Completion, extension, factorization, and lifting of operators with a negative index

The famous results of M.G. Kre\u{\i}n concerning the description of selfadjoint contractive extensions of a Hermitian contraction $T_1$ and the characterization of all nonnegative selfadjoint extensions $\widetilde A$ of a nonnegative operator $A$ via the inequalities $A_K\leq \widetilde A \leq A_F$, where $A_K$ and $A_F$ are the Kre\u{\i}n-von Neumann extension and the Friedrichs extension of $A$, are generalized to the situation, where $\widetilde A$ is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators $I-T_1^*T_1$ and $A$, respectively; these conditions are automatically satisfied if $T_1$ is contractive or $A$ is nonnegative, respectively. The approach developed in this paper starts by establishing first a generalization of an old result due to Yu.L. Shmul'yan on completions of $2\times 2$ nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M.G. Kre\u{\i}n and, in addition, to solve some related lifting problems for $J$-contractive operators in Hilbert, Pontryagin and Kre\u{\i}n spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.Comment: 29 page

Componentwise and Cartesian decompositions of linear relations

Let $A$ be a, not necessarily closed, linear relation in a Hilbert space \sH with a multivalued part \mul A. An operator $B$ in \sH with \ran B\perp\mul A^{**} is said to be an operator part of $A$ when A=B \hplus (\{0\}\times \mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of $A$. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation $A$ is said to have a Cartesian decomposition if A=U+\I V, where $U$ and $V$ are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of $A$ and the real and imaginary parts of $A$ is investigated

Representations of closed quadratic forms associated with Stieltjes and inverse Stieltjes holomorphic families of linear relations

In this paper holomorphic families of linear relations that belong to the Stieltjes or inverse Stieltjes class are studied. It is shown that in their domain of holomorphy C∖R+ the values of Stieltjes and inverse Stieltjes families are, up to a rotation, maximal sectorial. This leads to a study of the associated closed sesquilinear forms and their representations. In particular, it is shown that the Stieltjes and inverse Stieltjes holomorphic families of linear relations are of type (B) in the sense of Kato. These results are proved by using linear fractional transforms which connect these families to holomorphic functions that belong to a combined Nevanlinna-Schur class and a key tool then relies on a specific structure of contractive operators. Розглядаються голоморфні сім’ї лінійних відношень, які належать до класу Стілтьєса та оберненого класу Стілтьєса. Показано, що в їхній області голоморфності C∖R+ значення цих сімей є, з точністю до обертання, максимальними секторіальними. Із цим пов’язане дослідження відповідних замкнених півторалінійних форм та їхніх представлень. Зокрема, показано, що стілтьєсівські та обернені стілтьєсівські голоморфні сім’ї лінійних відношень належать до типу (В) у сенсі Като. Доведення базується на використанні дробово-лінійних перетворень, які переводять розглядувані сім’ї в голоморфні функції класу Неванлінни-Шура, псля чого використовується спеціальні структури операторів стиску.©2021 the Authors. Published by Methods of Functional Analysis and Topology (MFAT). The authors retain the copyright for their papers published in MFAT under the terms of the Creative Commons Attribution-ShareAlike License (CC BY-SA).fi=vertaisarvioitu|en=peerReviewed

Sequences of Operators, Monotone in the Sense of Contractive Domination

A sequence of operators Tn from a Hilbert space H to Hilbert spaces Kn which is nondecreasing in the sense of contractive domination is shown to have a limit which is still a linear operator T from H to a Hilbert space K. Moreover, the closability or closedness of Tn is preserved in the limit. The closures converge likewise and the connection between the limits is investigated. There is no similar way of dealing directly with linear relations. However, the sequence of closures is still nondecreasing and then the convergence is governed by the monotonicity principle. There are some related results for nonincreasing sequences.© The Author(s) 2024. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.fi=vertaisarvioitu|en=peerReviewed

Unitary boundary pairs for isometric operators in Pontryagin spaces and generalized coresolvents

An isometric operator V in a Pontryagin space H is called standard, if its domain and the range are nondegenerate subspaces in H. A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized coresolvents of non-standard Pontryagin space isometric operators are described. The methods used in this paper rely on a new general notion of boundary pairs introduced for isometric operators in a Pontryagin space setting. Even in the Hilbert space case this notion generalizes the earlier concept of boundary triples for isometric operators and offers an alternative approach to study operator valued Schur functions without any additional invertibility requirements appearing in the ordinary boundary triple approach.Comment: 42 page

Complementation and Lebesgue-type decompositions of linear operators and relations

In this paper, a new general approach is developed to construct and study Lebesgue-type decompositions of linear operators or relations T in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue-type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of T. The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue-type decomposition for pairs of quadratic forms. The approach built in this paper uses so-called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.© 2024 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed

Selfadjoint extensions of relations whose domain and range are orthogonal

© 2020 Authors. The authors retain the copyright for their papers published in MFAT under the terms of the Creative Commons Attribution-ShareAlike License (CC BY-SA).fi=vertaisarvioitu|en=peerReviewed