Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

Mousavi, Z.; Eskandari, Rasoul; Moslehian, Mohammad Sal; Mirzapour, F.

doi:10.1016/j.laa.2016.12.001

Cited by 10 publications

(3 citation statements)

References 12 publications

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“…Next, we give some conditions under which BB * ≤ AA * is equivalent to R(B) ⊆ R(A). To achieve it, we need the following lemma, which is a modification of [9,Theorem 2.4]. We give its proof for the sake of reader's convenience.…”

Section: Resultsmentioning

confidence: 99%

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Eskandari¹,

Fang²,

Moslehian³

et al. 2021

Preprint

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We extend a work of Pedersen and Takesaki by giving some equivalent conditions for the existence of a positive solution of the so-called Pedersen-Takesaki operator equation XHX = K in the setting of Hilbert C * -modules. It is known that the Douglas lemma does not hold in the setting of Hilbert C * -modules in its general form. In fact, if E is a Hilbert C * -module and A, B ∈ L(E ), then the operator inequality BB * ≤ λAA * with λ > 0 does not ensure that the operator equation AX = B has a solution, in general. We show that under a mild orthogonally complemented condition on the range of operators, AX = B has a solution if and only if BB * ≤ λAA * and R(A) ⊇ R(BB * ). Furthermore, we prove that if L(E ) is a W * -algebra, A, B ∈ L(E ), and R(A * ) = E , then BB * ≤ λAA * for some λ > 0 if and only if R(B) ⊆ R(A). Several examples are given to support the new findings.

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Section: Resultsmentioning

confidence: 99%

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Eskandari¹,

Fang²,

Moslehian³

et al. 2021

Preprint

Self Cite

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“…Unfortunately, a counterexample is constructed in [12] which indicates that the implication "(i)=⇒ (iv)" is false even if R(T * ) is orthogonally complemented. A gap is then contained in [3] to the proof of the implication "(ii)=⇒ (i)", see also [7].…”

Section: Introductionmentioning

confidence: 99%

On majorization and range inclusion of operators on Hilbert C*-modules

Fang

Moslehian

2018

Linear and Multilinear Algebra

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It is proved that for adjointable operators A and B between Hilbert C * -modules, certain majorization conditions are always equivalent without any assumptions on R(A * ), where A * denotes the adjoint operator of A and R(A * ) is the norm closure of the range of A * . In the case that R(A * ) is not orthogonally complemented, it is proved that there always exists an adjointable operator B whose range is contained in that of A, whereas the associated equation AX = B for adjointable operators is unsolvable.2010 Mathematics Subject Classification. 46L08, 47A05.

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“…[21, Theorem 2.4] Let A, C ∈ L(E ). Suppose that ran(A * ) and ran(B * ) are orthogonally complemented in E .…”

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confidence: 99%

Vector-valued reproducing kernel Hilbert $C^*$-modules

Moslehian¹

2021

Preprint

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The aim of this paper is to present a unified framework in the setting of Hilbert C * -modules for the scalar-and vector-valued reproducing kernel Hilbert spaces and C * -valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the C * -algebra of adjointable operators acting on a Hilbert C * -module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert C * -modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and present an interpolation theorem. Moreover, we study some basic properties of the so-called relative reproducing kernel Hilbert C * -modules and give a characterization of dual modules. Among other things, we prove that every conditionally negative definite kernel gives us a reproducing kernel Hilbert C * -module and a certain map. Several examples illustrate our investigation.

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Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

Cited by 10 publications

References 12 publications

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

Pedersen--Takesaki operator equation in Hilbert $C^*$-modules

On majorization and range inclusion of operators on Hilbert C*-modules

Vector-valued reproducing kernel Hilbert $C^*$-modules

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