Rank one and finite rank perturbations - survey and open problems

We survey the relationships of rank one self-adjoint and unitary perturbations as well as finite rank unitary perturbations with various branches of analysis and mathematical physics. We include the case of non-inner characteristic operator functions. For rank one perturbations and non-inner characteristic functions, we prove a representation formula for the adjoint of the Clark operator. Throughout we mention many open problems at varying levels of difficulty.Comment: 17 page

Regularizations of general singular integral operators

In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal, so the integral formally defining the operator or its bilinear form is not well defined (the integrand is not in L^1) even for nice functions. However, since the kernel only has singularities on the diagonal, the bilinear form is well defined say for bounded compactly supported functions with separated supports. One of the standard ways to interpret the boundedness of a singular integral operators is to consider regularized kernels, where the cut-off function is zero in a neighborhood of the origin, so the corresponding regularized operators with kernel are well defined (at least on a dense set). Then one can ask about uniform boundedness of the regularized operators. For the standard regularizations one usually considers truncated operators. The main result of the paper is that for a wide class of singular integral operators (including the classical Calderon-Zygmund operators in non-homogeneous two weight settings), the L^p boundedness of the bilinear form on the compactly supported functions with separated supports (the so-called restricted L^p boundedness) implies the uniform L^p-boundedness of regularized operators for any reasonable choice of a smooth cut-off of the kernel. If the kernel satisfies some additional assumptions (which are satisfied for classical singular integral operators like Hilbert Transform, Cauchy Transform, Ahlfors--Beurling Transform, Generalized Riesz Transforms), then the restricted L^p boundedness also implies the uniform L^p boundedness of the classical truncated operators.Comment: Introduced factor 1/2 in argument section 3.1, results unchange

Hyponormal Toeplitz operators with non-harmonic symbol acting on the Bergman space

The Toeplitz operator acting on the Bergman space $A^{2}(\mathbb{D})$, with symbol $\varphi$ is given by $T_{\varphi}f=P(\varphi f)$, where $P$ is the projection from $L^{2}(\mathbb{D})$ onto the Bergman space. We present some history on the study of hyponormal Toeplitz operators acting on $A^{2}(\mathbb{D})$, as well as give results for when $\varphi$ is a non-harmonic polynomial. We include a first investigation of Putnam's inequality for hyponormal operators with non-analytic symbols. Particular attention is given to unusual hyponormality behavior that arises due to the extension of the class of allowed symbols.Comment: 22 pages, 2 figure

Moment Representations of the Exceptional X1X_1-Laguerre Orthogonal Polynomials

Exceptional orthogonal Laguerre polynomials can be viewed as an extension of the classical Laguerre polynomials per excluding polynomials of certain order(s) from being eigenfunctions for the corresponding exceptional differential operator. We are interested in the (so-called) Type I $X_1$-Laguerre polynomial sequence $\{L_n^\alpha\}_{n=1}^\infty$, $\text{deg} \,p_n = n$ and $\alpha>0$, where the constant polynomial is omitted. We derive two representations for the polynomials in terms of moments by using determinants. The first representation in terms of the canonical moments is rather cumbersome. We introduce adjusted moments and find a second, more elegant formula. We deduce a recursion formula for the moments and the adjusted ones. The adjusted moments are also expressed via a generating function. We observe a certain detachedness of the first two moments from the others.Comment: 19 page

General Clark model for finite rank perturbations

All unitary perturbations of a given unitary operator $U$ by finite rank $d$ operators with fixed range can be parametrized by $(d\times d)$ unitary matrices $\Gamma$; this generalizes unitary rank one ($d=1$) perturbations, where the Aleksandrov--Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\subset\mathbb{C}$. For a purely contractive $\Gamma$ the resulting perturbed operator $T_\Gamma$ is a contraction (a completely non-unitary contraction under the natural assumption about cyclicity of the range), so they admit the functional model. In this paper we investigate the Clark operator, i.e. a unitary operator that intertwines $T_\Gamma$ (presented in the spectral representation of the non-perturbed operator $U$) and its model. We make no assumptions on the spectral type of the unitary operator $U$; absolutely continuous spectrum may be present. We find a representation of the adjoint Clark operator in the coordinate free Nikolski--Vasyunin functional model. This representation features a special version of the vector-valued Cauchy integral operator. Regularization of this singular integral operator yield representations of the adjoint Clark operator in the Sz.-Nagy--Foias transcription. In the special case of inner characteristic functions (purely singular spectral measure of $U$) this representation gives what can be considered as a natural generalization of the normalized Cauchy transform (which is a prominent object in the Clark theory for rank one case) to the vector-valued settings.Comment: 46 pages. Added Section 9 on the Clark operator, re-worded abstract and introduction, included heuristic explanation in Section 6, fixed a few minor error

Spectral Analysis, Model Theory and Applications of Finite-Rank Perturbations

This survey focuses on two main types of finite-rank perturbations: self-adjoint and unitary. We describe both classical and more recent spectral results. We pay special attention to singular self-adjoint perturbations and model representations of unitary perturbations.Comment: 30 page

Spectral Analysis of Iterated Rank-One Perturbations

The authors study the spectral theory of self-adjoint operators that are subject to certain types of perturbations. An iterative introduction of infinitely many randomly coupled rank-one perturbations is one of our settings. Spectral theoretic tools are developed to estimate the remaining absolutely continuous spectrum of the resulting random operators. Curious choices of the perturbation directions that depend on the previous realizations of the coupling parameters are assumed, and unitary intertwining operators are used. An application of our analysis shows localization of the random operator associated to the Rademacher potential. Obtaining fundamental bounds on the types of spectrum under rank-one perturbation, without restriction on its direction, is another main objective. This is accomplished by analyzing Borel/Cauchy transforms centrally associated with rank-one perturbation problems.Comment: 22 page

Singular integrals, rank one perturbations and Clark model in general situation

We start with considering rank one self-adjoint perturbations $A_\alpha = A+\alpha(\,\cdot\,,\varphi)\varphi$ with cyclic vector $\varphi\in \mathcal{H}$ on a separable Hilbert space $\mathcal H$. The spectral representation of the perturbed operator $A_\alpha$ is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators $A$ and $A_\alpha$. Similar results will be presented for unitary rank one perturbations of unitary operators, leading to singular integral operators on the circle. This motivates the study of abstract singular integral operators, in particular the regularization of such operator in very general settings. Further, starting with contractive rank one perturbations we present the Clark theory for arbitrary spectral measures (i.e. for arbitrary, possibly not inner characteristic functions). We present a description of the Clark operator and its adjoint in the general settings. Singular integral operators, in particular the so-called normalized Cauchy transform again plays a prominent role. Finally, we present a possible way to construct the Clark theory for dissipative rank one perturbations of self-adjoint operators. These lecture notes give an account of the mini-course delivered by the authors at the Thirteenth New Mexico Analysis Seminar and Afternoon in Honor of Cora Sadosky. Unpublished results are restricted to the last part of this manuscript.Comment: 36 pages. Lecture note

Properties of vector-valued submodules on the bidisk

In previous work, the authors studied the compressed shift operators $S_{z_1}$ and $S_{z_2}$ on two-variable model spaces $H^2(\mathbb{D}^2)\ominus \theta H^2(\mathbb{D}^2)$, where $\theta$ is a two-variable scalar inner function. Among other results, the authors used Agler decompositions to characterize the ranks of the operators $[S_{z_j}, S^*_{z_j}]$ in terms of the degree of rational $\theta.$ In this paper, we examine similar questions for $H^2(\mathbb{D}^2)\ominus \Theta H^2(\mathbb{D}^2)$ when $\Theta$ is a matrix-valued inner function. We extend several results our previous work connecting $\text{Rank} [S_{z_j}, S^*_{z_j}]$ and the degree of $\Theta$ to the matrix setting. When results do not clearly generalize, we conjecture what is true and provide supporting examples.Comment: 10 page

Clark model in general situation

For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter $\gamma, |\gamma|=1$. Namely all such unitary perturbations are $U_\gamma:=U+(\gamma-1) (., b_1)_{\mathcal H} b$, where $b\in\mathcal H, \|b\|=1, b_1=U^{-1} b, |\gamma|=1$. For $|\gamma|<1$ operators $U_\gamma$ are contractions with one-dimensional defects. Restricting our attention on the non-trivial part of perturbation we assume that $b$ is cyclic for $U$. Then the operator $U_\gamma$, $|\gamma|<1$ is a completely non-unitary contraction, and thus unitarily equivalent to its functional model $\mathcal M_\gamma$, which is the compression of the multiplication by the independent variable $z$ onto the model space $\mathcal K_{\theta_\gamma}$, where $\theta_\gamma$ is the characteristic function of the contraction $U_\gamma$. The Clark operator $\Phi_\gamma$ is a unitary operator intertwining $U_\gamma, |\gamma|<1$ and its model $\mathcal M_\gamma$, $\mathcal M_\gamma \Phi_\gamma = \Phi_\gamma U_\gamma$. If spectral measure of $U$ is purely singular (equivalently, $\theta_\gamma$ is inner), operator $\Phi_\gamma$ was described from a slightly different point of view by D. Clark. When $\theta_\gamma$ is extreme point of the unit ball in $H^\infty$ was treated by D. Sarason using the sub-Hardy spaces introduced by L. de Branges. We treat the general case and give a systematic presentation of the subject. We find a formula for the adjoint operator $\Phi^*_\gamma$ which is represented by a singular integral operator, generalizing the normalized Cauchy transform studied by A. Poltoratskii. We present a "universal" representation that works for any transcription of the functional model. We then give the formulas adapted for the Sz.-Nagy--Foias and de Branges--Rovnyak transcriptions, and finally obtain the representation of $\Phi_\gamma$.Comment: 34 pages. 8/17/2013: changed the arXiv abstract, so the symbols display correctly; no changes in the tex