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Sergei Treil

Publications and source records attributed to Sergei Treil.

At least 19 recordsLinked to original sources
arXiv2026

Analysis of the Singular Spectrum for General Perturbations

We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the ``shift'' of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the ``shift'' outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy--Foia\c s theory and the operator ${\bf A}_2$ condition.

math.SP↗
arXiv2026

The Poisson Matrix $\mathbf{A}_2$ characteristic and the 3/2 blow up of the Hilbert transform

Recently the matrix $A_2$ conjecture was disproved. Indeed, the growth of the vector Hilbert transform in the matrix weighted $L^2(W)$ space was shown to be at best a constant multiple of $[W]_{\mathbf{A}_2}^{3/2}$. This bound had previously been established and it was thus proved that it is sharp and the conjectured linear growth cannot be obtained. It is a natural question to see if the $3/2$ power persists if we replace the classical matrix $A_2$ characteristic by the "fattened", larger, so-called matrix Poisson $A_2$ characteristic. We show that the 3/2 power, even in this case, cannot be improved.

math.CA↗
arXiv2025

The spectral map for weighted Cauchy matrices is an involution

Let $N$ be a natural number. We consider weighted Cauchy matrices of the form \[ \mathcal{C}_{a,A}=\left\{\frac{\sqrt{A_j A_k}}{a_k+a_j}\right\}_{j,k=1}^N, \] where $A_1,\dots,A_N$ are positive real numbers and $a_1,\dots,a_N$ are distinct positive real numbers, listed in increasing order. Let $b_1,\dots,b_N$ be the eigenvalues of $\mathcal{C}_{a,A}$, listed in increasing order. Let $B_k$ be positive real numbers such that $\sqrt{B_k}$ is the Euclidean norm of the orthogonal projection of the vector \[ v_A=(\sqrt{A_1},\dots,\sqrt{A_N}) \] onto the $k$'th eigenspace of $\mathcal{C}_{a,A}$. We prove that the spectral map $(a,A)\mapsto (b,B)$ is an involution and discuss simple properties of this map.

math.RA↗
arXiv2025

Inverse spectral problems for positive Hankel operators

A Hankel operator $\Gamma$ in $L^2(\mathbb{R}_+)$ is an integral operator with the integral kernel of the form $h(t+s)$, where $h$ is known as the kernel function. It is known that $\Gamma$ is positive semi-definite if and only if $h$ is the Laplace transform of a positive measure $\mu$ on $\mathbb{R}_+$. Thus, positive semi-definite Hankel operators $\Gamma$ are parameterised by measures $\mu$ on $\mathbb{R}_+$. We consider the class of $\Gamma$ corresponding to \emph{finite} measures $\mu$. In this case it is possible to define the (scalar) spectral measure $\sigma$ of $\Gamma$ in a natural way. The measure $\sigma$ is also finite on $\mathbb{R}_+$. This defines the \emph{spectral map} $\mu\mapsto\sigma$ on finite measures on $\mathbb{R}_+$. We prove that this map is an involution; in particular, it is a bijection. We also consider a dual variant of this problem for measures $\mu$ that are not necessarily finite but have the finite integral \[ \int_0^\infty x^{-2}\mathrm{d}\mu(x); \] we call such measures \emph{co-finite}.

math.SP↗
arXiv2025

Unbounded integral Hankel operators

For a wide class of unbounded integral Hankel operators on the positive half-line, we prove essential self-adjointness on the set of smooth compactly supported functions.

math.SP↗
arXiv2023

An inverse spectral problem for non-compact Hankel operators with simple spectrum

We consider an inverse spectral problem for a class of non-compact Hankel operators $H$ such that the modulus of $H$ (restricted onto the orthogonal complement to its kernel) has simple spectrum. Similarly to the case of compact operators, we prove a uniqueness result, i.e. we prove that a Hankel operator from our class is uniquely determined by the spectral data. In other words, the spectral map, which maps a Hankel operator to the spectral data, is injective. Further, in contrast to the compact case, we prove the failure of surjectivity of the spectral map, i.e. we prove that not all spectral data from a certain natural set correspond to Hankel operators. We make some progress in describing the image of the spectral map. We also give applications to the cubic Szegő equation. In particular, we prove that not all solutions with initial data in BMOA are almost periodic; this is in a sharp contrast to the known result for initial data in VMOA.

math.FA↗
arXiv2023

A Dynamical System Approach to the Inverse Spectral Problem for Hankel Operators: A Model Case

We present an alternative proof of the result by P. Gerard and S. Grellier, stating that given two real sequences $(λ_n)_{n=1}^\infty$, $(μ_n)_{n=1}^\infty$ satisfying the intertwining relations \[ |λ_1| > |μ_1| > |λ_2| > |μ_2| > ...> |λ_n| > |μ_n|>\ldots >0 , \qquad λ_n\to 0, \] there exists a unique compact Hankel operator $Γ$ such that $λ_n$ are the (simple) eigenvalues of $Γ$ and $μ_n$ are the simple eigenvalues of its truncation $Γ_1$ obtained from $Γ$ by removing the first column. We use the dynamical systems approach originated in a paper by A. V. Megretski, V.V. Peller. S. R. Treil in 1995, and the proof is split into three independent parts. The first one, which is a slight modification of a result in that paper, is an abstract operator-theoretic statement reducing the problem to the asymptotic stability of some operators. The second one is the proof of the asymptotic stability, which is usually the hardest part, but in our case of compact operators it is almost trivial. And the third part is an abstract version of the Borg's two spectra theorem, which is essentially a simple exercise in graduate complex analysis.

math.FA↗
arXiv2023

A Dynamical System Approach To The Inverse Spectral Problem For Hankel Operators: The General Case

We study the inverse problem for the Hankel operators in the general case. Following the work of Gérard--Grellier, the spectral data is obtained from the pair of Hankel operators $Γ$ and $ΓS$, where $S$ is the shift operator. The theory of complex symmetric operators provides a convenient language for the description of the spectral data. We introduce the abstract spectral data for the general case, and use the dynamical system approach, to reduce the problem to asymptotic stability of some contraction, constructed from the spectral data. The asymptotic stability is usually the hard part of the problem, but in the investigated earlier by Gérard--Grellier case of compact operators we get it almost for free. For the case of compact operators we get a concrete representation of the abstract spectral data as two intertwining sequences of singular values, and two sequences of finitely supported probability measures. This representation is different from one treated by Gérard--Grellier, and we provide the translation from one language to the other; theory of Clark measures is instrumental there.

math.FA↗
arXiv2021

Preservation of absolutely continuous spectrum for contractive operators

We consider contractive operators $T$ that are trace class perturbations of a unitary operator $U$. We prove that the dimension functions of the absolutely continuous spectrum of $T$, $T^*$ and of $U$ coincide. In particular, if $U$ has a purely singular spectrum then the characteristic function $θ$ of $T$ is a two-sided inner function, i.e. $θ(ξ)$ is unitary a.e. on $\mathbb{T}$. Some corollaries of this result are related to investigations of the asymptotic stability of the operators $T$ and $T^*$ (convergence $T^n\to 0$ and $(T^*)^n\to 0$, respectively, in the strong operator topology). The proof is based on an explicit computation of the characteristic function.

math.FA↗
arXiv2021

Dyadic bi-parameter repeated commutator and dyadic product BMO

Consider a tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter repeated commutator its norm can be estimated from below by Chang-Fefferman $BMO$ norm pertinent to its symbol. See Theorems in Section 8 at the end of this article. But this is done below under an extra assumption on the Haar--Fourier side of the symbol. In Section 7 we carefully analyze what goes wrong in the absence of this extra assumption. At the end of this note we also list a counterexample to the existing proof of characterization of bi-parameter repeated commutator with the Hilbert transforms. This is a counterexample to the proof, and it is not a counterexample to the statement of factorization result in bi-disc, or to Nehari's theorem in bi-disc. To the best of our knowledge Nehari's theorem on bi-disc is still open. Moreover its dyadic bi-parameter version considered in the present paper is also still open for general symbol without any extra restrictions.

math.AP↗
arXiv2020

Dyadic bi-parameter simple commutator and dyadic little BMO

Let $\bfT$ is a certain tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter commutator the following equivalence holds $ \|\bfT b-b \bfT \| \asymp \|b\|_{bmo^d}$. This result is well-known for many types of bi-parameter commutators, see \cite{FS}, \cite{DLWY} and \cite{DPSK} for more details.

math.FA↗
arXiv2020

"Small step" remodeling and counterexamples for weighted estimates with arbitrarily "smooth" weights

For an $A_p$ weight $w$ the norm of the Hilbert Transform in $L^p(w)$, $1<p<\infty$ is estimated by $[w]_{A_p}^{s}$, where $[w]_{A_p}$ is the $A_p$ characteristic of the weight $w$ and $s = \max(1,1/(p-1))$; as simple examples with power weights show, these estimates are sharp. A natural question to ask, is whether it is possible to improve the exponent $s$ in the above estimate if one replaces the $A_p$ characteristic by its "fattened" version, where the averages are replaced by Poisson-like averages. For power weights (for example with $p=2$ and Poisson averages) one can see that there is indeed an improvement in the exponent: but is it true for general weights? In this paper we show that the optimal exponent $s$ remains the same by constructing counterexamples for arbitrarily "smooth" weights (in the sense that the doubling constant is arbitrarily close to $2$), so the "fattened" $A_p$ characteristic is equivalent to the classical one, and such that $\|T\|_{L^p(w)} \sim [w]_{A_p}^{s}$. We use the ideas from the unpublished manuscript by F. Nazarov disproving Sarason's conjecture. We start from simple classical counterexamples for dyadic models, and then by using what we call "small step construction" we transform them into examples with weights that are arbitrarily dyadically smooth. F.~Nazarov had used Bellman function method to prove the existence of such examples, but our construction gives a way to get such examples from the standard dyadic ones. We then use a modification of "remodeling", introduced by J.~Bourgain and developed by F.~Nazarov, to get from examples for dyadic models to examples for the Hilbert transform. As an added bonus, we present a proof that the $L^p$ analog of Sarason's conjecture is false for all $p$, $1<p<\infty$.

math.CA↗
arXiv2020

Dimension of the exceptional set in the Aronszajn-Donoghue theorem for finite rank perturbations

The classical Aronszajn-Donoghue theorem states that for a rank one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbolα} := A +\mathbf{B} \boldsymbolα \mathbf{B}^*$, $\mathbf{B}:\mathbb{C}^d\to \mathbf{H}$, with Ran$\,\mathbf{B}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices $\boldsymbolα$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbolα}$ are mutually singular for all $\boldsymbolα$ except for a small exceptional set $E$. It was shown earlier by the first two authors that $E$ is a subset of measure zero of the space $\mathbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim\mathbf{H}(d)-1 = d^2-1$.

math.FA↗
arXiv2020

Matrix-valued Aleksandrov--Clark measures and Carathéodory angular derivatives

This paper deals with families of matrix-valued Aleksandrov--Clark measures $\{\boldsymbolμ^α\}_{α\in\mathcal{U}(n)}$, corresponding to purely contractive $n\times n$ matrix functions $b$ on the unit disc of the complex plane. We do not make other apriori assumptions on $b$. In particular, $b$ may be non-inner and/or non-extreme. The study of such families is mainly motivated from applications to unitary finite rank perturbation theory. A description of the absolutely continuous parts of $\boldsymbolμ^α$ is a rather straightforward generalization of the well-known results for the scalar case ($n=1$). The results and proofs for the singular parts of matrix-valued $\boldsymbolμ^α$ are more complicated than in the scalar case, and constitute the main focus of this paper. We discuss matrix-valued Aronszajn--Donoghue theory concerning the singular parts of the Clark measures, as well as Carathéodory angular derivatives of matrix-valued functions and their connections with atoms of $\boldsymbolμ^α$. These results are far from being straightforward extensions from the scalar case: new phenomena specific to the matrix-valued case appear here. New ideas, including the notion of directionality, are required in statements and proofs.

math.FA↗
arXiv2020

Commutators in the two scalar and matrix weighted setting

In this paper we approach the two weighted boundedness of commutators via matrix weights. This approach provides both a sufficient and a necessary condition for the two weighted boundedness of commutators with an arbitrary linear operator in terms of one matrix weighted norm inequalities for this operator. Furthermore, using this approach, we surprisingly provide conditions that almost characterize the two matrix weighted boundedness of commutators with CZOs and completely arbitrary matrix weights, which is even new in the fully scalar one weighted setting. Finally, our method allows us to extend the two weighted Holmes/Lacey/Wick results to the fully matrix setting (two matrix weights and a matrix symbol), completing a line of research initiated by the first two authors.

math.CA↗
arXiv2019

Matrix Measures and Finite Rank Perturbations of Self-adjoint Operators

Matrix-valued measures provide a natural language for the theory of finite rank perturbations. In this paper we use this language to prove some new perturbation theoretic results. Our main result is a generalization of the Aronszajn--Donoghue theorem about the mutual singularity of the singular parts of the spectrum for rank one perturbations to the case of finite rank perturbations. Simple direct sum type examples indicate that an exact generalization is not possible. However, in this paper we introduce the notion of \emph{vector mutual singularity} for the matrix-valued measures and show that if we use this notion, the mutual singularity still holds for the finite rank perturbations. As for the scalar spectral measures and the classical mutual singularity, we show that the singular parts are mutually singular for almost all perturbations. One of the ways to prove that is to use a generalization of the Aleksandrov's spectral averaging to the matrix-valued measures, which is also one of the main results of this paper. Finally, the spectral representation of the perturbed operator is obtained. The matrix Muckenhoupt $A_2$ condition appears naturally there, and it plays an important role in establishing the vector mutual singularity of the spectral measures.

math.SP↗
arXiv2017

Superexponential estimates and weighted lower bounds for the square function

We prove the following superexponential distribution inequality: for any integrable $g$ on $[0,1)^{d}$ with zero average, and any $λ>0$ \[ |\{ x \in [0,1)^{d} \; :\; g \geqλ\}| \leq e^{- λ^{2}/(2^{d}\|S(g)\|_{\infty}^{2})}, \] where $S(g)$ denotes the classical dyadic square function in $[0,1)^{d}$. The estimate is sharp when dimension $d$ tends to infinity in the sense that the constant $2^{d}$ in the denominator cannot be replaced by $C2^{d}$ with $0 1$ they work with a special square function $S_\infty$, and their result does not imply the estimates for the classical square function. Using good $λ$ inequalities technique we then obtain unweighted and weighted $L^p$ lower bounds for $S$; to get the corresponding good $λ$ inequalities we need to modify the classical construction. We also show how to obtain our superexponential distribution inequality (although with worse constants) from the weighted $L^2$ lower bounds for $S$, obtained in [5].

math.AP↗