Adjoints and formal adjoints of matrices of unbounded operators

Möller, Manfred; Szafraniec, Franciszek Hugon

doi:10.1090/s0002-9939-08-09211-3

Cited by 22 publications

(19 citation statements)

References 15 publications

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“…The work [23] with Szafraniec has influenced further work along these lines: it suffices to mention [22], where the original Lebesgue decompositions have been extended to a more general context. Furthermore, the joint work [34] has had a direct influence on [21]. Columns, rows, and blocks are now introduced in [21] not only for operators but also for relations; a simple example of this was already encountered in [26].…”

Section: Decompositions and Extensions For Operators And Relationsmentioning

confidence: 99%

Mathematical work of Franciszek Hugon Szafraniec and its impacts

et al. 2020

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Section: Decompositions and Extensions For Operators And Relationsmentioning

confidence: 99%

Mathematical work of Franciszek Hugon Szafraniec and its impacts

et al. 2020

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“…The calculus of rows, columns, and block matrices, where the entries are unbounded operators, was studied in the paper of Möller and Szafraniec [13]; see also [8]. However, already in the case of 2 Â 2 blocks, it is natural that one of the entries is multivalued; see [2,6,11,18].…”

Section: Introductionmentioning

confidence: 99%

“…The basic idea in the context of linear relations is to first develop the notions of a row and a column consisting of a sequence of linear relations. This approach was taken in the study of matrices with unbounded operators in [13]. Rows and columns can be considered as the building blocks in the definition of a block matrix whose entries are linear relations.…”

Section: Introductionmentioning

confidence: 99%

“…The present paper is of an expository nature. It gives a brief survey of the above notions, along the lines of [13] where the case of operators was treated, but now in the context of linear relations, see also [4,12,[14][15][16]. However, the appearance of multivalued parts in the entries of the block produces some new obstacles in this study.…”

Section: Introductionmentioning

confidence: 99%

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Operational calculus for rows, columns, and blocks of linear relations

Hassi

Labrousse²,

Snoo

2020

Adv. Oper. Theory

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Columns and rows are operations for pairs of linear relations in Hilbert spaces, modelled on the corresponding notions of the componentwise sum and the usual sum of such pairs. The introduction of matrices whose entries are linear relations between underlying component spaces takes place via the row and column operations. The main purpose here is to offer an attempt to formalize the operational calculus for block matrices, whose entries are all linear relations. Each block relation generates a unique linear relation between the Cartesian products of initial and final Hilbert spaces that admits particular properties which will be characterized. Special attention is paid to the formal matrix multiplication of two blocks of linear relations and the connection to the usual product of the unique linear relations generated by them. In the present general setting these two products need not be connected to each other without some additional conditions. Keywords Linear relationDedicated to our friend Franek Szafraniec on the occasion of his eightieth birthday. Communicated by Lajos Molnar.

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“…Blocks of linear relations are built on the treatment of columns and rows of linear relations in [13]. For a related general treatment of blocks of linear operators, see [20]; see also [21]. A characterization of linear relations as block relations will be given later elsewhere; cf.…”

Section: Introductionmentioning

confidence: 99%

Selfadjoint extensions of relations whose domain and range are orthogonal

Hassi¹,

Labrousse²,

Snoo³

2020

Methods Funct. Anal. Topol.

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The selfadjoint extensions of a closed linear relation R from a Hilbert space H 1 to a Hilbert space H 2 are considered in the Hilbert space H 1 ⊕ H 2 that contains the graph of R. They will be described by 2 × 2 blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation S in H 1 ⊕ H 2 that is induced by R. Such a relation is characterized by the orthogonality property dom S ⊥ ran S and it is nonnegative. All nonnegative selfadjoint extensions A, in particular the Friedrichs and Kreȋn-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that A belongs to the class of extremal extensions of S if and only if dom A ⊥ ran A. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of S and semibounded parameters describing them if and only if the operator part of R is bounded.

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Adjoints and formal adjoints of matrices of unbounded operators

Cited by 22 publications

References 15 publications

Mathematical work of Franciszek Hugon Szafraniec and its impacts

Mathematical work of Franciszek Hugon Szafraniec and its impacts

Operational calculus for rows, columns, and blocks of linear relations

Selfadjoint extensions of relations whose domain and range are orthogonal

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