7,547 research outputs found
The effect of perturbations of linear operators on their polar decomposition
The effect of matrix perturbations on the polar decomposition has been
studied by several authors and various results are known. However, for
operators between infinite-dimensional spaces the problem has not been
considered so far. Here, we prove in particular that the partial isometry in
the polar decomposition of an operator is stable under perturbations, given
that kernel and range of original and perturbed operator satisfy a certain
condition. In the matrix case, this condition is weaker than the usually
imposed equal-rank condition. It includes the case of semi-Fredholm operators
with agreeing nullities and deficiencies, respectively. In addition, we prove a
similar perturbation result where the ranges or the kernels of the two
operators are assumed to be sufficiently close to each other in the gap metric.Comment: 13 page
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
Angles Between Infinite Dimensional Subspaces with Applications to the Rayleigh-Ritz and Alternating Projectors Methods
We define angles from-to and between infinite dimensional subspaces of a
Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general
canonical correlations of stochastic processes. The spectral theory of
selfadjoint operators is used to investigate the properties of the angles,
e.g., to establish connections between the angles corresponding to orthogonal
complements. The classical gaps and angles of Dixmier and Friedrichs are
characterized in terms of the angles. We introduce principal invariant
subspaces and prove that they are connected by an isometry that appears in the
polar decomposition of the product of corresponding orthogonal projectors.
Point angles are defined by analogy with the point operator spectrum. We bound
the Hausdorff distance between the sets of the squared cosines of the angles
corresponding to the original subspaces and their perturbations. We show that
the squared cosines of the angles from one subspace to another can be
interpreted as Ritz values in the Rayleigh-Ritz method, where the former
subspace serves as a trial subspace and the orthogonal projector of the latter
subspace serves as an operator in the Rayleigh-Ritz method. The Hausdorff
distance between the Ritz values, corresponding to different trial subspaces,
is shown to be bounded by a constant times the gap between the trial subspaces.
We prove a similar eigenvalue perturbation bound that involves the gap squared.
Finally, we consider the classical alternating projectors method and propose
its ultimate acceleration, using the conjugate gradient approach. The
corresponding convergence rate estimate is obtained in terms of the angles. We
illustrate a possible acceleration for the domain decomposition method with a
small overlap for the 1D diffusion equation.Comment: 22 pages. Accepted to Journal of Functional Analysi
Representation Theorems for Indefinite Quadratic Forms Revisited
The first and second representation theorems for sign-indefinite, not
necessarily semi-bounded quadratic forms are revisited. New straightforward
proofs of these theorems are given. A number of necessary and sufficient
conditions ensuring the second representation theorem to hold is proved. A new
simple and explicit example of a self-adjoint operator for which the second
representation theorem does not hold is also provided.Comment: This work is supported in part by the Deutsche Forschungsgemeinschaf
Eigenvalue estimates for non-selfadjoint Dirac operators on the real line
We show that the non-embedded eigenvalues of the Dirac operator on the real
line with non-Hermitian potential lie in the disjoint union of two disks in
the right and left half plane, respectively, provided that the of
is bounded from above by the speed of light times the reduced Planck
constant. An analogous result for the Schr\"odinger operator, originally proved
by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For
massless Dirac operators, the condition on implies the absence of nonreal
eigenvalues. Our results are further generalized to potentials with slower
decay at infinity. As an application, we determine bounds on resonances and
embedded eigenvalues of Dirac operators with Hermitian dilation-analytic
potentials
Dipoles in Graphene Have Infinitely Many Bound States
We show that in graphene charge distributions with non-vanishing dipole
moment have infinitely many bound states. The corresponding eigenvalues
accumulate at the edges of the gap faster than any power
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