Nevanlinna Functions,Perturbation Formulas,and Triplets of Hilbert Spaces

Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space ?? and let A be a selfadjoint operator extension of S in ??. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, 5 is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q - function of S and A belongs to the subclass No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space ?? the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space ?? is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q - function of S and A belongs to the subclass N₁ of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.     

Form sums of nonnegative selfadjoint operators

Summary The sum of two unbounded nonnegative selfadjoint operators is a nonnegative operator which is not necessarily densely defined. In general its selfadjoint extensions exist in the sense of linear relations (multivalued operators). One of its nonnegative selfadjoint extensions is constructed via the form sum associated with A and B. Its relations to the Friedrichs and Krein--von Neumann extensions of A+Bare investigated. For this purpose, the one-to-one correspondence between densely defined closed semibounded forms and semibounded selfadjoint operators is extended to the case of nondensely defined semibounded forms by replacing semibounded selfadjoint operators by semibounded selfadjoint relations. In particular, the inequality between two closed nonnegative forms is shown to be equivalent to a similar inequality between the corresponding nonnegative selfadjoint relations.</o:p>     

Q-functions of Hermitian Contractions of Krein-Ovcharenko Type

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space ℌ. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q_μ- and Q_M-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q_μ- and Q_M-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.     

Rank One Perturbations in a Pontryagin Space with One Negative Square

Let N₁ denote the class of generalized Nevanlinna functions with one negative square and let N_1, 0 be the subclass of functions Q(z)∈N₁ with the additional properties lim_y→∞ Q(iy)/y=0 and lim sup_y→∞ y |Im Q(iy)|<∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ)=A+τ[·, ω] ω in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions either belong to the subclass N_1, 0 or can be associated with the corresponding generalized Friedrichs extension. In this paper a spectral theoretical analysis of the perturbations A(τ) and the associated Friedrichs extension is carried out. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(τ), are based on a recent factorization result for generalized Nevanlinna functions.     

Spectral Estimations for Can nical Systems

Two–dimensional canonical systems are boundary value problems of the form with y₁(0) = 0 and Weyl's limit point case at L. The 2 × 2 matrix valued function H is real, symmetric and nonnegative, . The correspondence between canonical systems and their Titchmarsh–Weyl coefficients Q is a bijection between the class of all matrix functions H with tr H(x) = 1 a.e. on [0, L) and the class of the Nevanlinna functions ℕ augmented by the function Q ≡ 8. Each Titchmarsh–Weyl coefficient Q ∈ ℕ can be represented by means of a measure σ, the so–called spectral measure of the canonical system. In this note matrix functions H are specified whose corresponding spectral measures σ satisfy conditions of the form or . Herewith we generalize corresponding results of M.G. Krein and I. S. Kac for so–called vibrating strings.     

Extreme self‐adjoint extensions of a semibounded ‐difference operator

For a certain q‐difference operator introduced and studied in a series of articles by the same authors, we investigate its extreme self‐adjoint extensions, i.e., the so‐called Friedrichs and Kre?n extensions. We show that for the interval of parameters under consideration, the Friedrichs extension and the Kre?n extension are distinct and give values of the parameter in the von Neumann formulas that correspond to those extensions and describe their resolvent operators. A crucial rôle in our investigation plays the fact that both the Friedrichs and the Kre?n extensions are scale invariant.     

Study of Some Fundamental Projective Quadrics Associated with a Standard Pseudo-Hermitian Space H p,q

This self-contained short note deals with the study of the properties of some real projective compact quadrics associated with a a standard pseudo-hermitian space H _p,q , namely [(Q(p, q))\tilde], [(Q_2p+1,1)\tilde], [(Q_1,2q+1)\tilde], [(H_p,q)\tilde].  [(Q(p, q))\tilde]{\widetilde{Q(p, q)}, \widetilde{Q_{2p+1,1}}, \widetilde{Q_{1,2q+1}}, \widetilde{H_{p,q}}. \, \widetilde{Q(p, q)}} is the (2n – 2) real projective quadric diffeomorphic to (S ^2p–1 × S ^2q–1)/Z ₂. inside the real projective space P(E ₁), where E ₁ is the real 2n-dimensional space subordinate to H _p,q . The properties of [(Q(p, q))\tilde]{\widetilde{Q(p, q)}} are investigated. [(H_p,q)\tilde]{\widetilde{H_p,q}} is the real (2n – 3)-dimensional compact manifold-(projective quadric)- associated with H _p,q , inside the complex projective space P(H _p,q ), diffeomorphic to (S ^2p–1 × S ^2q–1)/S ¹. The properties of [(H_p,q)\tilde]{\widetilde{H_{p,q}}} are studied. [(Q_2p+1,1)\tilde]{\widetilde{Q_{2p+1,1}}} is a 2p-dimensional standard real projective quadric, and [(Q_1,2q+1)\tilde]{\widetilde{Q_{1,2q+1}}} is another standard 2q-dimensional projective quadric. [(Q_2p+1,1)\tilde] è[(Q_1,2q+1)\tilde]{\widetilde{Q_{2p+1,1}} \cup \widetilde{Q_{1,2q+1}}}, union of two compact quadrics plays a part in the understanding of the "special pseudo-unitary conformal compactification" of H _p,q . It is shown how a distribution y → D _y , where y ? H\{0},H{y \in H\backslash\{0\},H} being the isotropic cone of H _p,q allows to [(H_p+1,q+1)\tilde]{\widetilde{H_{p+1,q+1}}} to be considered as a "special pseudo-unitary conformal compactified" of H _p,q × R. The following results precise the presentation given in [1,c].     

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 H²-framework are obtained.     

Analysis of differential operators associated with boundary value problems

We consider boundary value problems in a two-component domain of the Euclidean space R ⁿ obtained by eliminating from R ⁿ the boundary G. Traces on both sides of G are defined without limit passages. In a Hilbert trace space, we introduce orthogonal projections, analogs of the Calderon projections, which are used for constructing operators whose continuous invertibility implies the solvability of the corresponding boundary value problems. For the resolvents we obtain representations similar to the Krein formula. For a symmetric differential operator we show that the constructed resolvents of boundary value problems correspond to closed (not necessarily self-adjoint) extensions of this operator in the sense of von Neumann. Bibliography: 9 titles. Dedicated to N. N. Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 7–48.     

Positive Decompositions of Selfadjoint Operators

Given a linear bounded selfadjoint operator a on a complex separable Hilbert space ${\mathcal{H}}Given a linear bounded selfadjoint operator a on a complex separable Hilbert space H{\mathcal{H}}, we study the decompositions of a as a difference of two positive operators whose ranges satisfy an angle condition. These decompositions are related to the canonical decompositions of the indefinite metric space (H,á ,  ?_a){(\mathcal{H},\langle\,, \,\rangle_a)}, associated to a. As an application, we characterize the orbit of congruence of a in terms of its positive decompositions.     

Closed embeddings of Hilbert spaces

Motivated by questions related to embeddings of homogeneous Sobolev spaces and to comparison of function spaces and operator ranges, we introduce the notion of closely embedded Hilbert spaces as an extension of that of continuous embedding of Hilbert spaces. We show that this notion is a special case of that of Hilbert spaces induced by unbounded positive selfadjoint operators that corresponds to kernel operators in the sense of L. Schwartz. Certain canonical representations and characterizations of uniqueness of closed embeddings are obtained. We exemplify these constructions by closed, but not continuous, embeddings of Hilbert spaces of holomorphic functions. An application to the closed embedding of a homogeneous Sobolev space on Rn in L₂(Rn), based on the singular integral operator associated to the Riesz potential, and a comparison to the case of the singular integral operator associated to the Bessel potential are also presented. As a second application we show that a closed embedding of two operator ranges corresponds to absolute continuity, in the sense of T. Ando, of the corresponding kernel operators.     

Hilbert functions and Betti numbers in a flat family

Summary This paper is dedicated to the study of Hilbert functions and Betti numbers of the projective varieties in a flat family. We prove that the Hilbert function H(X _y ,n),y Y-a parameter scheme-is lower semicontinuous for any fixed n. In case Y is integral and noetherian we obtain the well-known fact that the set V Y where H(X _y ,n)is maximal for all n's is open and nonempty. We show also that b_i(X _y )-the i- th Betti number of X_y—is upper semicontinuous for y V. The paper contains also a number of results concerning the relations among the various Betti numbers.Member of G.N.S.A.G.A.-C.N.R. Supported in part by M.P.I. (Italian Minstry of Education).     

C *-algebras generated by orthogonal projections and their applications

We study the following problem: Given a Hilbert spaceH and a set of orthogonal projectionsP, Q ₁, ..., Q_n on it, with the conditionsQ _j ·Q _k = _j,k Q _k , , describe theC ^*-algebraC ^*(P, Q ₁, ..., Q_n) generated by these projections.Applications to Naimark dilation theorems and to Toeplitz operators associated with the Heisenberg group are given.Dedicated to the memory of M. G. Krein.This work was partially supported by CONACYT Project 3114P-E9608, México.     

The Friedrichs Extension of the Aharonov–Bohm Hamiltonian on a Disc

We show that the Aharonov–Bohm Hamiltonian considered on a disc has a four-parameter family of self-adjoint extensions. Among the in- finitely many self-adjoint extensions, we determine to which parameters the Friedrichs extension H^F corresponds and its lowest eigenvalue is found. Moreover, we note that the diamagnetic inequality holds for H^F.     

On a class of J-self-adjoint operators with empty resolvent set

In the present paper we investigate the set ΣJ of all J-self-adjoint extensions of an operator S which is symmetric in a Hilbert space H with deficiency indices 〈2,2〉 and which commutes with a non-trivial fundamental symmetry J of a Krein space (H,[⋅,⋅]),

SJ=JS.