On operators which are adjoint to each other

Popovici, Dan; Sebestyén, Zoltán

doi:10.14232/actasm-012-857-7

Cited by 9 publications

(16 citation statements)

References 13 publications

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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%

“…We close the paper by characterizing orthogonal projections (cf. also [8,Corollary 3.7]): Theorem 9.3. Let P be a symmetric linear operator in a Hilbert space H. The following assertions (i)-(iii) are equivalent:…”

Section: Normal Unitary and Orthogonal Projection Operatorsmentioning

confidence: 98%

“…The importance of the role of the operator matrix M S,T initiates the recent papers of the authors [8,9,14,16,18]. The "flip" operator W : K × H → H × K will also be useful for our analysis, which is defined as follows (see [2]):…”

Section: Characterization Of the Adjoint Of A Linear Operatormentioning

confidence: 99%

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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%

Section: Normal Unitary and Orthogonal Projection Operatorsmentioning

confidence: 98%

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On the adjoint of Hilbert space operators

Sebestyén

Tarcsay

2018

Linear and Multilinear Algebra

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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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“…We also notice that the equivalence of (i) and (ii) below is taken from [11, Theorem 5.1], cf. also [8,Corollary 3.6]. For sake of brevity we also adopt the notation (iii) A is symmetric and there exists c ∈ R, c = 0, such that…”

Section: Perturbation Theorems For Selfadjoint and Essentially Selfadmentioning

confidence: 99%

Adjoint of sums and products of operators in Hilbert spaces

Sebestyén¹,

Tarcsay²

2016

Acta Sci. Math.

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Abstract. We provide sufficient and necessary conditions guaranteeing equations (A + B) * = A * + B * and (AB) * = B * A * concerning densely defined unbounded operators A, B between Hilbert spaces. We also improve the perturbation theory of selfadjoint and essentially selfadjoint operators due to Nelson, Kato, Rellich, and Wüst. Our method involves the range of two-by-two matrices of the form M S,T = I −T S I that makes it possible to treat real and complex Hilbert spaces jointly.

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On operators which are adjoint to each other

Cited by 9 publications

References 13 publications

On the adjoint of Hilbert space operators

On the adjoint of Hilbert space operators

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

Adjoint of sums and products of operators in Hilbert spaces

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