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The spectral mapping theorem, in conjunction with the concept of a spectral set, is utilized to construct reducing pairs of subspaces for a given operator. By leveraging the theory of functional calculus, we revisit the well-established spectral mapping theorem for various spectra, with a particular focus on the point spectrum. Concluding this discussion, we present an application wherein we collect and establish properties of the point spectrum of a compact exponentially m-isometry. Also, we examine the behavior of solutions of a system of differential equations associated with exponentially m-isometric matrices. [ABSTRACT FROM AUTHOR]

The Maxwell operator in a 3D cylinder is considered. The coefficients are assumed to be scalar functions depending on the longitudinal variable only. Such an operator is represented as a sum of a countable set of matrix differential operators of first order acting in $ L_2(\mathbb {R}) $ L 2 (R). Based on this representation we give a detailed description of the structure of the spectrum of the Maxwell operator in two particular cases: (1) in the case of coefficients stabilizing at infinity; and (2) in the case of periodic coefficients. [ABSTRACT FROM AUTHOR]

Given a densely defined and gapped symmetric operator with infinite deficiency index, it is shown how self-adjoint extensions admitting arbitrarily prescribed portions of the gap as essential spectrum are identified and constructed within a general extension scheme. The emergence of new spectrum in the gap by self-adjoint extension is a problem with a long history and recent deep understanding, and yet it remains topical in several recent applications. Whereas it is already an established fact that, in case of infinite deficiency index, any kind of spectrum inside the gap can be generated by a suitable self-adjoint extension, the present discussion has the virtue of showing the clean and simple operator-theoretic mechanism of emergence of such extensions. [ABSTRACT FROM AUTHOR]

Let E be a closed subset of the unit circle T , and let α ∈ (0 , 1) . Nikolski's result states that if the Hausdorff dimension of E is strictly greater than α , then for any operator T on a separable Hilbert space such that the point spectrum σ p (T) of T contains E, the series ∑ n n α - 1 ‖ T n ‖ - 2 converges. A partial converse of this result has been obtained by El-Fallah and Ransford. Namely they constructed, for any α strictly greater than the upper box dimension of E, an operator T on a separable Hilbert space such that σ p (T) contains E and 1 n ∑ k = 0 n - 1 T k 2 ≲ n α . In this paper, we improve on this latter result for regular sets. Indeed, for any Ahlfors–David regular set E and for any α strictly greater than the Hausdorff dimension of E there exists an operator T on a separable Hilbert space such that σ p (T) contains E and ‖ T n ‖ 2 ≍ n α . [ABSTRACT FROM AUTHOR]

A matrix model is considered related to a system describing two identical fermions and one particle of another nature on a lattice, interacting via annihilation and creation operators. The problem of the study of the spectrum of a block operator matrix is reduced to the investigation of the spectrum of block operator matrices of order three with a discrete variable, and the relations for the spectrum, essential spectrum, and point spectrum are established. Two-particle and three-particle branches of the essential spectrum of the block operator matrix are singled out. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spectrum of the classical Hilbert transform on Orlicz spaces Lσ over the real line R, extending Widom's and Jörgens's results in the context of Lp spaces [3, 8], since the classical Lebesgue spaces are particular examples of Orlicz spaces when the N-function σ = xp=p. Our motivation to do so is due to the classical result of Boyd [1] which says that the Hilbert transform is bounded on certain Orlicz spaces and the fact that the spectrum of the bounded linear operator is not an empty set. We first present an auxiliary result from the general theory of Banach algebras and results from general theory of Banach spaces, which further helps us to give a full decsription of the fine spectrum of the Hilbert transform on Orlicz spaces over the real line R. We also present a resolvent set of the Hilbert transform on Orlicz spaces over the real line R as well as its resolvent operator. [ABSTRACT FROM AUTHOR]

The spectrum of an infinite-dimensional doubly stochastic matrix, when considered as a bounded operator on the sequence space ep with 1 ≥ p < ∞, is contained within the closed unit disc D. In our work, we present an infinite doubly stochastic matrix that exhibits the entire closed unit disc as its spectrum. However, we prove that the points eiπr, where r is an irrational real number, cannot serve as eigenvalues for any doubly stochastic matrices, be it finite or infinite in size. On the other hand, we show that every other point within the closed unit disc can indeed be an eigenvalue of an infinite-dimensional doubly stochastic matrix. In fact, we construct a specific example of an infinite doubly stochastic matrix whose point spectrum precisely consists of D∪{eiπr: r ∈ Q}. Additionally, we investigate the behavior of doubly stochastic matrices in the context of the sequence space e∞, highlighting the contrasts with the ep setting for 1 ≥ p < ∞. [ABSTRACT FROM AUTHOR]

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A bounded linear operator T on a complex Banach space X is superconvex-cyclic, if for some vector x in the space, C. c o (o r b (T , x)) is dense. Here, co(orb(T, x)) means the convex hull of { T n x : n ≥ 0 }. In this paper, we give necessary conditions for the superconvex-cyclicity of an operator in terms of the point spectrum of its adjoint. Also, we prove that positive multiples of superconvex-cyclic operators are superconvex-cyclic. However, this concept is not necessarily transferred to positive integer powers of an operator. Moreover, we completely characterize superconvex-cyclic matrices whose spectra contain no real number or contain a positive real number. [ABSTRACT FROM AUTHOR]

In this paper, first we provide some necessary and sufficient conditions for quasi- normality and quasi- hyponormality of weighted conditional type operators. And then the spectrum, residual spectrum, point spectrum and spectral radius of weighted conditional type operators are computed. As an application, we give an equivalent conditions for weighted conditional type operators to be quasinilpotent. Also, some examples are provided to illustrate concrete applications of the main results. [ABSTRACT FROM AUTHOR]

A two-point algebra is a set of bounded analytic functions on the unit disk that agree at two distinct points a , b ∈ D . This algebra serves as a multiplier algebra for the family of Hardy Hilbert spaces H t 2 : = { f ∈ H 2 : f (a) = t f (b) } , where t ∈ C ∪ { ∞ } . We show that various spectra of certain Toeplitz operators acting on these spaces are connected. [ABSTRACT FROM AUTHOR]

The inĄnite lower triangular matrix B(r1, . . ., rl; s1, . . ., sl') is considered over the sequence space c0, where l and l' are positive integers. The diagonal and sub-diagonal entries of the matrix consist of the oscillatory sequences r = (rk(mod l)+1) and s = (sk(mod l')+1), respectively. The rest of the entries of the matrix are zero. It is shown that the matrix represents a bounded linear operator. Then the spectrum of the matrix is evaluated and partitioned into its Ąne structures: point spectrum, continuous spectrum, residual spectrum, etc. In particular, the spectra of the matrix B(r1, . . ., r4; s1, . . ., s6) are determined. Finally, an example is taken in support of the results. [ABSTRACT FROM AUTHOR]

In this study, we give another generalization of second order backward difference operator ∇ 2 by introducing its quantum analog ∇ q 2. The operator ∇ q 2 represents the third band infinite matrix. We construct its domains c 0 (∇ q 2) and c (∇ q 2) in the spaces c 0 and c of null and convergent sequences, respectively, and establish that the domains c 0 (∇ q 2) and c (∇ q 2) are Banach spaces linearly isomorphic to c 0 and c , respectively, and obtain their Schauder bases and α -, β - and γ -duals. We devote the last section to determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the operator ∇ q 2 over the Banach space c 0 of null sequences. [ABSTRACT FROM AUTHOR]

In this note, we prove a characterization of reflexive Banach spaces with unconditional basis and study the set theoretical complexity of certain sets associated to linear bounded operators acting on reflexive separable Banach spaces. [ABSTRACT FROM AUTHOR]

We show that near periodic rivet chains that connect two Kirchhoff plates and are modeled by point Sobolev transmission conditions, Rayleigh waves arise, propagate along the chains, and decay exponentially in the orthogonal direction. Under additional geometric conditions we discover the standing (periodic) waves that carry no energy. [ABSTRACT FROM AUTHOR]

Consider a tuple (Y 1 , ... , Y d) of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each Y i. We address the question what one can say about the sizes of the eigenspaces for any non-commutative polynomial P (Y 1 , ... , Y d) in those operators? We show that for each polynomial P there are unavoidable eigenspaces, which occur in P (Y 1 , ... , Y d) for any (Y 1 , ... , Y d) with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms - where it is given by realizations via matrices over the free skew field and via rank calculations - and in analytic terms - where it is given by freely independent random variables with prescribed atoms in their distributions. This allows us to draw many new conclusions about atoms in polynomials of free variables. In particular, we give a complete description of atoms in the free commutator and the free anti-commutator. Furthermore, our results do not only apply to polynomials, but more generally also to non-commutative rational functions. Since many random matrix models become asymptotically free in the large N limit, our results allow us to calculate the location and size of atoms in the asymptotic eigenvalue distribution of polynomials and rational functions in randomly rotated matrices. Applications to the Atiyah property and to the calculation of the spectrum of universal covers of graphs are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider an extended coined Szegedy model and discuss the existence of the point spectrum of induced quantum walks in terms of recurrence properties of the underlying birth and death process. We obtain that if the underlying random walk is not null recurrent, then the point spectrum exists in the induced quantum walks. As an application, we provide a simple computational way of the dispersion relation of the edge state part for the topological phase model driven by quantum walk using the recurrence properties of underlying birth and death process. [ABSTRACT FROM AUTHOR]

We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation. Then by using this result we prove that the essential growth bound of the C0-semigroup generated by the operator is 0, the C0-semigroup is not compact, not eventually compact, even not quasi-compact. Moreover, we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution. In addition, we obtain the spectral radius and essential spectral radius of the C0-semigroup. Lastly, we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum. [ABSTRACT FROM AUTHOR]

We study location of eigenvalues of one-dimensional discrete Schrödinger operators with complex ℓ p -potentials for 1 ≤ p ≤ ∞ . In the case of ℓ 1 -potentials, the derived bound is shown to be optimal. For p > 1 , two different spectral bounds are obtained. The method relies on the Birman–Schwinger principle and various techniques for estimations of the norm of the Birman–Schwinger operator. [ABSTRACT FROM AUTHOR]

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire's sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schrödinger operators. [ABSTRACT FROM AUTHOR]

We investigate the point spectrum associated with travelling wave solutions in a Keller–Segel model for bacterial chemotaxis with small diffusivity of the chemoattractant, a logarithmic chemosensitivity function and a constant, sublinear or linear consumption rate. We show that, for constant or sublinear consumption, there is an eigenvalue at the origin of order two. This is associated with the translation invariance of the model and the existence of a continuous family of solutions with varying wave speed. These point spectrum results, in conjunction with previous results in the literature, imply that in these cases the travelling wave solutions are absolute unstable if the chemotactic coefficient is above a certain critical value, while they are transiently unstable otherwise. • Travelling wave solutions are transiently unstable for a range of parameter values. • The point eigenvalue of order two at the origin persists for small diffusion. • The constant consumption model is spectrally stable in the weighted function space. • The spectral gap vanishes for linear consumption. • Eigenfunctions for the origin are contained in the weighted function space. [ABSTRACT FROM AUTHOR]

The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component of the dynamics. They factor the dynamics, which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a method through which these eigenfunctions can be obtained from a kernel integral operator, which also annihilates the continuous spectrum. We show that incorporating a large number of delay coordinates in constructing the kernel of that operator results, in the limit of infinitely many delays, in the creation of a map into the point spectrum subspace of the Koopman operator. This enables efficient approximation of Koopman eigenfunctions in systems with pure point or mixed spectra. We illustrate our results with applications to product dynamical systems with mixed spectra. [ABSTRACT FROM AUTHOR]

In this paper, a criterion for the existence of large linear algebras consisting, except for zero, of one-to-one operators on an infinite dimensional Banach space is provided. As a consequence, it is shown that every separable infinite dimensional Banach space supports a commutative infinitely generated free linear algebra of operators all of whose nonzero members are one-to-one. In certain cases, the assertion holds for nonseparable Banach spaces.

A linear system of mixed-type differential difference equations is studied. A step derivation method and some conditions on an initial function are used to guarantee existence, uniqueness and smoothness of the solution. Further, a transformation group is defined on a complete countably normed space of initial functions and the spectrum of the infinitesimal generator of this group is studied. The same technique applies to a linear system of retarded differential difference equations. A problem to extend solutions of such a system to the left on the real line is solved.

Let (X, d) be a locally compact separable ultrametric space. Given a measure m on X and a function C defined on the set B of all balls B ⊂ X, we consider the hierarchical Laplacian L = LC. The operator L acts in L²(X, m), is essentially self-adjoint, and has a purely point spectrum. Choosing a family {ε(B)}B∈B of i.i.d. random variables, we define the perturbed function C(B) = C(B)(1 + ε(B)) and the perturbed hierarchical Laplacian L = LC. All outcomes of the perturbed operator L are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process M defined in terms of L-eigenvalues. Under some natural assumptions, M can be approximated by a Poisson point process. Using a result of Arratia, Goldstein, and Gordon based on the Chen--Stein method, we provide total variation convergence rates for the Poisson approximation. We apply our theory to random perturbations of the operator Dα, the p-adic fractional derivative of order α > 0. [ABSTRACT FROM AUTHOR]

Abstract The space of entire functions which are integrable with respect to the Gaussian weight, known also as the Fock space, is one of the preferred functional Hilbert spaces for modeling and experimenting harmonic analysis, quantum mechanics or spectral analysis phenomena. This space of entire functions carries a three parameter family of canonical isometric involutions. We characterize the linear differential operators acting on Fock space which are complex symmetric with respect to these conjugations. In parallel, as a basis of comparison, we discuss the structure of self-adjoint linear differential operators. The computation of the point spectrum of some of these operators is carried out in detail. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectra and Fredholm properties of Ornstein-Uhlenbeck operators Lv(x) :=AΔv(x) + + Df (vΔ(x))v(x), x ∈ Rd, d ≥ 2, where vΔ : Rd→Rm is the profile of a rotating wave satisfying vΔ(x)→v∞ ∈ Rm as x →∞, the map f : Rm→Rm is smooth and the matrix A ∈ Rm,m has eigenvalues with positive real parts and commutes with the limit matrix Df (v∞). The matrix S ∈ Rd,d is assumed to be skew-symmetric with eigenvalues (λ1, . . . , λd)=(±iσ1, . . . ,±iσk, 0, . . . , 0). The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction-diffusion systems. We prove under appropriate conditions that every λ ∈ C satisfying the dispersion relation det(λIm + η2A - Df (v∞) + iIm)=0 for some η ∈ R and n ∈ Zk, σ =(σ1, . . . , σk)T ∈ Rk belongs to the essential spectrum σess(L) in Lp. For values Re λ to the right of the spectral bound for Df (v∞), we show that the operator λI - L is Fredholm of index 0, solve the identification problem for the adjoint operator (λI - L)∗ and formulate the Fredholm alternative. Moreover, we show that the set σ(S) ∪ {λi + λj : λi, λj ∈ σ(S), 1 ≤ i

We extend a result concerning λ-commuting normal operators with empty point spectrum. More precisely, we prove that for a hyponormal operator T with empty point spectrum for which there exists a Hilbert-Schmidt operator K such that TK = λKT + μK for some |λ| < 1 and μ ∈ C, implies K = 0.

The research included in the paper concerns a class of symmetric block Jacobi matrices. The problem of the approximation of eigenvalues for a class of a self-adjoint unbounded operators is considered. We estimate the joint error of approximation for the eigenvalues, numbered from \(1\) to \(N\), for a Jacobi matrix \(J\) by the eigenvalues of the finite submatrix \(J_n\) of order \(pn \times pn\), where \(N = \max \{k \in \mathbb{N}: k \leq rpn\}\) and \(r \in (0,1)\) is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of \(J\) in the case \(p=3\).

We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space \(l^2(\mathbb{N})\) by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with \(n\to \infty\)) of the joint error of approximation for the first \(n\) eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order \(n \times n\). The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [H. Volkmer, Error Estimates for Rayleigh-Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation, Constr. Approx. 20 (2004), 39-54]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.

A linear system of mixed-type differential difference equations is studied. A step derivation method and some conditions on an initial function are used to guarantee existence, uniqueness and smoothness of the solution. Further, a transformation group is defined on a complete countably normed space of initial functions and the spectrum of the infinitesimal generator of this group is studied. The same technique applies to a linear system of retarded differential difference equations. A problem to extend solutions of such a system to the left on the real line is solved. [ABSTRACT FROM AUTHOR]

The spectrum of the Dirac operator is investigated on a Riemannian manifold one of whose end is asymptotically flat in some sense. It is proved that there is no non-zero eigenvalue and the essential spectrum is equal to R . [ABSTRACT FROM AUTHOR]

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if X is an infinite-dimensional complex Banach space then every operator T ∈ L ( X ) admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well. [ABSTRACT FROM AUTHOR]

This paper deals with general $${n\times n}$$ upper-triangular operator matrices with given diagonal entries. The characterizations of perturbations of their point spectra, residual spectra and continuous spectra are given, based on the space decomposition method. [ABSTRACT FROM AUTHOR]

We will study the linear spectral properties of steady Saffman–Taylor bubbles with surface tension in a Hele-Shaw cell. It was known [21] that the Saffman–Taylor exact solution without surface tension is linearly unstable and the point spectrum is a continuum set. In this paper, we rigorously conclude that the point spectrum of the stability operator of any branch of Saffman–Taylor bubbles is a discrete set when surface tension is not zero. [ABSTRACT FROM AUTHOR]

Aq-analog expression of the determinantal polynomial of a weighted shift operator is introduced and applied to find the numerical radius of the operator. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundedness of the Libera operator on Dirichlet spaces in terms of the Schur test. Moreover, we get its point spectrum and norm. [ABSTRACT FROM AUTHOR]

The notion of atomic observable was introduced by S.Gudder for effect test spaces in 1997. In this paper an observable is a σ-homomorphism from the Borel algebra on a line to some logic. Roughly, an observable on a logic is atomic, if it is completely determined by its restriction to one-element subsets of its point spectrum. In particular, every discrete observable is atomic. We study some elementary properties of such observables, and discuss a possible notion of functional dependency between them. Algebraically, a dependency is a certain preorder relation on the set of all atomic observables, which induces an order relation on the set of all maximal orthogonal subsets of the logic. Several properties, as well as characteristics in terms of the underlying logic, of these relations are stated. [ABSTRACT FROM AUTHOR]

This work consists of two parts. The first one contains a characterization (localization) of the point spectrum of one sided, infinite and periodic Jacobi matrices with scalar entries. The second one deals with the same questions about one sided, infinite periodic Jacobi matrices with matrix entries. In particular, an example illustrating the difference between the above localization property in scalar and matrix entries cases is given. [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum of the operator D(r, 0, 0, s) on a sequence space bv0, the space of all sequences x such that limn→∞ xn = 0 and ∥ x ∥ = Σ n=0∞ ∣ x n+1 - xn ∣ < ∞. [ABSTRACT FROM AUTHOR]

In this paper we study the structural, topological, metric and fractal properties of the distribution of the complex-valued random variable ζ = Σ∞k=1 2/3k εηk ≡ Δη1…ηk…, where the indices ηk are independent random variables taking the values 0, 1, 2, 3, 4 with probabilities p0k, p1k, p2k, p3k, p4k respectively, pik ≥ 0, Σ4i=0 pik = 1, Σ4i=0 pik = 1 for any k ∊ ℕ, εm = emπi/2 = im (m = 0, 1, 2, 3) are 4th roots of unity, ε4 = 0. We prove that the distribution of ζ is supported on a self-similar fractal curve such that the branching index of every point is equal to 2 or 4. We obtain conditions for the distribution of ζ to be of pure type: discrete or singularly continuous with respect to two-dimensional Lebesgue measure. In the case of discrete distribution, the point spectrum is described. In the case of singular distribution, we describe topological, metric and fractal properties of minimal closed support (spectrum). We study in detail the case of identically distributed indices. In particular, the mathematical expectation and variance of the random variable are found. [ABSTRACT FROM AUTHOR]

In this paper we determine the spectrum of $$D(r$$ , 0, $$s$$ , 0, $$t)$$ regarded as an operator on the sequence space $$\ell _{p}$$ and bv $$_{p}$$ (1 $$\le p<\infty )$$ . [ABSTRACT FROM AUTHOR]

In this paper, we study the spectrum of the operator D(r, 0, 0, s) on a sequence space bv0, the space of all sequences x such that limn→∞ xn = 0 and ∥ x ∥ = Σ n=0∞ ∣ x n+1 - xn ∣ < ∞. [ABSTRACT FROM AUTHOR]

The paper is concerned with essential spectra of singular linear Hamiltonian systems of arbitrary order with arbitrary equal defect indices. Several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding Hamiltonian system, and a sufficient and necessary condition for the essential spectral points is obtained for Hamiltonian systems of even-order with one singular endpoint. An advantage of these results is that one can determine the essential spectral points of Hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations. In addition, two illustrative examples are provided to show how to get some information about the essential spectrum by our results. [ABSTRACT FROM AUTHOR]

In this paper, some $$*$$ -classes of weighted conditional expectation type operators, such as $$A$$ -class, $$*$$ - $$A$$ -class and quasi- $$*$$ - $$A$$ classes on $$L^{2}(\Sigma )$$ are investigated. Also, the spectrum, point spectrum and spectral radius of these operators are computed. [ABSTRACT FROM AUTHOR]