Semibounded Operators Research Articles

Strict domain monotonicity of the principal eigenvalue and a characterization of lower boundedness for the Friedrichs extension of four-coefficient Sturm–Liouville operators

Using the variational characterization of the principal (i.e., smallest) eigenvalue below the essential spectrum of a lower semibounded self-adjoint operator, we prove strict domain monotonicity (with respect to changing the finite interval length) of the principal eigenvalue of the Friedrichs extension \( T_F\) of the minimal operator for regular four-coefficient Sturm–Liouville differential expressions. In the more general singular context, these four-coefficient differential expressions act according to $$\tau f= \frac{1}{r}(-(f^{[1]})'+sf^{[1]}+qf)\, {\rm with}\, f^{[1]}=p[f'+sf]\, {\rm on}\, (a,b) \subseteq \mathbb{R}, $$ where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p > 0, r > 0 a.e. on (a, b), and \( p^{-1},q,r,s \in L^{1}_{loc}((a,b);dx)\), and \( f\) is supposed to satisfy $$f\in AC_{loc}((a,b)),p[f'+sf] \in AC_{loc}((a,b)). $$ This setup is sufficiently general so that \(\tau \) permits certain distributional potential coefficients q, including potentials in \(H^{-1}_{loc}((a,b)) \).As a consequence of the strict domain monotonicity of the principal eigenvalue of the Friedrichs extension in the regular case, and on the basis of oscillation theory in the singular context, in our main result, we characterize all lower bounds of \(T_F \) as those \(\lambda \in \mathbb{R} \) for which the differential equation \(\tau u =\lambda u \) has a strictly positive solution u > 0 on (a, b).

Cwikel estimates and negative eigenvalues of Schrödinger operators on noncommutative tori

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension n ≥ 2. We use them to derive Cwikel–Lieb–Rozenblum inequalities and Lieb–Thirring inequalities for negative eigenvalues of fractional Schrödinger operators on noncommutative tori. The latter leads to a Sobolev inequality for noncommutative tori. On the way, we establish a “borderline version” of the abstract Birman–Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue.

О резольвенте комплексного оператора

A normed algebra of bounded linear complex operators acting in a complex normed space consisting of elements of the Cartesian square of a real Banach space is constructed. In this algebra, it is singled out a set of operators for each of which the real and imaginary parts commute with each other. It is proved that in this set, any operator for which the sum of squares of its real and imaginary parts is a continuously invertible operator, is invertible itself; a formula for the inverse operator is found. For an operator from the indicated set, the form of its regular points is investigated: conditions under which a complex number is a regular point of the given operator are found; a formula for the resolvent of a complex operator is obtained. The set of unbounded linear complex operators acting in the above complex normed space is considered. In this set, a subset of those operators for each of which the domains of the real and imaginary parts coincide is distinguished. For an operator from the specified subset, conditions on a complex number under which this number belongs to the resolvent set of the given operator are found; a formula for the resolvent of the operator is obtained. The concept of a semi-bounded complex operator as an operator in which one component is a bounded and the other is an unbounded operator is introduced. It is noted that the first and second resolvent identities for complex operators can be proved similarly to the case of real operators.

New notions and constructions of the boundary control method

<p style='text-indent:20px;'>The paper provides a list of mathematical objects, constructions and notions, which were revealed and introduced in the course of creation and development of the boundary control method. Among them, there are: amplitude integral, equidistant transforms of 3d vector fields, <inline-formula><tex-math id="M1">\begin{document}$ s $\end{document}</tex-math></inline-formula>-points in acoustic scattering, formulas for Betti numbers via DN-map, eikonal algebras, wave spectrum of symmetric semi-bounded operator. We describe these objects and explain how they work for solving inverse problems.</p>

Pusz–Woronowicz Functional Calculus and Extended Operator Convex Perspectives

In this article, we first study, in the framework of operator theory, Pusz and Woronowicz’s functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $$[0,\infty )^2$$ . Our construction has special features that functions on $$[0,\infty )^2$$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in $$(-\infty ,\infty ]$$ . Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on $$(0,\infty )$$ . We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Ando’s theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.

An abstract Logvinenko-Sereda type theorem for spectral subspaces

We provide an abstract framework for a Logvinenko-Sereda type theorem, where the classical compactness assumption on the support of the Fourier transform is replaced by the assumption that the functions under consideration belong to a spectral subspace associated with a finite energy interval for some lower semibounded self-adjoint operator on a Euclidean L2-space. Our result then provides a bound for the L2-norm of such functions in terms of their L2-norm on a thick subset with a constant explicit in the geometric and spectral parameters. This recovers previous results for functions on the whole space, hyperrectangles, and infinite strips with compact Fourier support and for finite linear combinations of Hermite functions and allows to extend them to other domains. The proof follows the approach by Kovrijkine and is based on Bernstein-type inequalities for the respective functions, complemented with a suitable covering of the underlying domain.

Oscillation theory and semibounded canonical systems

Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems Ju'=-zHu and then use it to discuss semibounded operators from this point of view. Our main new result is a characterization of systems with purely discrete spectrum in terms of the asymptotics of their coefficient functions; we also discuss the exponential types of the transfer matrices.

Self-Adjoint Extensions with Friedrichs Lower Bound

We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.

On strong stability of explicit Runge–Kutta methods for nonlinear semibounded operators

AbstractExplicit Runge–Kutta methods are classical and widespread techniques in the numerical solution of ordinary differential equations (ODEs). Considering partial differential equations, spatial semidiscretizations can be used to obtain systems of ODEs that are solved subsequently, resulting in fully discrete schemes. However, certain stability investigations of high-order methods for hyperbolic conservation laws are often conducted only for the semidiscrete versions. Here, strong stability (also known as monotonicity) of explicit Runge–Kutta methods for ODEs with nonlinear and semibounded (also known as dissipative) operators is investigated. Contrary to the linear case it is proven that many strong-stability-preserving (SSP) schemes of order 2 or greater are not strongly stable for general smooth and semibounded nonlinear operators. Additionally, it is shown that there are first-order-accurate explicit SSP Runge–Kutta methods that are strongly stable (monotone) for semibounded (dissipative) and Lipschitz continuous operators.

Решение обратных спектральных задач для дискретных полуограниченных операторов, заданных на геометрических графах

Решение обратных спектральных задач для дискретных полуограниченных операторов, заданных на геометрических графах

Asymptotic completeness and S-matrix for singular perturbations

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple (A0,A), where A0 and A are semi-bounded self-adjoint operators in L2(M,B,m) such that the set {u∈dom(A0)∩dom(A):A0u=Au} is dense. No sort of trace-class condition on resolvent differences is required. Applications to the case in which A0 corresponds to the free Laplacian in L2(Rn) and A describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

On the Difference of Spectral Projections

For a semibounded self-adjoint operator T and a compact self-adjoint operator S acting on a complex separable Hilbert space of infinite dimension, we study the difference $$ D(\lambda ) := E_{(-\infty , \lambda )}(T+S) - E_{(-\infty , \lambda )}(T), \, \lambda \in \mathbb {R} $$ , of the spectral projections associated with the open interval $$ (-\infty , \lambda ) $$ . In the case when S is of rank one, we show that $$ D(\lambda ) $$ is unitarily equivalent to a block diagonal operator $$ \Gamma _{\lambda } \oplus 0 $$ , where $$ \Gamma _{\lambda } $$ is a bounded self-adjoint Hankel operator, for all $$ \lambda \in \mathbb {R} $$ except for at most countably many $$ \lambda $$ . If, more generally, S is compact, then we obtain that $$ D(\lambda ) $$ is unitarily equivalent to $$ \Gamma _{\lambda } + \Lambda _{\lambda } $$ for all $$ \lambda \in \mathbb {R} $$ except for at most countably many $$ \lambda $$ , where $$ \Gamma _{\lambda } $$ is a bounded self-adjoint Hankel operator and $$ \Lambda _{\lambda } $$ is a compact self-adjoint operator.

Wave model of the Sturm–Liouville operator on the half-line

The notion of the wave spectrum of a semi-bounded symmetric operator was introduced by one of the authors in 2013. The wave spectrum is a topological space determined by the operator in a canonical way. The definition uses a dynamical system associated with the operator: the wave spectrum is constructed from its reachable sets. In the paper we give a description of the wave spectrum of the operator $L_0=-\frac{d^2}{dx^2}+q$ which acts in the space $L_2(0,\infty)$ and has defect indices $(1,1)$. We construct a functional (wave) model of the operator $L_0^*$ in which the elements of the original $L_2(0,\infty)$ are realized as functions on the wave spectrum. It turns out to be identical to the original $L_0^*$. The latter is fundamental in solving inverse problems: the wave model is determined by their data, which allows for reconstruction of the original.

The Friedrichs extension for elliptic wedge operators of second order

Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such that $C^\infty_{{\mathscr T}_F}({\mathcal M};E)={\mathfrak P} (C^\infty({\mathcal Y};{\mathscr T}_F)) +\dot C^\infty({\mathcal M};E)\subset {\mathcal D}_{\max}(A)$. It is shown that the closure of $A$ when given the domain $C^\infty_{{\mathscr T}_F}({\mathcal M};E)$ is the Friedrichs extension of (†) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.

The Friedrichs extension for elliptic wedge operators of second order

Let ${\mathcal M}$ be a smooth compact manifold whose boundary is the total space of a fibration ${\mathcal N}\to {\mathcal Y}$ with compact fibers, let $E\to{\mathcal M}$ be a vector bundle. Let \begin{equation} A:C_c^\infty( \overset{\,\,\circ} {\mathcal M};E)\subset x^{-\nu} L^2_b({\mathcal M};E)\to x^{-\nu} L^2_b({\mathcal M};E) $ \tag*{(†)} \end{equation} be a second order elliptic semibounded wedge operator. Under certain mild natural conditions on the indicial and normal families of $A$, the trace bundle of $A$ relative to $\nu$ splits as a direct sum ${\mathscr T}={\mathscr T}_F\oplus{\mathscr T}_{aF}$ and there is a natural map ${\mathfrak P} :C^\infty({\mathcal Y};{\mathscr T}_F)\to C^\infty( \overset{\,\,\circ} {\mathcal M};E)$ such that $C^\infty_{{\mathscr T}_F}({\mathcal M};E)={\mathfrak P} (C^\infty({\mathcal Y};{\mathscr T}_F)) +\dot C^\infty({\mathcal M};E)\subset {\mathcal D}_{\max}(A)$. It is shown that the closure of $A$ when given the domain $C^\infty_{{\mathscr T}_F}({\mathcal M};E)$ is the Friedrichs extension of (†) and that this extension is a Fredholm operator with compact resolvent. Also given are theorems pertaining the structure of the domain of the extension which completely characterize the regularity of its elements at the boundary.

Toeplitz versus Hankel: semibounded operators

Our goal is to compare various results for Toeplitz \(T\) and Hankel \(H\) operators. We consider semibounded operators and find necessary and sufficient conditions for their quadratic forms to be closable. This property allows one to define \(T\) and \(H\) as self-adjoint operators under minimal assumptions on their matrix elements. We also describe domains of the closed Toeplitz and Hankel quadratic forms.

$$L_2$$ L 2 Stability of Explicit Runge–Kutta Schemes

Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial differential equations, spatial semidiscretisations result in large systems of ODEs that are solved subsequently. However, stability investigations of high-order methods for transport equations are often conducted only in the semidiscrete setting. Here, strong-stability of semidiscretisations for linear transport equations, resulting in ODEs with semibounded operators, are investigated. For the first time, it is proved that the fourth-order, ten-stage SSP method of Ketcheson (SIAM J Sci Comput 30(4):2113–2136, 2008) is strongly stable for general semibounded operators. Additionally, insights into fourth-order methods with fewer stages are presented.

Unbounded operators in Hilbert space, duality rules, characteristic projections, and their applications

Our main theorem is in the generality of the axioms of Hilbert space, and the theory of unbounded operators. Consider two Hilbert spaces whose intersection contains a fixed vector space $$\mathscr {D}$$ . In the case when $$\mathscr {D}$$ is dense in one of the Hilbert spaces (but not necessarily in the other), we make precise an operator-theoretic linking between the two Hilbert spaces. No relative boundedness is assumed. Nonetheless, under natural assumptions (motivated by potential theory), we prove a theorem where a comparison between the two Hilbert spaces is made via a specific selfadjoint semibounded operator. Applications include physical Hamiltonians, both continuous and discrete (infinite network models), and the operator theory of reflection positivity.

Coupling of definitizable operators in Krein spaces

Indefinite Sturm-Liouville operators defined on the real line are often considered as a coupling of two semibounded symmetric operators defined on the positive and the negative half axis. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension. In this paper we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm-Liouville problems on the real line.

Book Review: Semi-bounded differential operators, contractive semigroups, and beyond

Book Review: Semi-bounded differential operators, contractive semigroups, and beyond