Finite Rank Perturbations Research Articles

On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \), Im z ≠ 0, where Bz are finite-rank operators such that dom Bz ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\), we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\). We prove the uniqueness of \(\tilde A\) under the condition that rank Bz = n.

Singular Perturbations of Self-Adjoint Operators

Singular finite rank perturbations of an unbounded self-adjoint operator A 0 in a Hilbert space ℌ0 are defined formally as A (α)=A 0+GαG *, where G is an injective linear mapping from ℋ=ℂ d to the scale space ℌ-k(A0)k ∈ ℕ, k∈N, of generalized elements associated with the self-adjoint operator A 0, and where α is a self-adjoint operator in ℋ. The cases k=1 and k=2 have been studied extensively in the literature with applications to problems involving point interactions or zero range potentials. The scalar case with k=2n>1 has been considered recently by various authors from a mathematical point of view. In this paper, singular finite rank perturbations A (α) in the general setting ran G⊂ ℌ− k (A 0), k∈N, are studied by means of a recent operator model induced by a class of matrix polynomials. As an application, singular perturbations of the Dirac operator are considered.

Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator

We consider the dispersion properties in L p spaces of Schrödinger hamiltonians with a large number of obstacles modelled by rank one perturbations. We obtain both for the dispersion an Strichartz estimates nonperturbative results with respect to the coupling constants.

On Field Theory Methods in Singular Perturbation Theory

Singular and supersingular finite rank perturbations of self-adjoint operators are studied using methods from renormalization theory for quantum fields. It is shown that the ideas from dimensional and Pauli-Villars regulatizations can be applied to determine uniquely certain finite rank supersingular perturbations. Approach is based on the regularization of homogeneous singular quadratic forms.

Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceBp and the Dirichlet spaceDp. In the case ofBp we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.

On semi B-Fredholm operators

An operator T on a Banach space is called ‘semi B-Fredholm’ if for some n \in {\tf=&quot;times-b&quot;N} the range R(T\;\!^n) of T\;\!^n is closed and the induced operator T_n on R(T\;\!^n) semi-Fredholm. Semi B-Fredholm operators are stable under finite rank perturbation, and subject to the spectral mapping theorem; on Hilbert spaces they decompose as sums of nilpotent and semi-Fredholm operators. In addition some recent generalizations of the punctured neighborhood theorem turn out to be consequences of Grabiner's theory of ‘topological uniform descent’.

On generalized Weyl operators

The generalized Weyl operators between two Hilbert spaces are taken to be those with closed range for which the null space and that of the adjoint are of equal Hilbert space dimension. We show that products of two of these which happen to have closed range, and finite rank perturbation of these, are also generalized Weyl.

On Matrix–Valued Herglotz Functions

We provide a comprehensive analysis of matrix-valued Herglotz functions and illustrate their applications in the spectral theory of self-adjoint Hamiltonian systems including matrix-valued Schrodinger and Dirac-type operators. Special emphasis is devoted to appropriate matrix-valued extensions of the well-known Aronszajn-Donoghue theory concerning support properties of measures in their Nevanlinna-Riesz-Herglotz representation. In particular, we study a class of linear fractional transformations M_A(z) of a given n \times n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuos part of the measure associated to M_A(z) is invariant under these linear fractional transformations. Additional applications discussed in detail include self-adjoint finite-rank perturbations of self-adjoint operators, self-adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.

On Matrix-Valued Herglotz Functions

We provide a comprehensive analysis of matrix–valued Herglotz functions and illustrate their applications in the spectral theory of self–adjoint Hamiltonian systems including matrix–valued Schrodinger and Dirac–type operators. Special emphasis is devoted to appropriate matrix–valued extensions of the well–known Aronszajn–Donoghue theory concerning support properties of measures in their Nevanlinna–Riesz–Herglotz representation. In particular, we study a class of linear fractional transformations MA(z) of a given n × n Herglotz matrix M(z) and prove that the minimal support of the absolutely continuous part of the measure associated to MA(z) is invariant under these linear fractional transformations. Additional applications discussed in detail include self–adjoint finite–rank perturbations of self–adjoint operators, self–adjoint extensions of densely defined symmetric linear operators (especially, Friedrichs and Krein extensions), model operators for these two cases, and associated realization theorems for certain classes of Herglotz matrices.

Finite rank perturbations of contractions

We study finite rank perturbations of contractions of classC .0 with finite defect indices. The completely nonunitary part of such a perturbation is also of classC .0, while the unitary part is singular. When the defect indices of the original contraction are not equal, it can be shown that almost always (with respect to a suitable measure) the perturbation has no unitary part.

Finite rank perturbations and distribution theory

Perturbations A T A_T of a selfadjoint operator A A by symmetric finite rank operators T T from H 2 ( A ) \mathcal {H}_2 (A) to H − 2 ( A ) \mathcal {H}_{-2} (A) are studied. The finite dimensional family of selfadjoint extensions determined by A T A_T is given explicitly.

Subnormal operators of finite type II. Structure theorems

This paper concerns pure subnormal operators with finite rank self-commutator, which we call subnormal operators of finite type. We analyze Xia's theory of these operators [21–23] and give its alternative exposition. Our exposition is based on the explicit use of a certain algebraic curve in \mathbb C^2 , which we call the discriminant curve of a subnormal operator, and the approach of dual analytic similarity models of [26]. We give a complete structure result for subnormal operators of finite type which corrects and strenghtens the formulation that Xia gave in [23]. Xia claimed that each subnormal operator of finite type is unitarily equivalent to the operator of multiplication by z on a weighted vector H^2 -space over a “quadrature Riemann surface” (with a finite rank perturbation of the norm. We explain how this formulation can be corrected and show that, conversely, every “quadrature Riemann surface” gives rise to a family of subnormal operators. We prove that this family is parametrized by the so-called characters. As a departing point of our study, we formulate a kind of scattering scheme for normal operators, which includes Xia's model as a particular case.

Dilation to the unilateral shifts

The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is aC0 contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.

Finite rank singular perturbations and distributions with discontinuous test functions

Point interactions for the $n$-th derivative operator in one dimension are investigated. Every such perturbed operator coincides with a selfadjoint extension of the $n$-th derivative operator restricted to the set of functions vanishing in a neighborhood of the singular point. It is proven that the selfadjoint extensions can be described by the planes in the space of boundary values which are Lagrangian with respect to the symplectic form determined by the adjoint operator. A distribution theory with discontinuous test functions is developed in order to determine the selfadjoint operator corresponding to the formal expression \[ L=\left (i\frac d{dx}\right )^n+\sum ^{n-1}_{l,m=0}c_{lm}\delta ^{(m)}(\cdot ) \delta ^{(l)},\qquad c_{lm}=\overline {c_{ml}},\] representing a finite rank perturbation of the $n$-th derivative operator with the support at the origin.

Automatic continuity of perturbations of casual operators

We obtain automatic continuity results for finite-rank perturbations of causal sequence space operators, and provide examples to illustrate cases where automatic continuity does not hold.

Semi-Fredholm perturbations and commutators

AbstractThe Laffey–West theorem concerning finite rank perturbations of bounded Fredholm operators is extended to closed densely defined operators on Banach Spaces.

Mixed-sensitivity optimization for a class of unstable infinite-dimensional systems

In this paper we solve the H ∞ mixed-sensitivity minimization problem for a class of unstable distributed systems. The solution is based on an extension of the skew Toeplitz methodology. The key mathematical fact used is that the skew Toeplitz operators arising in the unstable case are finite-rank perturbations of the classical skew Toeplitz operators obtained from compressions of rational functions. A system of linear equations (the singular system) is derived for this class of operators, from which one can compute the associated singular values and vectors. An example is included to illustrate the results.

Economical finite rank perturbations of semi-Fredholm operators

Economical finite rank perturbations of semi-Fredholm operators