Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces

A definition of Bessel’s sequences in spaces with an indefinite metric is introduced as a generalization of Bessel’s sequences in Hilbert spaces. Moreover, a complete characterization of Bessel’s sequences in the Hilbert space associated to a space with an indefinite metric is given. The fundamental tools of Bessel’s sequences theory are described in the formalism of spaces with an indefinite metric. It is shown how to construct a Bessel’s sequences in spaces with an indefinite metric starting from a pair of Hilbert spaces, a condition is given to decompose a Bessel’s sequences into in spaces with an indefinite metric so that this decomposition generates a pair of Bessel’s sequences for the Hilbert spaces corresponding to the fundamental decomposition. In spaces where there was no norm, it seemed impossible to construct Bessel’s sequences. The fact that in [1] frame were constructed for Krein spaces motivated us to construct Bessel’s sequences for spaces of indefinite metric.

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Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space

Theoretical and Mathematical Physics ◽

10.1007/bf01017080 ◽

1992 ◽

Vol 92 (3) ◽

pp. 1032-1037 ◽

Cited By ~ 2

Author(s):

Yu. G. Shondin

Keyword(s):

Metric Space ◽

Differential Operators ◽

Extension Theory ◽

Indefinite Metric ◽

Linear Relations ◽

High Codimension ◽

Indefinite Metric Space

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Essentially self-adjoint linear relations in Hilbert spaces

Periodica Mathematica Hungarica ◽

10.1007/s10998-020-00373-8 ◽

2021 ◽

Author(s):

Marcel Roman ◽

Adrian Sandovici

Keyword(s):

Hilbert Spaces ◽

Linear Relations

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Canonical Graph Contractions of Linear Relations on Hilbert Spaces

Complex Analysis and Operator Theory ◽

10.1007/s11785-020-01066-3 ◽

2021 ◽

Vol 15 (1) ◽

Author(s):

Zsigmond Tarcsay ◽

Zoltán Sebestyén

Keyword(s):

Hilbert Spaces ◽

Linear Relation ◽

Regular Part ◽

Linear Relations ◽

Closed Linear Relation ◽

Canonical Graph

AbstractGiven 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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