Dimension Of Eigenspace Research Articles

Automorphisms of graphs corresponding to conjugacy classes of finite-rank self-adjoint operators

We consider the graph whose vertex set is a conjugacy class C consisting of finite-rank self-adjoint operators on a complex Hilbert space H. The dimension of H is assumed to be not less than 3. In the case when operators from C have two eigenvalues only, we obtain the Grassmann graph formed by k-dimensional subspaces of H, where k is the smallest dimension of eigenspaces. Chow's theorem describes automorphisms of this graph for k>1. Under the assumption that operators from C have more than two eigenvalues we show that every automorphism of the graph is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces with the same dimensions. In contrast to this result, Chow's theorem states that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality if C is formed by operators with precisely two eigenvalues.

Commutativity preserving transformations on conjugacy classes of finite rank self-adjoint operators

Let H be a complex Hilbert space and let C be a conjugacy class of rank k self-adjoint operators on H with respect to the action of the group of unitary operators. Under the assumption that dim⁡H≥4k we describe all bijective transformations of C preserving the commutativity in both directions. In particular, it follows from this description that every such transformation is induced by a unitary or anti-unitary operator only in the case when for every operator from C the dimensions of eigenspaces are mutually distinct.

A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces

In this paper, we show that for Tin B({mathcal {H}}), if {mathcal {M}} is almost-invariant for T, then every maximal almost-invariant subspace of {mathcal {M}} is of codimension 1 in {mathcal {M}}, where {mathcal {H}} is a separable, infinite-dimensional Hilbert space. We also describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most 1 acting on {mathcal {H}}. Our result is that for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension 1 in it.

Exact spectrum of the Laplacian on a domain in the Sierpinski gasket

For a certain domain Ω in the Sierpinski gasket SG whose boundary is a line segment, a complete description of the eigenvalues of the Laplacian, with an exact count of dimensions of eigenspaces, under the Dirichlet and Neumann boundary conditions is presented. The method developed in this paper is a weak version of the spectral decimation method due to Fukushima and Shima, since for a lot of “bad” eigenvalues the spectral decimation method can not be used directly. Let ρ0(x), ρΩ(x) be the eigenvalue counting functions of the Laplacian associated to SG and Ω respectively. We prove a comparison between ρ0(x) and ρΩ(x) says that 0≤ρ0(x)−ρΩ(x)≤Cxlog⁡2/log⁡5log⁡x for sufficiently large x for some positive constant C. As a consequence, ρΩ(x)=g(log⁡x)xlog⁡3/log⁡5+O(xlog⁡2/log⁡5log⁡x) as x→∞, for some (right-continuous discontinuous) log⁡5-periodic function g:R→R with 0<infR⁡g<supR⁡g<∞. Moreover, we explain that the asymptotic expansion of ρΩ(x) should admit a second term of the order log⁡2/log⁡5, that becomes apparent from the experimental data. This is very analogous to the conjectures of Weyl and Berry.

The absolute continuity of convolutions of orbital measures in symmetric spaces

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rank G/K +1 is needed.

Spectral Decomposition of Cyclic Operators in Discrete Harmonic Analysis

For the operators of the discrete Fourier transform, the discrete Vilenkin–Christenson transform, and all linear transpositions of the discrete Walsh transform, we obtain their spectral decompositions and calculate the dimensions of eigenspaces. For complex operators, namely, the discrete Fourier transform and the Vilenkin–Christenson transform, we obtain real projectors on eigenspaces. For the discrete Walsh transform, we consider in detail the Paley and Walsh orderings and a new ordering in which the matrices of operators are symmetric. For operators of linear transpositions of the discrete Walsh transforms with nonsymmetric matrices, we obtain a spectral decomposition with complex projectors on eigenspaces. We also present the Parseval frame for eigenspaces of the discrete Walsh transform.

Boundary Value Problems on a Half Sierpinski Gasket

We study boundary value problems for the Laplacian on a domain \Omega consisting of the left half of the Sierpinski Gasket ( SG ), whose boundary is essentially a countable set of points X . For harmonic functions we give an explicit Poisson integral formula to recover the function from its boundary values, and characterize those that correspond to functions of finite energy. We give an explicit Dirichlet to Neumann map and show that it is invertible. We give an explicit description of the Dirichlet to Neumann spectra of the Laplacian with an exact count of the dimensions of eigenspaces. We compute the exact trace spaces on X of the L^2 and L^\infty domains of the Laplacian on SG . In terms of the these trace spaces, we characterize the functions in the L^2 and L^\infty domains of the Laplacian on \Omega that extend to the corresponding domains on SG , and give an explicit linear extension operator in terms of piecewise biharmonic functions.

Eigenvalue analysis for acoustic problem in 3D by boundary element method with the block Sakurai–Sugiura method

Abstract This paper presents accurate numerical solutions for nonlinear eigenvalue analysis of three-dimensional acoustic cavities by boundary element method (BEM). To solve the nonlinear eigenvalue problem (NEP) formulated by BEM, we employ a contour integral method, called block Sakurai–Sugiura (SS) method, by which the NEP is converted to a standard linear eigenvalue problem and the dimension of eigenspace is reduced. The block version adopted in present work can also extract eigenvalues whose multiplicity is larger than one, but for the complex connected region which includes a internal closed boundary, the methodology yields fictitious eigenvalues. The application of the technique is demonstrated through the eigenvalue calculation of sphere with unique homogenous boundary conditions, cube with mixed boundary conditions and a complex connected region formed by cubic boundary and spherical boundary, however, the fictitious eigenvalues can be identified by Burton–Miller's method. These numerical results are supported by appropriate convergence study and comparisons with close form.

Classification of Image Database Using Independent Principal Component Analysis

The paper presents a modified approach of Principal Component Analysis (PCA) for an automatic classification of image database. Principal components are the distinctive or peculiar features of an image. PCA also holds information regarding the structure of data. PCA can be applied to all training images of different classes together forming universal subspace or to an individual image forming an object subspace . But if PCA is applied independently on the different classes of objects, the main direction will be different for them. Thus, they can be used to construct a classifier which uses them to make decisions regarding the class. Also the dimension reduction of feature vector is possible. Initially training image set is chosen for each class. PCA, using eigen vector decomposition, is applied to an individual class forming an individual and independent eigenspace for that class. If there are n classes of training images, we get n eigenspaces. The dimension of eigenspace depends upon the number of selected eigen vectors. Each training image is projected on the corresponding eigenspace giving its feature vector. Thus n sets of training feature vectors are produced. In testing phase, new image is projected on all eigenspaces forming n feature vectors. These feature vectors are compared with training feature vectors in corresponding eigenspace. Feature vector nearest to new image in each eigenspace is found out. Classification of new image is accomplished by comparing the distances between the nearest feature vector and training image feature vector in each eigenspace. Two distance criteria such as Euclidean and Manhattan distance are used. The system is tested on COIL-100 database. Performance is tested and tabulated for different sizes of training image database.

Structure and dimension of the eigenspace of a concave Monge matrix

The eigenspace structure for a given n × n concave Monge matrix in a max-plus algebra is described. Based on the description, an O ( n ) algorithm for computing the eigenspace dimension is formulated, which is faster than the previously known algorithms. Analogous results for convex Monge matrices have been published in [M. Gavalec, J. Plavka, Structure of the eigenspace of a Monge matrix in max-plus algebra, Discrete Appl. Math. 156 (2008) 596–606].

Structure of the eigenspace of a Monge matrix in max-plus algebra

A complete description of the eigenspace structure for a given n × n Monge matrix in a max-plus algebra is presented. Based on the description, an O ( n 2 ) algorithm for computing the eigenspace dimension is formulated, which is faster than the previously known algorithms.

Zeta regularized determinant of the Laplacian for classes of spherical space forms

We derive the spectral zeta function in terms of certain Dirichlet series for a variety of spherical space forms M G . Extending results in [C. Nash, D. O’Connor, Determinants of Laplacians on lens spaces, J. Math. Phys. 36 (3) (1995) 1462–1505] the zeta-regularized determinant of the Laplacian on M G is obtained explicitly from these formulas. In particular, our method applies to manifolds of dimension higher than 3 and it includes the case where G arises from the dihedral group of order 2 m . As a crucial ingredient in our analysis we determine the dimension of eigenspaces of the Laplacian in form of some combinatorial quantities for various infinite classes of manifolds from the explicit form of the generating function in [A. Ikeda, On the spectrum of a Riemannian manifold of positive constant curvature, Osaka J. Math. 17 (1980) 75–93].

On approximately simultaneously diagonalizable matrices

A collection A 1, A 2, …, A k of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer–Sethuraman on bounds for the dimension of the algebra generated by A 1, A 2, …, A k . Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish dim C [ A 1 , … , A k ] ⩽ 5 n / 4 for commuting matrices A 1, …, A k when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)

Spectral gap of positive operators and applications

Let E be a Polish space equipped with a probability measure μ on its Borel σ-field B , and π a non-quasi-nilpotent positive operator on L p ( E , B , μ ) with 1 < p < ∞ . Using two notions, tail norm condition (TNC for short) and uniformly positive improving property ( UPI / μ for short) for the resolvent of π, we prove a characterization for the existence of spectral gap of π, i.e., the spectral radius r sp ( π ) of π being an isolated point in the spectrum σ ( π ) of π. This characterization is a generalization of M. Hino's result for exponential convergence of π n , where the assumption of existence of the ground state, i.e., of a nonnegative eigenfunction of π for eigenvalue r sp ( π ) , in M. Hino's result, is removed. Indeed, under TNC only, we prove the existence of ground state of π. Furthermore, under the TNC, we also establish the finiteness of dimension of eigenspace of π for eigenvalue r sp ( π ) and a interesting finite triangularization of π, which generalizes L. Gross' famous result by removing his assumption of symmetry and weakening his assumption of hyperboundedness. Finally, we give several applications of the characterization for spectral gap to Schrödinger operators, some invariance principles of Markov processes, and Girsanov semigroups respectively. In particular, we present a sharp condition to guarantee the existence of spectral gap for Girsanov semigroups.

Finite-dimensional perturbations of linear operators and some applications to boundary integral equations

Finite-dimensional perturbing operators are constructed using some incomplete information about eigen-solutions of an original and/or adjoint generalized Fredholm operator equation (with zero index). Adding such a perturbing operator to the original one reduces the eigen-space dimension and can, particularly, lead to an unconditionally and uniquely solvable perturbed equation. For the second kind Fredholm operators, the perturbing operators are analyzed such that the spectrum points for an original and the perturbed operators coincide except a spectrum point considered, which can be removed for the perturbed operator. A relation between resolvents of original and perturbed operators is obtained. Effective procedures are described for calculation of the undetermined constants in the right-hand side of an operator equation for the case when these constants must be chosen to satisfy the solvability conditions not written explicitly. Implementation of the methods is illustrated on a boundary integral equation of elasticity.

Effects of powder characteristics on simulation of density distribution variations: S. Shima and M.A.E. Saleh (Kyoto University, Kyoto, Japan)

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rank G/K +1 is needed.

Effects of presintering parameters on double pressed and sintered steels: I. Donaldson (PresMet Corp, Worcester, USA)

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rank G/K +1 is needed.

SPECTRAL APPROXIMATION THEORY FOR MULTIGROUP NEUTRON TRANSPORT OPERATORS

In this paper, by using operator theory in Banach space, we mainly study spectral approximation theory for the case of an inhomogeneous, anisotropically scattering, and arbitrarily finite convex body with any cavity in an Lp context for general p, 1⩽p<∞. We prove that solutions of the time-dependent multigroup systems converge, in a suitable sense, to the corresponding solution of the exact time-dependent transport equation; the eigenvalues and the corresponding eigenfunctions and generalized eigenfunctions of the exact transport operator are approximated respectively by the eigenvalues and the corresponding eigenfunctions and generalized eigenfunctions of the multigroup transport operators; the dominant eigenvalues and the corresponding positive eigenfunctions of the multigroup transport operators (the existence is proved, too) accumulate to those of the exact transport operators; the orders of magnitude of the rate of convergence axe given respectively. Other results are obtained to compare the dimensions of eigenspaces and generalized eigenspaces of the transport operator with those of the multigroup transport operators.

Localization of Discrete Spectrum of Multiparticle Schrödinger Operators

An upper bound on the dimension of eigenspaces of multiparticle Schrödinger operators is given. Its relation to upper and lower bounds on the eigenvalues is discussed.

Modification of Müller's variational principle for Schrödinger operators with spherical symmetric potential

An upper bound on the dimension of eigenspaces of Schrödinger operators with spherical symmetric potentials is given. The use of the bound to localize the discrete spectrum is discussed.