤ W and R(X) ⊆ S ⊥ }.Krein used this extremal operator in his theory of extension of symmetric operators. Years later, Anderson and Trapp [2] studying the same problem, called this maximum the shorted operator of W to S (in the following denoted W /S ) and showed interesting properties of this operator and its connections with electrical circuit theory. The shorted operator have shown to be useful in many applications [5], [32].The pair (W, S) is said to be compatible if there exists a bounded linear (not necessarily selfadjoint) projection Q onto S such that W Q is selfadjoint. Thus, ifthen (W, S) is compatible if and only if P(W, S) is not empty. In [15], it was shown that there exists a strong relationship between compatibility, the projections of P(W, S) and the shorted operator W /S . Later, in [5], the notion of shorted operator was generalized to that of bilateral shorted operator, for an operator W ∈ L(H) (not necessarly positive) and a pair of closed subspaces. The proposed definition comes from the notion of weak complementability, which is a refinement of a finite dimensional notion due to T. Ando [3]. In that paper, it was proven that given an operator W ∈ L(H) positive an...

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Section: Definition Givenmentioning

confidence: 92%

“…The left minus order was defined in [18] for operators in L(H). Using Theorem 3.3, it is easy to see that this notion is weaker than the minus order, in the infinite dimensional setting.…”

Section: Minus Order and Compatibilitymentioning

confidence: 99%

Section: Weighted Star Order and Applicationsmentioning

confidence: 99%

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Shorted operators and minus order

Contino

Giribet

Maestripieri

2018

Linear and Multilinear Algebra

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“…Different definitions where given for the minus order, for example, using generalized inverses in the matrix case, see [33]. We give the following definition, equivalent to those appearing in [38] and [20]. Definition.…”

Section: Applications: Schur Complements Of Selfadjoint Operatorsmentioning

confidence: 99%

Semiclosed projections and applications

Contino

Maestripieri

Marcantognini

2021

Journal of Mathematical Analysis and Applications

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“…Finally, given T ∈ P • L h with R(T )+N (T ) = H we shall prove that A N is optimal in A T with respect to the minus order in L(H). For this we use the following result due to Dijić, Fongi and Maestripieri [13,Proposition 3.2]). where the minimum is taken with respect to the minus order.…”

Section: Optimal Decompositionsmentioning

confidence: 99%