Operator-valued Functions Research Articles

Ergodicity and stability for (b, l)-regularized resolvent operator families

This paper is concerned with the ergodicity and strong stability for (b, l)-regularized resolvent operator families. First, we study Abel ergodicity and Cesaro ergodicity of (b, l)-regularized resolvent operator families by using the methods of operator theory and complex Tauberian theorem. And then, by constructing a new operator-valued function, we obtain some sufficient conditions on the strong stability of bounded (b, l)-regularized resolvent operator families by means of Cauchy theorem and Riemann–Lebesgue lemma. Our results generalize the related conclusions for operator semigroups and resolvent operator families.

Some spectral properties of operator-valued positive-real functions

We consider operator-valued positive-real functions H and show that the intersections of the point, continuous and residual spectra of H ( s ) with the imaginary axis do not depend on s . In particular, if H is positive real and H ( z ) is invertible for some z in the open right-half plane, then H ( s ) is invertible for all s in the open right-half plane. Furthermore, we prove that the eigenspace of H ( s ) corresponding to an imaginary eigenvalue does not depend on s . It is also shown that the intersection of the numerical range of H ( s ) with the imaginary axis is independent of s . Finally, we prove that, under suitable assumptions, application of a “sufficiently positive-real” static output feedback to a positive-real transfer function leads to a strictly positive-real closed-loop system.

Paley_ Wiener Theorems for The Fourier Transform depending on the Heisenberg group

عرّفنا في بحثنا هذا زمرة هايزنبرغ، وهي الزمرة الأكثر شهرةً من زمر لي. ثمّ ناقشنا نظريّة التمثيل لهذه الزمرة، إضافةً إلى العلاقة بين نظريّة التمثيل لزمرة هايزنبرغ، ومؤثرات كميّة الحركة والموضع. وهذا ما يُبيّن لنا كيفية تحقيق الترابط بين زمرة هايزنبرغ والفيزياء. وسنأخذ في بحثنا هذا بوجهة نظر شرودنجر Schr dinger، التي تعتبر أنّ جميع التمثيلات التي نفرضها تتحقق على فضاء هيلبرت . ثمّ نُعرّف زمرة تحويل فورييه على زمرة هايزنبرغ. إضافةً إلى مفاهيم أخرى ذات أهميّة في الأبحاث المشابهة. وقد اعتمدنا في بحثنا هذا على صيغ جديدة لبعض المفاهيم الرياضيّة مثل تحويل فورييه وتحويل وايل. إنَ الهدف الرئيس من بحثنا هذا هو تقديم مبرهنة Paley_ Wiener لأجل تحويل فورييه على زمرة هايزنبرغ. وسوف نُبيّن أنّ مبرهنة Paley_ Wiener التقليديّة لتحويل فورييه التقليدي، تُميّز الدوال ذات الدعامة المتراصّة من خلال سلوك تحويل فورييه لها. وسنهتم بإنشاء النتائج لأجل زمرة تحويل فورييه، وتحويل وايل.

Compressed Resolvents, Schur Functions, Nevanlinna Families and Their Transformations

We study certain transformations of Nevanlinna families and Schur class operator-valued functions, construct their realizations by means of compressed resolvents and by the transfer functions of conservative systems, and find fixed points of such transformations. In particular, we characterize some automorphic Schur functions and corresponding them Nevanlinna families, including periodic and scale invariant.

Nonlinear thermal coherent states

Coherent states and their generalizations are normally appropriate candidates for describing radiation fields. Despite this, the effect of thermal noise on generalized coherent states has not been extensively studied. In fact, thermal effects are unavoidable at finite temperatures, and they should be taken into account to have a better agreement with experimental results. In this regard, we use the concept of thermal coherent states, which are indeed the standard coherent states including thermal effects. So, in this paper, by using a nonlinear coherent states approach, we generalize the thermal coherent states to their nonlinear counterparts. In other words, we find a natural link between the thermal coherent states and the nonlinear coherent states associated with nonlinear oscillator algebra. Afterwards, the nonclassicality features of the obtained states are numerically investigated to explore the roles of both nonlinearity and thermal noise in physical properties. The results show that the thermal effects lead to the transition from nonclassical states to classical ones. Moreover, it is seen that the operator-valued intensity-dependent function plays a leading role in controlling the depth as well as the domain of nonclassicality aspects.

E-integrability of trace-class operator-valued functions

Here, we show the equivalent conditions for integrability and being almost everywhere zero of trace-class operator-valued functions with respect to some spectral measure. Furthermore, we investigate the relation between the convergence of these functions and the one of their integrals.

On the extension of positive definite kernels to topological algebras

We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras and describe their associated reproducing kernel spaces. The case of entire functions is of special interest, and we give a precise meaning to some power series expansions of analytic functions that appears in many algebras.

Realizations of holomorphic and slice hyperholomorphic functions: The Krein space case

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.

Toeplitz Operators with Positive Operator-Valued Symbols on Vector-Valued Generalized Fock Spaces

In this article, we study some characterizations of Toeplitz operators with positive operator-valued function as symbols on the vector-valued generalized Bargmann-Fock spaces $$F_\varphi^2$$ . Main results including Fock-Carleson condition, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten-p class are all considered.

Periodic Solutions to Partial Neutral Functional Differential Equations in Admissible Spaces on a Half-Line

We prove the existence and uniqueness of periodic solutions to partial neutral function differential equation of the form $\frac {\partial Fu_{t}}{\partial t}= A(t)Fu_{t} +g(t,u_{t}),~t\in [0,\infty )$, on a Banach space X, where the operator-valued function t↦A(t) and the nonlinear delay operator g(t,v) are both T-periodic with respect to t, whereas g is φ-Lipschitz with respect to v for φ belonging to an admissible space. Then, in the case that the family (A(t))t≥ 0 generates an evolution family having an exponential dichotomy, we apply our abstract results to study the existence, uniqueness of periodic solutions. We also prove the conditional stability, and the existence of local stable manifolds around such periodic solutions.

Mathematical and Numerical Analysis of the Generalized Complex-Frequency Eigenvalue Problem for Two-Dimensional Optical Microcavities

The current paper proposes a parametric eigenvalue problem for the Helmholtz equation on the plane based on two well-known physical models of emission from two-dimensional (2-D) microcavity lasers. The first model has been developed for passive, either lossy or lossless, open cavities and is usually referred to as the Complex-Frequency Eigenvalue Problem. The second model, named the Lasing Eigenvalue Problem (LEP), has been tailored to characterize emission from open cavities, filled in with gain material, on the threshold of nonattenuating in time radiation. Our generalized model contains both of them as special cases. We reduce the original problem to a nonlinear eigenvalue problem for a set of boundary integral equations with weakly singular kernels and formulate it as a parametric eigenvalue problem for a holomorphic Fredholm operator-valued function, which is convenient for mathematical and numerical analysis. Using this formulation and fundamental results of the theory of holomorphic operator-valued functions, we study the properties of the spectrum. For numerical solution of the problem, we propose a Nyström method, prove its convergence, and derive error estimates in the eigenvalue approximation. Combining this discretization with the residual inverse iteration technique, we compute approximate solutions of LEP and compare them with the exact ones and with the results obtained using other numerical methods.

Perturbations of discrete spectra of holomorphic operator-valued functions

Let X be a complex Banach space and L(X) be the Banach space of all bounded linear operators on X. Let $$\Omega \subset {{\mathbb {C}}}$$ be open and connected. Let $$T, V : \Omega \longrightarrow L(X)$$ be holomorphic operator-valued functions. We consider the one parameter family of operator-valued functions $$W(\lambda , t) := T(\lambda ) + t V(\lambda )$$, for $$t \in {{\mathbb {C}}}$$, and analyze evolution of the discrete eigenvalues of $$W(\lambda , t)$$ when t varies in $${{\mathbb {C}}}.$$ We provide a brief review of the discrete spectrum of $$T(\lambda )$$ and present several equivalent characterizations for discrete eigenvalues of $$T(\lambda ).$$ We also prove Rouche’s theorem for operator-valued functions under a weaker assumption, which we utilize to derive perturbation bounds for the discrete eigenvalues of $$W(\lambda , t)$$ when |t| is small.

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Let [Formula: see text] be a separable Hilbert space and [Formula: see text] be a self-adjoint bounded linear operator on [Formula: see text] with norm [Formula: see text], satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator [Formula: see text] determines a [Formula: see text]-deformed Fock space [Formula: see text]. We start with reviewing and extending the known results about the structure of the [Formula: see text]-particle spaces [Formula: see text] and the commutation relations satisfied by the corresponding creation and annihilation operators acting on [Formula: see text]. We then choose [Formula: see text], the [Formula: see text]-space of [Formula: see text]-valued functions on [Formula: see text]. Here [Formula: see text] and [Formula: see text] with [Formula: see text]. Furthermore, we assume that the operator [Formula: see text] acting on [Formula: see text] is given by [Formula: see text]. Here, for a.a. [Formula: see text], [Formula: see text] is a linear operator on [Formula: see text] with norm [Formula: see text] that satisfies [Formula: see text] and the spectral quantum Yang–Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function [Formula: see text] in the case [Formula: see text] determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its [Formula: see text]-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

Mixed Fractional Differential Equations and Generalized Operator-Valued Mittag-Leffler Functions

We introduce the most general mixed fractional derivatives and integrals from three points of views: probability, the theory of operator semigroups, and the theory of generalized functions. The solutions to the resulting mixed fractional PDEs turned out to be representable in terms of of completely monotone functions in a certain class generalizing the usual Mittag-Leffler functions.

Diffusion on a Hilbert Space Equipped with a Shift- and Rotation-Invariant Measure

We study measures on a real separable Hilbert space E that are invariant with respect to both shifts by arbitrary vectors of the space and orthogonal transformations. In particular, our first concern is a finitely additive analog of the Lebesgue measure. We present such an analog; namely, we construct a nonnegative finitely additive measure that is invariant with respect to shifts and rotations and is defined on the minimal ring of subsets of E that contains all infinite-dimensional rectangles such that the products of their side lengths converge absolutely. We also define a Hilbert space $${\mathcal H}$$ of complex-valued functions on E that are square integrable with respect to a shift- and rotation-invariant measure. For random vectors whose distributions are given by families of Gaussian measures on E that form semigroups with respect to convolution, we define expectations of the corresponding shift operators. We establish that such expectations form a semigroup of self-adjoint contractions in $${\mathcal H}$$ that is not strongly continuous, and find invariant subspaces of strong continuity for this semigroup. We examine the structure of an arbitrary semigroup of self-adjoint contractions of the Hilbert space, which may not be strongly continuous. Finally, we show that the method of Feynman averaging of strongly continuous semigroups based on the notion of Chernoff equivalence of operator-valued functions is also applicable to discontinuous semigroups.

Spectral stability of small amplitude solitary waves of the Dirac equation with the Soler-type nonlinearity

We study the point spectrum of the linearization at a solitary wave solution ϕω(x)e−iωt to the nonlinear Dirac equation in Rn, for all n≥1, with the nonlinear term given by f(ψ⁎βψ)βψ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with positive real part, in the non-relativistic limit ω→m−0, in the case when f∈C1(R∖{0}), f(τ)=|τ|κ+O(|τ|K) for τ→0, with 0<κ<K. For n≥1, we prove the spectral stability of small amplitude solitary waves (ω⪅m) for the charge-subcritical cases κ⪅2/n (in particular, 1<κ≤2 when n=1) and for the “charge-critical case” κ=2/n (with K>4/n).An important part of the stability analysis is the proof of the absence of bifurcation of nonzero-real-part eigenvalues from the embedded threshold points at ±2mi. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model and on the analysis of the behavior of “nonlinear eigenvalues” (characteristic roots of holomorphic operator-valued functions).

Operator-valued zeta functions and Fourier analysis

The Riemann zeta function is defined as the infinite sum , which converges when . The Riemann hypothesis asserts that the nontrivial zeros of lie on the line . Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex s for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by .

Nonuniversality of fluctuations of outliers for Hermitian polynomials in a complex Wigner matrix and a spiked diagonal matrix

We study the fluctuations associated to the a.s. convergence of the outliers established by Belinschi–Bercovici–Capitaine of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the nonuniversality phenomenon established by Capitaine–Donati-Martin–Féral for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The result is described using the operator-valued subordination functions of free probability theory.

Stability of abstract linear switched impulsive differential equations

Linear impulsive switched differential equations in Hilbert space are studied. We propose a new method of stability investigation, which is based in constructing an equivalent impulsive system without switching. Using operator-valued and scalar Lyapunov functions, sufficient conditions for Lyapunov stability of linear switched impulsive equations in a Banach space are established. The examples of studies of this class of equations with periodic switching are given.

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Abstract This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.