On selfadjoint extensions of symmetric operator with exit from space

Operators having selfadjoint squares

The main goal of this paper is to show that a (not necessarily densely defined or closed) symmetric operator $A$ acting on a real or complex Hilbert space is selfadjoint exactly when $I+A^2$ is a full range operator.

Scattering theory for a class of non-selfadjoint extensions of symmetric operators

This work deals with the functional model for a class of extensions of symmetric operators and its applications to the theory of wave scattering. In terms of Boris Pavlov's spectral form of this model, we find explicit formulae for the action of the unitary group of exponentials corresponding to almost solvable extensions of a given closed symmetric operator with equal deficiency indices. On the basis of these formulae, we are able to construct wave operators and derive a new representation for the scattering matrix for pairs of such extensions in both self-adjoint and non-self-adjoint situati…

Selfadjoint extensions of a singular differential operator

In this work, firstly in the Hilbert space of vector-functions L^2 (H,(-\infty,a)\bup(b,+\infty)),a<b all selfadjoint extensions of the minimal operator generated by linear singular symmetric differential expression l(\cdot)=i d/dt+A with a selfadjoint operator coefficient A in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.

Selfadjoint Extensions of a Singular Multipoint Differential Operator for First Order

In this work, firstly in the direct sum of Hilbert spaces of vector-functions L^2 (H,(-{\infty},a_1)){\Box}L^2 (H,(a_2,b_2)){\Box}L^2 (H,(a_3,+{\infty})),- {\infty}<a_1<a_2<b_2<a_3<+{\infty} all selfadjoint extensions of the minimal operator generated by linear singular multipoint symmetric differential expression l=(l_1,l_2,l_3),l_k=i d/dt+A_k with a selfadjoint operator coefficient A_k k=1,2,3 in any Hilbert space H, are described in terms of boundary values. Later structure of the spectrum of these extensions is investigated.

Minimal spectral functions of an ordinary differential operator

Let $l[y]$ be a formally selfadjoint differential expression of an even order on the interval $[0,b> \;(b\leq \infty)$ and let $L_0$ be the corresponding minimal operator. By using the concept of a decomposing boundary triplet we consider the boundary problem formed by the equation $l[y]-ły=f\;(f\in L_2[0,b>)$ and the Nevanlinna $ł$-depending boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the $m$-function, which in the case of selfadjoint decomposing boundary conditions coincides with the classical characteristic (Titchmarsh-W…

Selfadjoint extensions of relations whose domain and range are orthogonal

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in ${\mathfrak H}_1 \oplus {\mathfrak H}_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative self…

Commutation properties of the form sum of positive, symmetric operators

A new construction for the form sum of positive, selfadjoint operators is given in this paper. The situation is a bit more general, because our aim is to add positive, symmetric operators. With the help of the used method, some commutation properties of the form sum extension are observed.

Spectral functions of non essentially selfadjoint operators

One of the many problems to which J.S. Dowker devoted his attention is the effect of a conical singularity in the base manifold on the behavior of the quantum fields. In particular, he studied the small-$t$ asymptotic expansion of the heat-kernel trace on a cone and its effects on physical quantities, as the Casimir energy. In this article we review some peculiar results found in the last decade, regarding the appearance of non-standard powers of $t$, and even negative integer powers of $\log{t}$, in this asymptotic expansion for the selfadjoint extensions of some symmetric operators with sing…

A Birman-Krein-Vishik-Grubb theory for sectorial operators

We consider densely defined sectorial operators $A_\pm$ that can be written in the form $A_\pm=\pm iS+V$ with $\mathcal{D}(A_\pm)=\mathcal{D}(S)=\mathcal{D}(V)$, where both $S$ and $V\geq \varepsilon>0$ are assumed to be symmetric. We develop an analog to the Birmin-Krein-Vishik-Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions $A_D$ of $A_+$ with the property that $\overline{A_+}\subset A_D\subset A_-^*$. Here, $D$ is an auxiliary operator from $\ker(A_-^*)$ to $\ker(A_+^*)$ that parametriz…

Eigenvalue estimates for singular left-definite Sturm-Liouville operators

The spectral properties of a singular left-definite Sturm-Liouville operator $JA$ are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart $A$ which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the $J$-selfadjoint operator $JA$ is real and it follows that an interval $(a,b)\subset\mathbb R^+$ is a gap in the essential spectrum of $A$ if and only if both intervals $(-b,-a)$ and $(a,b)$ are gaps in the essential spectrum of the $J$-selfadjoint operator $JA$. As one of the main results it is …

Mathematical and physical aspects of complex symmetric operators

Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, $C$-selfadjoint extensions of $C$-symmetric unbounded operators, resolvent estimates, reality of spectrum, bases of $C$-orthonormal vectors, and conjugate-linear symmetric operators. The main results are complemented by a variety of natural examples arising in field theory, quantum physics, and complex variables.

Spectral flow and winding number in von Neumann algebras

We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips' analytical definition and derive integral formulas for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey--Phillips. Furthermore we prove the homotopy invariance of the real-valued index. As an example w…

The Closed Extensions of a Closed Operator

Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed with respect to the graph norm of $A^*$ and satisfy certain conditions. In particular, this will allow us to characterize all densely defined and closed restrictions of $A^*$. After this, we will express our results using the language of Gel'fand triples generalizing the well-known results for the selfadjoint case. As applications we construct: (i) a sequence o…

On compressions of self-adjoint extensions of a symmetric linear relation

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with equal deficiency indices $n_\pm (A)\leq\infty$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an exit space extension of $A$; such an extension is called finite-codimensional if $\dim (\wt\gH\ominus\gH)< \infty$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of exit space extensions $\wt A=\wt A^*$. For a certain class of extensions $\wt A$ we parameterize the compressions $C (\wt A)$ by means of abstract boundary conditions. This enables us to characterize vari…

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.

On a theorem of Livsic

The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators such as Schrodinger operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions such as model subspaces of Hardy spaces, deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators (including Schrodinger operators from quantum mechanics), Toeplitz operator…

Generalized boundary triples, Weyl functions and inverse problems

With a closed symmetric operator $A$ in a Hilbert space ${\mathfrak H}$ a triple $Π=\{{\mathcal H},Γ_0,Γ_1\}$ of a Hilbert space ${\mathcal H}$ and two abstract trace operators $Γ_0$ and $Γ_1$ from $A^*$ to ${\mathcal H}$ is called a generalized boundary triple for $A^*$ if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions $M$ are investigated. The most important ones for applications are specific classes of (essentially) unitary boundary triples which guarantee that the Weyl functions of bo…

用表示映射构造 Friedrichs 与 Kreĭn 型扩张

Friedrichs and Kre\uın type extensions in terms of representing maps

本文利用表示映射构造 Hilbert 空间中下半有界算子或关系的 Friedrichs 扩张、Kreĭn 型扩张及所有极值扩张，并研究其性质。

Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators

The classical concept of $Q$-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized $Q$-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an $H^2$-framework are obtained.

The Non-Proper Dissipative Extensions of a Dual Pair

We consider dissipative operators $A$ of the form $A=S+iV$, where both $S$ and $V\geq 0$ are assumed to be symmetric but neither of them needs to be (essentially) selfadjoint. After a brief discussion of the relation of the operators $S\pm iV $to dual pairs with the so called common core property, we present a necessary and suffcient condition for any extension of $A$ with domain contained in $\mathcal{D}((S-iV)^*)$ to be dissipative. We will discuss several special situations in which this condition can be expressed in a particularly nice form -- accessible to direct computations. Examples in…

Operators which preserve a positive definite inner product

Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal H}$ and $\langle f,g\rangle_A=\langle Af,g\rangle$, $f,g\in {\cal H}$, the $A$-inner product. This paper studies the geometry of the set $$ {\cal I}_A^a:=\{\hbox{ adjointable isometries for } \langle \ , \ \rangle_A\}. $$ It is proved that ${\cal I}_A^a$ is a submanifold of the Banach algebra of adjointable operators, and a homogeneous space of the group of invertible operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth curves in ${\cal I}_A^a$ with given initial conditions, which are minima…

On eigenfunction expansions of first-order symmetric systems and ordinary differential operators of an odd order

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b\rangle $ with the regular endpoint $a$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the minimal relation $\Tmi$ satisfy $n_+(\Tmi)< n_-(\Tmi)$. We define $ł$-depending boundary conditions which are analogs of separated self-adjoint boundary conditions for Hamiltonian systems. With a boundary value problem involving such conditions we associate an exit space self-adjoint extension $\wt T$ of $\Tmi$ and the $m$-function $m(\cd)$, which is an analog of the T…

On Titchmarsh-Weyl functions and eigenfunction expansions of first-order symmetric systems

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'(t)-B(t)y(t)=\D(t) f(t)$ on an interval $\cI=[a,b> $ with the regular endpoint $a$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the minimal relation $\Tmi$ in $\LI$ satisfy $n_-(\Tmi)\leq n_+(\Tmi)$. By using a Nevanlinna boundary parameter $τ=τ(ł)$ at the singular endpoint $b$ we define self-adjoint and $ł$-depending Nevanlinna boundary conditions which are analogs of separated self-adjoint boundary conditions for Hamiltonian systems. With a boundary value problem involving such conditions we asso…

Resolvent and propagation estimates for Klein-Gordon equations with non-positive energy

We study in this paper an abstract class of Klein-Gordon equations: \[ \p_{t}^{2}ϕ(t)- 2ık \p_{t}ϕ(t)+ h ϕ(t)=0, \] where $ϕ: \rr\to \cH$, $\cH$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\cH$. We consider their generators $H$ (resp. $K$) in the two natural spaces of Cauchy data, the energy (resp. charge) spaces. We do not assume that the dynamics generated by $H$ or $K$ has any positive conserved quantity, in particular these operators may have complex spectrum. Assuming conditions on $h$ and $k$ which allow to use the theory of selfadjoint oper…