180 research outputs found
Linear Fractional Transformations of Nevanlinna Functions Associated with a Nonnegative Operator
No full textIn the present paper a subclass of scalar Nevanlinna functions is studied, which coincides with the class of Weyl functions associated to a nonnegative symmetric operator of defect one in a Hilbert space. This class consists of all Nevanlinna functions that are holomorphic on (−∞, 0) and all those Nevanlinna functions that have one negative pole a and are injective on (−∞, a) ∪ (a, 0). These functions are characterized via integral representations and special attention is paid to linear fractional transformations which arise in extension and spectral problems of symmetric and selfadjoint operators.
Contributions to management science, mathematics and modelling : essays in honour of professor Ilkka Virtanen
No full textKirjoittaja: Bräysy Olli, Dullaert Wout,
Van de Weyer Geert, Hassi Seppo, Laaksonen Matti, Luhta Irma, Luoma-Martti, Nikkinen Jussi,
Sahlström Petri, Malaska Pentti, Kasanen Eero,
Moisio Marko J., Okko Paavo, Perttunen Jukka, Pihlanto Pekka, Pynnönen Seppo, Hogan Warren,
Batten Jonathan, Reponen Tapio, Salmi Timo,
Töyli Juuso, Kanto Antti, Vuolle-Apiala Juhafi=vertaisarvioitu|en=peerReviewed|ei tietoa saavutettavuudest
Contributions to Mathematics and Statistics : Essays in honor of Seppo Hassi
No full textThis Festschrift contains thirteen articles in honor of the sixtieth birthday of Professor Seppo Hassi (University of Vaasa). It centers on three topics: functional analysis and operator theory, boundary value problems, and statistics, stochastics, and the history of mathematics.
The collection contains four papers on the topic of functional analysis and operator theory. More precisely, it includes a paper treating the transformation of operator-valued Nevanlinna functions and the congruence of their associated realizing operators, a paper treating Parseval frames in the setting of Krein spaces, a paper treating algebraic inclusions of relations as well as the generalized inverses of relations, and a paper treating Krein-von Neumann and Friedrichs extensions by means of energy spaces.
Boundary value problems are considered in six of the contributions. In particular, singular perturbations of the Dirac operator are treated by means of the technique of boundary triplets, the connection between sectorial Schrödinger L-systems and certain classes of Weyl-Titchmarsh functions is considered, PT-symmetric Hamiltonians are treated from the perspective of couplings of dual pairs, the Riesz basis property of indefinite Sturm-Liouville problems is considered, the stability properties of spectral characteristics of boundary value problems are investigated, and the completeness and minimality of systems of eigenfunctions and associated functions of ordinary differential operators are treated.
Finally, the collection also contains three contributions connected with the topics of statistics, stochastics, and the history of mathematics. More precisely, a new statistic is introduced for the testing of cumulative abnormal returns in the case of partially overlapping event windows, a new characterization of Brownian motion is established, and, finally, a history of (the department of) mathematics and statistics at the University of Vaasa is presented.fi=vertaisarvioimaton|en=nonPeerReviewed|ei tietoa saavutettavuudest
Probability Error Bounds for Approximation of Functions in Reproducing Kernel Hilbert Spaces
No full textWe find probability error bounds for approximations of functions f in a separable reproducing kernel Hilbert space H with reproducing kernel K on a base space X, firstly in terms of finite linear combinations of functions of type Kxi and then in terms of the projection πxn on spanKxii=1n, for random sequences of points x=xii in X. Given a probability measure P, letting PK be the measure defined by dPKx=Kx,xdPx, x∈X, our approach is based on the nonexpansive operator L2X;PK∋λ↦LP,Kλ≔∫XλxKxdPx∈H, where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by HP, that is the operator range of LP,K. Our main result establishes bounds, in terms of the operator LP,K, on the probability that the Hilbert space distance between an arbitrary function f in H and linear combinations of functions of type Kxi, for xii sampled independently from P, falls below a given threshold. For sequences of points xii=1∞ constituting a so-called uniqueness set, the orthogonal projections πxn to spanKxii=1n converge in the strong operator topology to the identity operator. We prove that, under the assumption that HP is dense in H, any sequence of points sampled independently from P yields a uniqueness set with probability 1. This result improves on previous error bounds in weaker norms, such as uniform or Lp norms, which yield only convergence in probability and not almost certain convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space H2D are presented as well
Functional models for Nevanlinna families
No full textTyt. z nagł.References p. 244.Dostępny również w formie drukowanej.ABSTRACT: The class of Nevanlinna families consists of R-symmetric holomorphic multivalued functions on
Extension theory for elliptic partial differential operators with pseudodifferential methods
No full textThis is a short survey on the connection between general extensiontheories and the study of realizations of elliptic operators A on smooth domainsin R^n, n >1. The theory of pseudodifferential boundary problems has turnedout to be very useful here, not only as a formulational framework, but alsofor the solution of specific questions. We recall some elements of that theory,and show its application in several cases (including new results), namely tothe lower boundedness question, and the question of spectral asymptotics fordifferences between resolvents
Lebesgue type decompositions and Radon–Nikodym derivatives for pairs of bounded linear operators
No full textFor a pair of bounded linear Hilbert space operators A and B one considers the Lebesgue type decompositions of B with respect to A into an almost dominated part and a singular part, analogous to the Lebesgue decomposition for a pair of measures in which case one speaks of an absolutely continuous and a singular part. A complete parametrization of all Lebesgue type decompositions will be given, and the uniqueness of such decompositions will be characterized. In addition, it will be shown that the almost dominated part of B in a Lebesgue type decomposition has an abstract Radon–Nikodym derivative with respect to the operator A.© 2022 The Authors. Published by Springer. 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
Holomorphic operator-valued functions generated by passive selfadjoint systems
No full textLet M be a Hilbert space. In this paper we study a class RS(m) of operator functions that are holomorphic in the domain C∖{(−∞,−1] ∪ [1,+∞)} and whose values are bounded linear operators in m . The functions in RS(m) are Schur functions in the open unit disk D and, in addition, Nevanlinna functions in C+∪C− . Such functions can be realized as transfer functions of minimal passive selfadjoint discrete-time systems.We give various characterizations for the class RS(m) and obtain an explicit form for the inner functions from the class RS(m) as well as an inner dilation for any function from RS(m) . We also consider various transformations of the class RS(m) , construct realizations of their images, and find corresponding fixed points.fi=vertaisarvioitu|en=peerReviewed
On Rank One Perturbations of Selfadjoint Operators
No full textLet A be a selfadjoint operator in a Hilbert space h. Its rank one perturbations A+τ(·,ω)ω, τ ∈ R, are studied when ω belongs to the scale space h-2 associated with h+2 = dom A and (·,·) is the corresponding duality. If A is nonnegative and ω belongs to the scale space h-1, it is proven that the spectral measures of A(τ), τ ∈ R, converge weakly to the spectral measure of the limiting perturbation A(∞). In fact A(∞) can be identified as a Friedrichs extension. Further results for nonnegative operators A were obtained by allowing ω ∈ h-2. Our purpose is to show that most of these results are valid for rank one perturbations of selfadjoint operators, which are not necessarily semibounded. We use the fact that rank one perturbations constitute selfadjoint extensions of an associated symmetric operator. The use of so-called Q-functions facilitates the descriptions. In the special case that ω belongs to the scale space h-1 associated with h+1 = dom |A|^½, the limiting perturbation A(∞) is shown to be the generalized Friedrichs extension.
Spectral Decompositions of Selfadjoint Relations in Pontryagin Spaces and Factorizations of Generalized Nevanlinna Functions
No full textSelfadjoint relations in Pontryagin spaces do not possess a spectral family completely characterizing them in the way that selfadjoint relations in Hilbert spaces do. Here it is shown that a combination of a factorization of generalized Nevanlinna functions with the standard spectral family of selfadjoint relations in Hilbert spaces can function as a spectral family for selfadjoint relations in Pontryagin spaces. By this technique additive decompositions are established for generalized Nevanlinna functions and selfadjoint relations in Pontryagin spaces.© Springer Nature Switzerland AG 2020.fi=vertaisarvioitu|en=peerReviewed
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