Self-adjoint Linear Relation Research Articles

Reducibility of Self-Adjoint Linear Relations and Application to Generalized Nevanlinna Functions

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{\kappa_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{\Gamma } )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},\Gamma_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $\kappa =\kappa_{1}+\kappa_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$.

A matrix formula for Schur complements of nonnegative selfadjoint linear relations

If a nonnegative selfadjoint linear relation A in a Hilbert space and a closed subspace S are assumed to satisfy that the domain of A is invariant under the orthogonal projector onto S, then A admits a particular matrix representation with respect to the decomposition S⊕S⊥. This matrix representation of A is used to give explicit formulae for the Schur complement of A on S as well as the S-compression of A.

Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space is decomposed by means of the reducing subspaces of A . The sum of two functions Q i ∈ N κ i ( ℋ ) , i = 1,2 , minimally represented by the triplets ( 𝒦 i , A i , Γ i ) is also studied. For this purpose, we create a model ( 𝒦 ˜ , A ˜ , Γ ˜ ) to represent Q : = Q 1 + Q 2 in terms of ( 𝒦 i , A i , Γ i ) . By using this model, necessary and sufficient conditions for κ = κ 1 + κ 2 are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and the decomposition of the corresponding function Q .

Singular boundary conditions for Sturm–Liouville operators via perturbation theory

AbstractWe show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in \{1,2\}$ . This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as $$ \begin{align*} \boldsymbol{A}_{ {}_{\scriptstyle \Theta}}=\boldsymbol{A}_0+\mathbf{B}\Theta\mathbf{B}^*, \end{align*} $$ where $\boldsymbol {A}_0$ is a distinguished self-adjoint extension and $\Theta $ is a self-adjoint linear relation in $\mathbb {C}^d$ . The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form-bounded with respect to $\boldsymbol {A}_0$ , i.e., it belongs to $\mathcal {H}_{-1}(\boldsymbol {A}_0)$ , with possible “infinite coupling.” A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure that the perturbation is well defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations $\Theta $ .The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

On Self-Adjoint Linear Relations

A linear operator on a Hilbert space , in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if .In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

Essentially self-adjoint linear relations in Hilbert spaces

The main aim of this paper is to provide some range-type criteria for the essentially self-adjointness of symmetric linear relations in real or complex Hilbert spaces. The main tool is a matrix whose entries are certain linear relations.

Shorting, parallel addition and form sums of nonnegative selfadjoint linear relations

We extend the operations of shorting and parallel addition from the cone of bounded nonnegative selfadjoint operators in a Hilbert space to the set of all nonnegative selfadjoint linear relations. New properties of these operations and connections with the Cayley transforms are established. It is shown that for a pair of nonnegative selfadjoint linear relations there exist, in general, two arithmetic–harmonic means. Applications of the arithmetic, harmonic, and arithmetic–harmonic means to the theory of nonnegative selfadjoint extensions of nonnegative symmetric linear relations are given.

On Compressions of Self-Adjoint Extensions of a Symmetric Linear Relation with Unequal Deficiency Indices

Let A be a symmetric linear relation in the Hilbert space ℌ with unequal deficiency indices n−A < n+(A). A self-adjoint linear relation \( \tilde{A}\supset A \)in some Hilbert space \( \tilde{\mathrm{\mathfrak{H}}}\supset \mathrm{\mathfrak{H}} \)is called an (exit space) extension of A. We study the compressions \( C\left(\tilde{A}\right)={P}_{\mathrm{\mathfrak{H}}}\tilde{A}\upharpoonright \mathrm{\mathfrak{H}} \) of extensions \( \tilde{A}=\tilde{A^{\ast }}. \) Our main result is a description of compressions \( C\left(\tilde{A}\right) \) by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter τ(λ) from the Krein formula for generalized resolvents. We describe also all extensions \( \tilde{A}=\tilde{A^{\ast }}. \) of A with the maximal symmetric compression \( C\left(\tilde{A}\right) \) and all extensions \( \tilde{A}=\tilde{A^{\ast }}. \) of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators A with equal deficiency indices n+(A) = n−(A).

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A &lt;n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

Spectral theory for self-adjoint linear relation (SALR) on a Hilbert space and its application in homogenous abstract cauchy problem

A spectral theory studies eigenvalues and eigenvectors of SALR on H. SALR on Hilbert space H is a linear relation satisfying A = A*. Many applications of SALR on quantum theory, such as the homogenous abstract Cauchy problem.If M is an operator that has an inverse then eigenvalues and eigenvectors are easily determined, but If M is an operator that does not have an inverse then eigenvalues and eigenvectors are quite difficult determined. One way that can be done is to use a linear relation. Furthermore, there are some properties of spectral theoryof linear operator that can not apply to SALR. This paper aims to give a spectral theory for SALR and its application in a homogenous abstract Cauchy problem.

Additional conditions of self-adjoint operator to be applied self-adjoint linear relation on a Hilbert space

Let ℌ is a Hilbert space over field real number ℝ. An operator T on ℌ is a function from ℌ to ℌ. A Self-adjoint operator is an operator that satisfies T = T*. Furthermore, a linear relation ε on ℌ is the set of pairs of elements w and x with w,x ∈ ℌ. A self-adjoint linear relation is a relation that meets ε = ε*. Some properties of a Self-adjoint operator on ℌ is not applicable in self-adjoint linear relation. This paper aims to determine the properties of a self-adjoint linear relation based on linear operators.

The square root of nonnegative selfadjoint linear relations in Hilbert spaces

An elementary construction of the square root of nonnegative selfadjoint linear relations in Hilbert spaces is presented.

On Compressions of Self-Adjoint Extensions of a Symmetric Linear Relation

Let A be a symmetric linear relation in the Hilbert space $${\mathfrak {H}}$$ with equal deficiency indices $$n_\pm (A)\le \infty $$ . A self-adjoint linear relation $$\widetilde{A}\supset A$$ in some Hilbert space $$\widetilde{\mathfrak {H}}\supset {\mathfrak {H}}$$ is called an exit space extension of A; such an extension is called finite-codimensional if $$\dim (\widetilde{\mathfrak {H}}\ominus {\mathfrak {H}})< \infty $$ . We study the compressions $$C (\widetilde{A})=P_{\mathfrak {H}}\widetilde{A}\upharpoonright {\mathfrak {H}}$$ of exit space extensions $$\widetilde{A}=\widetilde{A}^*$$ . For a certain class of extensions $$\widetilde{A}$$ we parameterize the compressions $$C (\widetilde{A})$$ by means of abstract boundary conditions. This enables us to characterize various properties of $$C (\widetilde{A})$$ (in particular, self-adjointness) in terms of the parameter for $$\widetilde{A}$$ in the Krein formula for resolvents. We describe also the compressions of a certain class of finite-codimensional extensions. These results develop the results by A. Dijksma and H. Langer obtained for a densely defined symmetric operator A with finite and equal deficiency indices.

Von Neumann’s theorem for linear relations

Assume that T is a linear relation acting between two real or complex Hilbert spaces and , respectively. T is shown to be closed if and only if the linear relations and are both selfadjoint linear relations on the corresponding Hilbert spaces and , respectively.

Finite-Codimensional Compressions of Symmetric and Self-Adjoint Linear Relations in Krein Spaces

Theorems due to Stenger (Bull Am Math Soc 74:369–372, 1968) and Nudelman (Int Equ Oper Theory 70:301–305, 2011) in Hilbert spaces and their generalizations to Krein spaces in Azizov and Dijksma (Int Equ Oper Theory 74(2):259–269, 2012) and Azizov et al. (Linear Algebra Appl 439:771–792, 2013) generate additional questions about properties a finite-codimensional compression $${T_0}$$ of a symmetric or self-adjoint linear relation $${T}$$ may or may not inherit from $${T}$$ . These questions concern existence of invariant maximal nonnegative subspaces, definitizability, singular critical points and defect indices.

Functional Calculus for Definitizable Self-adjoint Linear Relations on Krein Spaces

In the present note a functional calculus $${\phi \mapsto \phi(A)}$$ for self-adjoint definitizable linear relations on Krein spaces is developed. This functional calculus is the proper analogue of $${\phi \mapsto \int \phi \, dE}$$ in the Hilbert space situation where $${\phi}$$ is a bounded and measurable function on $${\sigma(A)}$$ and $${\int \phi \, dE}$$ is defined in the weak sense. The derived functional calculus also comprises the Spectral Theorem for self-adjoint definitizable operators on Krein spaces showing the existence of spectral projections.

Stability of essential spectra of self-adjoint subspaces under compact perturbations

This paper studies stability of essential spectra of self-adjoint subspaces (i.e., self-adjoint linear relations) under finite rank and compact perturbations in Hilbert spaces. Relationships between compact perturbation of closed subspaces and relatively compact perturbation of their operator parts are first established. This gives a characterization of compact perturbation in terms of difference between the operator parts of perturbed and unperturbed subspaces. It is shown that a self-adjoint subspace is still self-adjoint under either relatively bounded perturbation with relative bound less than one or relatively compact perturbation or compact perturbation with a certain additional condition. By using these results, invariance of essential spectra of self-adjoint subspaces is proved under relatively compact and compact perturbations, separately. As a special case, finite rank perturbation is discussed. The results obtained in this paper generalize the corresponding results for self-adjoint operators to self-adjoint subspaces.

Compressions of maximal dissipative and self-adjoint linear relations and of dilations

In this paper we generalize results from Stenger (1968) [30], Nudelman (2011) [28] and Azizov and Dijksma (2012) [7] to maximal dissipative and self-adjoint linear relations and discuss related results for nonnegative self-adjoint extensions of nonnegative symmetric linear relations and self-adjoint dilations of maximal dissipative linear relations.

A functional model associated with a generalized Nevanlinna pair

Let \( \mathcal{L} \) be a Hilbert space, and let \( \mathcal{H} \) be a Pontryagin space. For every self-adjoint linear relation \( \tilde{A} \) in \( \mathcal{H} \oplus \mathcal{L} \), the pair \( \{ I + \lambda \psi (\lambda ),\,\psi (\lambda )\} \) where \( \psi (\lambda ) \) is the compressed resolvent of \( \tilde{A} \), is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some self-adjoint linear relation \( \tilde{A} \) in the above sense. A functional model for this selfadjoint linear relation \( \tilde{A} \) is constructed.

Operational calculus of linear relations

(unbounded spectrum, domain Φ X) in finite-dimens ional spaces which linear operators exhibit only in infinite-dime nsional spaces. We present an outline of the paper. In § 2 we define p(T) where p is a polynomial with coefficients in the field Φ involved in X. We prove that (pq)(T) = p(T)q(T), (poq)(T) = p{q{T)), and point out that sometimes (p + q){T) Φ p(T) + q(T), etc. In § 3 we turn to relations in dual pairs. In this situation, adjoints can be defined. We build an automorphism λ —> λ of Φ into the theory of dual pairs, so as not to exclude the Hubert space situation, which dual pairs are intended to imitate. (Thus the transpose is a special kind of adjoint.) Closedness is defined algebraically, but in a way compatible with the topological concept. Closure of T 7* and other algebraic properties of * are established. Finally, it is shown that if T is closed and its resolvent is not void then p(T) is also closed. Section 4 considers the self-dual case. We give a simple condition (4.3) always true in Hubert space, that T*T be self-adjoint, T being closed. In § 5 we give the spectral analysis of self-ad joint linear relations in Hubert space. In a 1:1 manner these correspond to the unitary operators, via the Cay ley transform. However, it can be shown directly that X is the direct sum of orthogonal subspaces Y, Z which reduce T (= T7*) giving in Za self-ad joint operator and in Fthe inverse of the zero-operator. 2 Linear relations* A relation T between members of a set X and members of a set Y is merely a subset of X x Y. For x e X, T(x) = {y (x, y) e T}. The domain of T consists of those x such that T(x) is not void. T is called single-valued if T(x) never contains more than one element. The range of T is the union of all T(x).