15,394 research outputs found

### 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

### 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

### 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 $X_1$-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

### 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

### 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

### 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

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