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

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

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

Passive systems with a normal main operator and quasi-selfadjoint systems

Passive systems $\tau={T,M,N,H}$ with $M$ and $N$ as an input and output space and $H$ as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system $\tau$ with $M=N$ is said to be quasi-selfadjoint if $ran(T-T^*)\subset N$. The subclass $S^{qs}$ of the Schur class $S$ is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass $S^{qs}$ is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass $S^{qs}$ and the $Q$-function of $T$ is given.Comment: 29 page

Boundary relations and generalized resolvents of symmetric operators

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in \dC_+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page