Adjoint of sums and products of operators in Hilbert spaces

Sebestyén, Zoltán; Tarcsay, Zsigmond

doi:10.14232/actasm-015-809-3

Cited by 16 publications

(9 citation statements)

References 17 publications

(32 reference statements)

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“…We adapt the terminology of M. H. Stone [19] and say that S and T are adjoint to each other if they satisfy (1.1) and write S ∧ T, in that case (cf. also [8,14,18]). Our main purpose in this paper is to provide a method to verify whether the operators S and T under the weaker condition S ∧ T satisfy the stronger property S * = T , or the much stronger one of being adjoint of each other, i.e., S * = T and T * = S. In this direction our main results are Theorem 2.2 and Theorem 3.1 which give necessary and sufficient conditions by means of the operator matrix M S,T := I −T S I acting on the product Hilbert space H × K.…”

Section: Introductionmentioning

confidence: 92%

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On the adjoint of Hilbert space operators

Sebestyén

Tarcsay

2018

Linear and Multilinear Algebra

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In general, it is a non trivial task to determine the adjoint S * of an unbounded operator S acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator T to be identical with S * . In our considerations, a central role is played by the operator matrix M S,T = I −T S I . Our approach has several consequences such as characterizations of closed, normal, skew-and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that T * T always has a positive selfadjoint extension.1991 Mathematics Subject Classification. Primary 47A05, 47B25. Key words and phrases. Adjoint, closed operator, selfadjoint operator, positive operator, symmetric operator.

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Section: Introductionmentioning

confidence: 92%

“…A well-known assumption for (4.1) is that any of the operators be bounded (see [10]). For more general results the reader may consult [2,3,5,9,18,21]. In the next theorem we provide necessary and sufficient conditions in order that (4.1) be satisfied (cf.…”

Section: Adjoint Of Sums and Productsmentioning

confidence: 99%

On the adjoint of Hilbert space operators

Sebestyén

Tarcsay

2018

Linear and Multilinear Algebra

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“…for cf. also [10,14]. The general case of linear relations was discussed in [11] in the same spirit.…”

Section: Linear Relations Adjoint To Each Othermentioning

confidence: 97%

“…A different approach to the the same issue was treated in the recent paper [15] of the authors. The interested reader may also consult with [10,11,13,14,16]. In Sect.…”

Section: Introductionmentioning

confidence: 99%

Canonical Graph Contractions of Linear Relations on Hilbert Spaces

Tarcsay

Sebestyén

2021

Complex Anal. Oper. Theory

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Given a closed linear relation T between two Hilbert spaces $$\mathcal {H}$$ H and $$\mathcal {K}$$ K , the corresponding first and second coordinate projections $$P_T$$ P T and $$Q_T$$ Q T are both linear contractions from T to $$\mathcal {H}$$ H , and to $$\mathcal {K}$$ K , respectively. In this paper we investigate the features of these graph contractions. We show among other things that $$P_T^{}P_T^*=(I+T^*T)^{-1}$$ P T P T ∗ = ( I + T ∗ T ) - 1 , and that $$Q_T^{}Q_T^*=I-(I+TT^*)^{-1}$$ Q T Q T ∗ = I - ( I + T T ∗ ) - 1 . The ranges $${\text {ran}}P_T^{*}$$ ran P T ∗ and $${\text {ran}}Q_T^{*}$$ ran Q T ∗ are proved to be closely related to the so called ‘regular part’ of T. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed.

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“…The equality in (2.8) has received attention in [17][18][19][20][21][22][23][24] recently. The results in the present paper are closely related and can be used in these considerations; cf.…”

Section: Lemma 22mentioning

confidence: 99%

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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Adjoint of sums and products of operators in Hilbert spaces

Cited by 16 publications

References 17 publications

On the adjoint of Hilbert space operators

On the adjoint of Hilbert space operators

Canonical Graph Contractions of Linear Relations on Hilbert Spaces

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

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