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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]

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]

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]

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]

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]

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]

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]

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]

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, 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]

If A is a self-adjoint operator that is bounded below in a Hilbert space H, Littlejohn and Wellman (J Diff Equ 181(2):280-339, ) showed that, for each r > 0, there exists a unique Hilbert space H and a unique self-adjoint operator A in H satisfying certain conditions dependent on H and A. The space H and the operator A are called, respectively, the rth left-definite space and rth left-definite operator associated with ( H, A). In this paper, we show that the operators A, A, and A ( r, s > 0) are isometrically isomorphically equivalent and that the spaces H, H, and H ( r, s > 0) are isometrically isomorphic. These results are then used to reproduce the left-definite spaces and left-definite operators. Furthermore, we will see that our new results imply that the spectra of A and A are equal, giving us another proof of this phenomenon that was first established in Littlejohn and Wellman (J Diff Equ 181(2):280-339, ). [ABSTRACT FROM AUTHOR]

Let A be a bounded linear operator on a Banach space X. We investigate the conditions of existing rank-one operator B such that I + f(A)B is invertible for every analytic function f on (A). Also, we compare the invariant subspaces of f(A)B and B. This work is motivated by an operator method on the Banach space l² for solving some PDEs, extended to general operator space under some conditions here. [ABSTRACT FROM AUTHOR]

In this paper, we study the perturbation of spectra for 2 × 2 operator matrices such as M = () and M = () on the Hilbert space H ⊕ K and the sets $\bigcap\limits_{X \in \mathcal{B}(K,H)} {P_\sigma (M_X )} ,\bigcap\limits_{X \in \mathcal{B}(K,H)} {R_\sigma (M_X )} $ and $\bigcap\limits_{Z \in \mathcal{B}(H,K)} {\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {P_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {R_\sigma (M_Z )} ,\bigcap\limits_{Z \in \mathcal{B}(H,K)} {C_\sigma (M_Z )} $, where R( C) is a closed subspace, are characterized [ABSTRACT FROM AUTHOR]

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σ( H) = σ( A) ∪ σ(− A*). Using the characteristic of the set σ(− A*), we divide the point spectrum σ( A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σ(− A*) and one part of σ( A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σ( H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations. [ABSTRACT FROM AUTHOR]

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class denoted by l-*- A, of operators satisfying T*| T| T ≥ T*| T*| T, and we prove the basic properties of these operators. Using these results, we also prove that if T or T* ∈ l-*- A, then w( f( T)) = f( w( T)), σ( f( T)) = f( σ( T)) for every f ∈ H( σ( T)), where H( σ( T)) denotes the set of all analytic functions on an open neighborhood of σ( T). [ABSTRACT FROM AUTHOR]

We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{ k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23]. [ABSTRACT FROM AUTHOR]

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich’s estimate for manifolds with positive scalar curvature as well as the author’s estimate on surfaces. [ABSTRACT FROM AUTHOR]

Let Ω = [ a, b] ν and let T be a partially integral operator defined in L 2(Ω2) as follows: In the article, we study the solvability of the partially integral Fredholm equations f − ℵ Tf = g, where g ∈ L 2(Ω2) is a given function and ℵ ∈ ℂ. The notion of determinant (which is a measurable function on Ω) is introduced for the operator E − ℵ T, with E is the identity operator in L 2(Ω2). Some theorems on the spectrum of a bounded operator T are proven. [ABSTRACT FROM AUTHOR]

Let Ω= [ a, b] × [ c, d] and T 1, T 2 be partial integral operators in $$ C $$ (Ω): ( T 1 f)( x, y) = $$ \mathop \smallint \limits_a^b $$ k 1( x, s, y) f( s, y) ds, ( T 2 f)( x, y) = $$ \mathop \smallint \limits_c^d $$ k 2( x, ts, y) f( t, y) dt where k 1 and k 2 are continuous functions on [ a, b] × Ω and Ω × [ c, d], respectively. In this paper, concepts of determinants and minors of operators E− τT 1, τ ∈ ℂ and E− τT 2, τ ∈ ℂ are introduced as continuous functions on [ a, b] and [ c, d], respectively. Here E is the identical operator in C(Ω). In addition, Theorems on the spectra of bounded operators T 1, T 2, and T = T 1 + T 2 are proved. [ABSTRACT FROM AUTHOR]

It is known that relatively bounded compact perturbations of analytic semigroups yield again generators of analytic semigroups. On the other hand, for each non-analytic semigroup there is a relatively bounded rank one perturbation which is no longer a generator. In this note we show that in fact any non-analytic semigroup admits a relatively bounded rank one perturbation with a sequence of eigenvalues zj such that the real parts R(zj) converge to infinity. [ABSTRACT FROM AUTHOR]

LetX be a complex Banach space and letT be a bounded linear operator onX. Denote by σp (T) the point spectrum ofT and by the unit circle. We investigate how the growth of the sequence ‖Tn‖ is influenced by the size of the set (T) and by the geometry of the spaceX. We also prove analogous results forC0-semigroups(Tt)t≥0. [ABSTRACT FROM AUTHOR]

We find all elastic systems with point spectrum. For almost elastic nonconservative systems we obtain large time asymptotics for the velocities of the balls. This one-dimensional ball model was invoked in order to elucidate some problems of gas kinetics and turbulence but turned out hard to study in its own right. [ABSTRACT FROM AUTHOR]

We characterize eigenspaces of restrictions of the operatorFgiven byto some subspaces of the vector space [ABSTRACT FROM AUTHOR]

We investigate the spectral properties of a class of Jacobi matrices resulting from periodic pertubations of Jacobi operators with smooth coefficients. [ABSTRACT FROM AUTHOR]

We develop a spectral framework for testing the hypothesis of complete spatial randomness (CSR) for a spatial point pattern. Five formal tests based on the periodogram (sample spectrum) proposed in Mugglestone (1990) are considered. A simulation study is used to evaluate and compare the power of the tests against clustered, regular and inhomogeneous alternatives to CSR. A subset of the tests is shown to be uniformly more powerful than the others against the alternatives considered. The spectral tests are also compared with three widely used space-domain tests that are based on the mean nearest-neighbor distance, the reduced second-order moment function (K-function), and a bivariate Cramér-von Mises statistic. The test based on the scaled cumulative R-spectrum is more powerful than the space-domain tests for detecting clustered alternatives to CSR, especially when the number of events is small. [ABSTRACT FROM AUTHOR]