Finite Algebraic Multiplicity Research Articles

Spectral Properties of the Operator in the Problem of Oscillations in a Mixture of Viscous Compressible Fluids

We study the problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids that fills a bounded domain in three-dimensional space with infinitely smooth boundary. We prove that the essential spectrum of the problem is a finite set of segments located on the real axis. The remaining spectrum consists of isolated eigenvalues of finite algebraic multiplicity and is located on the real axis, with the possible exception of finitely many complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

Spectral bounds for indefinite singular Sturm–Liouville operators with uniformly locally integrable potentials

The non-real spectrum of a singular indefinite Sturm–Liouville operatorA=1r(−ddxpddx+q) with a sign changing weight function r consists (under suitable additional assumptions on the real coefficients 1/p,q,r∈Lloc1(R)) of isolated eigenvalues with finite algebraic multiplicity which are symmetric with respect to the real line. In this paper bounds on the absolute values and the imaginary parts of the non-real eigenvalues of A are proved for uniformly locally integrable potentials q and potentials q∈Ls(R) for some s∈[1,∞]. The bounds depend on the negative part of q, on the norm of 1/p, and in an implicit way on the sign changes and zeros of the weight function.

The Spectrum of a Transport Operator with Abstract Boundary Conditions

The objective of this paper is to research transport equations with anisotropic continuous energy homogeneous slab geometry for abstract boundary conditions in L_p(1≤p+∝) space.By resolvent operator approach,it is to obtain that the spectrum of the transport operator only consist of finitely many isolate eigenvalues which have a finite algebraic multiplicity in the trip(?)_e.

Spectral Analysis of the General Singular Transport Operator with Periodic Boundary Conditions

This paper researches the general singular transport equations in L_p(1p∞) space for anisotropic,continuous energy,nonhomogeneous with periodic boundary conditions in slab geometry. It proves that the transport operator A generates a strongly continuous C_0 semigroup V(t)(t≥0) and the compactness of the second-order remained term of the D yson-Phillips expansion for the C_0 semigroup V(t)(t≥0),and it obtains the results that the spectrum of the transport operator only consist of finitely many isolate eigenvalues which have a finite algebraic multiplicity in tripΓ.

On the spectrum of Euler–Bernoulli beam equation with Kelvin–Voigt damping

The spectral property of an Euler–Bernoulli beam equation with clamped boundary conditions and internal Kelvin–Voigt damping is considered. The essential spectrum of the system operator is rigorously identified to be an interval on the left real axis. Under some assumptions on the coefficients, it is shown that the essential spectrum contains continuous spectrum only, and the point spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The asymptotic behavior of eigenvalues is presented.

Spectral analysis of a wave equation with Kelvin‐Voigt damping

AbstractA vibrating system with some kind of internal damping represents a distributed or passive control. In this article, a wave equation with clamped boundary conditions and internal Kelvin‐Voigt damping is considered. It is shown that the spectrum of the system operator is composed of two parts: point spectrum and continuous spectrum. The point spectrum consists of isolated eigenvalues of finite algebraic multiplicity, and the continuous spectrum that is identical to the essential spectrum is an interval on the left real axis. The asymptotic behavior of eigenvalues is presented.

Spectral Theory of the Klein–Gordon Equation in Krein Spaces

AbstractIn this paper the spectral properties of the abstract Klein–Gordon equation are studied. The main tool is an indefinite inner product known as the charge inner product. Under certain assumptions on the potential V, two operators are associated with the Klein–Gordon equation and studied in Krein spaces generated by the charge inner product. It is shown that the operators are self-adjoint and definitizable in these Krein spaces. As a consequence, they possess spectral functions with singularities, their essential spectra are real with a gap around 0 and their non-real spectra consist of finitely many eigenvalues of finite algebraic multiplicity which are symmetric to the real axis. One of these operators generates a strongly continuous group of unitary operators in the Krein space; the other one gives rise to two bounded semi-groups. Finally, the results are applied to the Klein–Gordon equation in ℝn.

General Adiabatic Evolution with a Gap Condition

We consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times: the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity, away from the rest of the spectrum. The restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable. The presence of eigenilpotents in the spectral decomposition of the generator forbids the evolution to follow the instantaneous eigenprojectors of the generator in the adiabatic limit. Making use of superadiabatic renormalization, we construct a different set of time-dependent projectors, close to the instantaneous eigeprojectors of the generator in the adiabatic limit, and an approximation of the evolution semigroup which intertwines exactly between the values of these projectors at the initial and final times. Hence, the evolution semigroup follows the constructed set of projectors in the adiabatic regime, modulo error terms we control.

Spectral Theory of the Klein–Gordon Equation in Pontryagin Spaces

In this paper we investigate an abstract Klein–Gordon equation by means of indefinite inner product methods. We show that, under certain assumptions on the potential which are more general than in previous works, the corresponding linear operator A is self-adjoint in the Pontryagin space $$\mathcal{K}$$ induced by the so-called energy inner product. The operator A possesses a spectral function with critical points, the essential spectrum of A is real with a gap around 0, and the non-real spectrum consists of at most finitely many pairs of complex conjugate eigenvalues of finite algebraic multiplicity; the number of these pairs is related to the ‘size’ of the potential. Moreover, A generates a group of bounded unitary operators in the Pontryagin space $$\mathcal{K}$$ . Finally, the conditions on the potential required in the paper are illustrated for the Klein–Gordon equation in $$\mathbb{R}^n$$ ; they include potentials consisting of a Coulomb part and an L p -part with n ≤ p < ∞.

Kato type operators and Weyl's theorem

A Banach space operator T satisfies Weyl's theorem if and only if T or T ∗ has SVEP at all complex numbers λ in the complement of the Weyl spectrum of T and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity. If T ∗ (respectively, T) has SVEP and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all λ ∈ iso σ ( T ) ), then T satisfies a-Weyl's theorem (respectively, T ∗ satisfies a-Weyl's theorem).

SPECTRAL PROPERTIES OF TRANSPORT EQUATIONS FOR SLAB GEOMETRY IN L1WITH REENTRY BOUNDARY CONDITIONS

The time dependent neutron transport equation in a nonhomogeneous slab with generalized boundary conditions in the setting of L 1is studied. For the transport operator Aand the C0semigroup T(t) generated by A, it is shown under fairly general assumptions that the accumulation points of Pas(A) : σ(A) ∩ {λ : Reλ > −λ*}, if they exist, can only appear on the line Reλ = −λ*, where λ*is the essential infimum of the total collision frequency. Further, the spectrum of T(t) outside the disk {λ : |λ| ≤ exp(−λ* t)} consists of isolated eigenvalues of T(t) with finite algebraic multiplicity, and the accumulation points of σ(T(t)) ∩ {λ : |λ| > exp(−λ* t)}, if they exist, can only appear on the circle {λ : |λ| = exp(−λ* t)}.

Spectral Properties of a Streaming Operator with Diffuse Reflection Boundary Condition

In this paper, the spectrum of a streaming operator with diffuse reflection boundary condition arising in transport theory is observed in L1 space. We associate this unbounded operator with bounded integral operators which are determined by the boundary condition. It is shown that the spectrum of the streaming operator is completely determined by these integral operators and consists of countable isolated eigenvalues, each of which has finite algebraic multiplicity. Furthermore, under the assumption that the diffusion on the boundary is of Maxwell type, it is proved that the streaming operator has only one real eigenvalue which is simple and that each of the complex eigenvalues has geometry multiplicity one. A formula for computing these eigenvalues is also derived, by means of which the algebraic multiplicity of the complex eigenvalues is discussed.

A note on spectral approximation of linear operations

This work deals on sufficient conditions for the spectral convergence of a sequence of linear operators. The general context is a complex separable Banach space and the pointwise limit of the sequence is a continuous linear operator which is not supposed to be compact. By spectral convergence is meant the self-range-uniform convergence of the approximate spectral projections. This implies the gap convergence of the approximate maximal invariant subspaces to those of the limit operator corresponding to a nonzero isolated eigenvalue (or a subset of close nonzero isolated eigenvalues) with finite algebraic multiplicity. Neither the exact nor the approximate eigenvalues are supposed to be semisimple.

A perturbation theorem for the essential spectral radius of strongly continuous semigroups

We generalize the Jorgens-Vidav semigroup perturbation theorem: If (U (t)), (V (t)) are s. c. semigroups, the generator of (V (t)) a bounded perturbation of the generator of (U (t)), and some remainder in the iteration series ofV (t) is strictly power compact, then the spectrum ofV (t) outside the spectral disc ofU (t) consists of eigenvalues of finite algebraic multiplicity. As a prerequisite, we show the invariance of components of the essential resolvent set of an operator under relatively power compact pertubations.

Erratum to "Essential Spectrum for a Hilbert Space Operator"

TIEOREM 3. If T is a closed operator which is reduced by each finitedimensional subspace ker (T z) for any complex z then (2), (3), (4), and (5), from the following sets are identical. Furthermore, if T is also reduced by each finite-dimensional subspace ker (T* z) then all five sets are identical: (1) aei(T), (2) Ue2( 7), (3) Oe3((), (4) re4(T), (5) the points of a(T) which are not isolated eigenvalues of finite algebraic multiplicity.