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Seppo Hassi

Publications and source records attributed to Seppo Hassi.

27 records · Page 2Linked to original sources
arXiv2014

Factorization, majorization, and domination for linear relations

Let ${\mathfrak H}_A$, ${\mathfrak H}_B$, and ${\mathfrak H}$ be Hilbert spaces. Let $A$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_A$ and let $B$ be a linear relation from ${\mathfrak H}$ to ${\mathfrak H}_B$. If there exists an operator $Z \in \mathbf{B}({\mathfrak H}_B,{\mathfrak H}_A)$ such that $ZB \subset A$, then $B$ is said to dominate $A$. This notion plays a major role in the theory of Lebesgue type decompositions of linear relations and operators. There is a strong connection to the majorization and factorization in the well-known lemma of Douglas, when put in the context of linear relations. In this note some aspects of the lemma of Douglas are discussed in the context of linear relations and the connections with the notion of domination will be treated.

math.FA↗
arXiv2013

Q-functions and boundary triplets of nonnegative operators

Operator-valued $Q$-functions for special pairs of nonnegative selfadjoint extensions of nonnegative not necessarily densely defined operators are defined and their analytical properties are studied. It is shown that the Kre\uın-Ovcharenko statement announced in \cite{KrO2} is valid only for $Q$-functions of densely defined symmetric operators with finite deficiency indices. A general class of boundary triplets for a densely defined nonnegative operator is constructed such that the corresponding Weyl functions are of Kre\uın-Ovcharenko type.

math.FA↗
arXiv2012

Unitary equivalence of proper extensions of a symmetric operator and the Weyl function

Let $A$ be a densely defined simple symmetric operator in $\gH$, let $Π=\bt$ be a boundary triplet for $A^*$ and let $M(\cd)$ be the corresponding Weyl function. It is known that the Weyl function $M(\cd)$ determines the boundary triplet $Π$, in particular, the pair ${A,A_0}$, where $A_0:= A^*\lceil\ker\G_0 (= A^*_0)$, uniquely up to unitary similarity. At the same time the Weyl function corresponding to a boundary triplet for a dual pair of operators defines it uniquely only up to weak similarity. In this paper we consider symmetric dual pairs ${A,A}$ generated by $A\subset A^*$ and special boundary triplets $\wtΠ$ for ${A,A}$. We are interested whether the result on unitary similarity remains valid provided that the Weyl function corresponding to $\wtΠ$ is $\wt M(z)= K^*(B-M(z))^{-1} K,$ where $B$ is some non-self-adjoint bounded operator in $\cH$. We specify some conditions in terms of the operators $A_0$ and $A_B= A^*\lceil \ker(\G_1-B\G_0)$, which determine uniquely (up to unitary equivalence) the pair ${A,A_B}$ by the Weyl function $\wt M(\cd)$. Moreover, it is shown that under some additional assumptions the Weyl function $M_Π(\cdot)$ of the boundary triplet $Π$ for the dual pair $\DA$ determines the triplet $Π$ uniquely up to unitary similarity. We obtain also some negative results demonstrating that in general the Weyl function $\wt M(\cd)$ does not determine the operator $A_B$ even up to similarity.

math.FA↗
arXiv2010

On J-Self-Adjoint Operators with Stable C-Symmetry

The paper is devoted to a development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. The main attention is paid to the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results are specialized further by studying in detail the case where S has defect numbers <2,2>.

math.FA↗
arXiv2010

Truncated moment problems in the class of generalized Nevanlinna functions

Truncated moment problems in the class of generalized Nevanlinna functions are investigated. General solvability criteria will be established, covering both the even and odd problems, including complete parametrizations of solutions. The main new results concern the case where the corresponding Hankel matrix of moments is degenerate. One of the new effects which reveals in the indefinite case is that the degenerated moment problem may have infinitely many solutions. However, with a careful application of an indefinite analogue of a step-by-step Schur algorithm a complete description of the set of solutions will be obtained.

math.FA↗
arXiv2008

On Symmetries in the Theory of Finite Rank Singular Perturbations

For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $\mathfrak{H}$ singular perturbations of the form $A_0+V, \ V=\sum_{1}^{n}{b}_{ij}<ψ_j,\cdot>ψ_i$ are studied under some additional requirements of symmetry imposed on the initial operator $A_0$ and the singular elements $ψ_j$. A concept of symmetry is defined by means of a one-parameter family of unitary operators $\sU$ that is motivated by results due to R. S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials $V$ and the corresponding self-adjoint realizations of $A_0+V$. The results are applied for the investigation of singular perturbations of the Schrödinger operator in $L_2(\dR^3)$ and for the study of a (fractional) \textsf{p}-adic Schrödinger type operator with point interactions.

math.SP↗
arXiv2006

Boundary relations and generalized resolvents of symmetric operators

The Kre\uın-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\uın-Naimark formula is investigated and explained.

math.SP↗
arXiv2005

A general realization theorem for matrix-valued Herglotz-Nevanlinna functions

New special types of stationary conservative impedance and scattering systems, the so-called non-canonical systems, involving triplets of Hilbert spaces and projection operators, are considered. It is established that every matrix-valued Herglotz-Nevanlinna function of the form V(z)=Q+Lz+\int_{\dR}(\frac{1}{t-z}-\frac{t}{1+t^2})dΣ(t) can be realized as a transfer function of such a new type of conservative impedance system. In this case it is shown that the realization can be chosen such that the main and the projection operators of the realizing system satisfy a certain commutativity condition if and only if L=0. It is also shown that $V(z)$ with an additional condition (namely, $L$ is invertible or L=0), can be realized as a linear fractional transformation of the transfer function of a non-canonical scattering $F_+$-system. In particular, this means that every scalar Herglotz-Nevanlinna function can be realized in the above sense. Moreover, the classical Livsic systems (Brodskii-Livsic operator colligations) can be derived from $F_+$-systems as a special case when $F_+=I$ and the spectral measure $dΣ(t)$ is compactly supported. The realization theorems proved in this paper are strongly connected with, and complement the recent results by Ball and Staffans.

math.SP↗