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 singular perturbations and distributions with discontinuous test functions

Point interactions for the -th derivative operator in one dimension are investigated. Every such perturbed operator coincides with a selfadjoint extension of the -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

representing a finite rank perturbation of the -th derivative operator with the support at the origin.

     

Finite rank,relatively bounded perturbations of semigroups generators

Summary This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator A_F=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b H. While Part I studied the question of generation of a s.c. semigroup on H by A_F and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of A_F, given A and a H, via a suitable vector b H; alternatively, given A, via a suitable pair of vectors a, b H; (ii) spectrality of A_F—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {_n} of A, the given vector a H and a given suitable sequence {_n} of nonzero complex numbers, which guarantee the existence of a suitable vector b H such that A_F possesses the following two desirable properties: (i) the eigenvalues of A_F are precisely equal to _n+_n; (ii) the corresponding eigenvectors of A_F form a Riesz basis (a fortiori, A_F is spectral). While finitely many _ns can be preassigned arbitrarily, it must be however that _n 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A₀ acting on a Hilbert space H singular perturbations of the form A₀+V, are studied under some additional requirements of symmetry imposed on the initial operator A₀ and the singular elements ψj. A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A₀+V. The results are applied for the investigation of singular perturbations of the Schrödinger operator in L₂(R³) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.     

Perturbation theory of selfadjoint matrices and sign characteristics under generic structured rank one perturbations

For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, the largest Jordan block of the eigenvalue is destroyed and (in case the eigenvalue is real) all other Jordan blocks keep their sign characteristic. The new eigenvalues, i.e. those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix, are typically simple, and in some cases information is provided about their sign characteristic (if the new eigenvalue is real). The main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky canonical form and on general results concerning rank one perturbations obtained.     

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ_∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ_∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ_∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied.     

Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations

udy the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in [18, 28, 30, 31]. Thus a generic structured perturbation would not be generic if considered as an unstructured perturbation. In other settings of structured matrices, the generic perturbation behavior of the Jordan structures, within the confines imposed by the structure, follows the pattern of that of unstructured perturbations.     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.

     

On finite rank perturbations of definitizable operators

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.     

Eigenvalue perturbation theory of symplectic,orthogonal, and unitary matrices under generic structured rank one perturbations

We study the perturbation theory of structured matrices under structured rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic with respect to an indefinite inner product. The rank one perturbations are not necessarily of arbitrary small size (in the sense of norm). In the case of sesquilinear forms, results on selfadjoint matrices can be applied to unitary matrices by using the Cayley transformation, but in the case of real or complex symmetric or skew-symmetric bilinear forms additional considerations are necessary. For complex symplectic matrices, it turns out that generically (with respect to the perturbations) the behavior of the Jordan form of the perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan form of complex symplectic matrices are accounted for. For instance, the number of Jordan blocks of fixed odd size corresponding to the eigenvalue 1 or ?1 have to be even. For complex orthogonal matrices, it is shown that the behavior of the Jordan structures corresponding to the original eigenvalues that are not moved by perturbations follows again the pattern established earlier for unstructured matrices, taking into account the specifics of Jordan forms of complex orthogonal matrices. The proofs are based on general results developed in the paper concerning Jordan forms of structured matrices (which include in particular the classes of orthogonal and symplectic matrices) under structured rank one perturbations. These results are presented and proved in the framework of real as well as of complex matrices.     

Singular perturbations of finite rank. Point spectrum

We establish necessary and sufficient conditions for the appearance of an additional point spectrum under singular perturbations of finite rank. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1186–1194, September, 1997.     

Eigenvalues of rank one perturbations of unstructured matrices

Let A be a fixed complex matrix and let $u\text{,}v$ be two vectors. The eigenvalues of matrices $A+\tau {\mathit{uv}}^{?}$ $(\tau \in \mathbb{R})$ form a system of intersecting curves. The dependence of the intersections on the vectors $u\text{,}v$ is studied.     

On hypercyclic rank one perturbations of unitary operators

Recently, S. Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. Another construction was suggested later by the first and the third authors. Here, using a functional model for rank one perturbations of singular unitary operators, we give yet another construction of hypercyclic rank one perturbation of a unitary operator. In particular, we show that any countable union of perfect Carleson sets on the circle can be the spectrum of a perturbed (hypercyclic) operator.     

Finite rank multivariable Hankel forms

We obtain a several variables generalization of Kronecker's well-known result on finite rank Hankel matrices.     

Point spectrum and mixed spectral types for rank one perturbations

We consider examples of rank one perturbations with a cyclic vector for . We prove that for any bounded measurable set , an interval, there exist so that
eigenvalue agrees with up to sets of Lebesgue measure zero. We also show that there exist examples where has a.c. spectrum for all , and for sets of 's of positive Lebesgue measure, also has point spectrum in , and for a set of 's of positive Lebesgue measure, also has singular continuous spectrum in .

     

Finite rank operators with large trace

We give some estimates for how well the normalized trace functional exposes the identity map on a finite-dimensional space. Research supported by NSF Grant DMS-8320632. Research supported by NSF Grant DMS-8400960.