On the nonnegativity of operator products

Summary A bounded, not necessarily everywhere defined, nonnegative operator A in a Hilbert space <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathfrak{H}$ is assumed to intertwine in a certain sense two bounded everywhere defined operators B and C. If the range of A is provided with a natural inner product then the operators B and C induce two new operators on the completion space. This construction is used to show the existence of selfadjoint and nonnegative extensions of B*A and C*A.     

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.     

Characterizations of positive selfadjoint extensions

The set of all positive selfadjoint extensions of a positive operator (which is not assumed to be densely defined) is described with the help of the partial order which is relevant to the theory of quadratic forms. This enables us to improve and extend a result of M. G. Krein to the case of not necessarily densely defined operators .

     

Extremal extensions for the sum of nonnegative selfadjoint relations

The sum of two nonnegative selfadjoint relations (multi-valued operators) and is a nonnegative relation. The class of all extremal extensions of the sum is characterized as products of relations via an auxiliary Hilbert space associated with and . The so-called form sum extension of is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with and . Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on and .

     

Extremal maximal sectorial extensions of sectorial relations

The extremal maximal sectorial extensions of a not necessarily densely defined sectorial relation (multivalued linear operator) in a Hilbert space are characterized in terms of a construction which goes back to Sebestyén and Stochel. In particular the two extreme maximal sectorial extensions, namely the Friedrichs extension and the Kreĭn extension, are characterized. For this purpose a survey is given of the connection between closed sectorial forms and maximal sectorial relations.     

A characterization of semibounded selfadjoint operators

For a class of closed symmetric operators with defect numbers it is possible to define a generalization of the Friedrichs extension, which coincides with the usual Friedrichs extension when is semibounded. In this paper we provide an operator-theoretic interpretation of this class of symmetric operators. Moreover, we prove that a selfadjoint operator is semibounded if and only if each one-dimensional restriction of has a generalized Friedrichs extension.

     

Operator extensions with closed range

We characterize those positive (not necessarily densely defined) operators whose Krein–von Neumann extension, the smallest among all positive selfadjoint extensions, has closed range. In addition, we construct their Moore–Penrose pseudoinverse by employing factorization via an auxiliary Hilbert space. Other extremal extensions, in particular the Friedrichs extension, are also investigated from this point of view. As an application, new characterizations of essentially selfadjoint positive operators are presented.     

On von Neumann's problem in extension theory of nonnegative operators

The solution of von Neumann's problem about parametrization of all nonegative selfadjoint extensions of a nonnegative densely defined operator in terms of his formulas is obtained.

     

Ultracompleteness of hyperspaces of compact sets

A nonnegative linear relation S in a Hilbert space ? is assumed to intertwine in a certain sense two bounded everywhere defined operators B and?C. A?related quotient of the range of S is then provided with a natural inner product and the operators B and C induce two operators on the completion space. This construction is used to show the existence of self-adjoint and nonnegative extensions of the linear relations B ^? S and C ^? S, respectively.     

On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.     

Definitizability of a Class of J-Selfadjoint Operators with Applications

We formulate an abstract result concerning the definitizability of J-selfadjoint operators which, roughly speaking, differ by at most finitely many dimensions from the orthogonal sum of a J-selfadjoint operator with finitely many negative squares and a semibounded selfadjoint operator in a Hilbert space. The general perturbation result is applied to a class of singular Sturm–Liouville operators with indefinite weight functions. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extremal Extensions of the Tensor Product of Nonnegative Operators

For densely defined nonnegative operators A and B acting in separable Hilbert spaces we describe the Friedrichs and the Krein-von Neumann extension of the tensor product . Moreover, we discuss the class of extremal extensions of . An application is presented as well. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Star partial order on B(H)

Let H be an infinite-dimensional complex Hilbert space and let B(H) be the algebra of all bounded linear operators on (H). In the paper the equivalent definition of the star partial order on B(H), using selfadjoint idempotent operators, is introduced. Also some properties of the generalized concept of order relations on B(H), defined with the help of idempotent operators, are investigated.     

Rank one perturbations of not semibounded operators

Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension.     

M ‐functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems

In this paper, we combine results on extensions of operators with recent results on the relation between the M ‐function and the spectrum, to examine the spectral behaviour of boundary value problems. M ‐functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M ‐function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Iterative solution of linear equations with unbounded operators

A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space, including equations with unbounded, closed, densely defined linear operators. The method is proved to be stable towards small perturbation of the data. Some abstract results are established and used in an analysis of variational regularization method for equations with unbounded linear operators. The dynamical systems method (DSM) is justified for unbounded, closed, densely defined linear operators. The stopping time is chosen by a discrepancy principle. Equations with selfadjoint operators are considered separately. Numerical examples, illustrating the efficiency of the proposed method, are given.     

Linear extensions associated with abstract functional operators

Abstract functional operators are defined as elements of a C*-algebra B with a structure consisting of a closed C*-subalgebra A ? B and a unitary element T ? B such that the mapping \(\hat T:a \to TaT^{ - 1} \) is an automorphism of A and the set of finite sums \(\sum {a_k T^k } ,a_k \in A\), is norm dense in B.We give a new construction of a linear extension associated with the abstract weighted shift operator aT and obtain generalizations of known theorems about the relationship between the invertibility of operators and the hyperbolicity of the associated linear extensions to the case of abstract functional operators.