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Constanze Liaw

Publications and source records attributed to Constanze Liaw.

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↗
arXiv2025

Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra

We develop a Clark theory for commuting compressed shift operators on model spaces $K_{\phi}$ associated with inner functions $\phi$ on the bidisk, which exhibits both similarities and marked differences compared to the classical one-variable version. We first identify the adjoint of the embedding operator $J_{\alpha} \colon K_{\phi}\to L^2(\sigma_{\alpha})$ as a weighted Cauchy transform of the Clark measure $\sigma_{\alpha}$. Under natural assumptions, which generically include the case when $\phi$ is rational inner, we obtain commuting unitaries on $K_{\phi}$ that are (often infinite-dimensional) perturbations of the compressed shift operators $K_{\phi}$. We prove that these unitaries are unitarily equivalent to multiplication by the coordinate functions on $L^2(\sigma_\alpha)$ and then establish a number of related properties and simplified results in special cases. Finally, we show that the Taylor joint spectrum of these Clark unitaries coincides with level sets of $\phi$ when $\phi$ is a rational inner function.

math.CV↗
arXiv2022

Spectral Properties of Singular Sturm-Liouville Operators via Boundary Triples and Perturbation Theory

We apply both the theory of boundary triples and perturbation theory to the setting of semi-bounded Sturm-Liouville operators with two limit-circle endpoints. For general boundary conditions we obtain refined and new results about their eigenvalues and eigenfunctions. In the boundary triple setup, we obtain simple criteria for identifying which self-adjoint extensions possess double eigenvalues when the parameter is a matrix. We also identify further spectral properties of the Friedrichs extension and (when the operator is positive) the von Neumann-Krein extension. Motivated by some recent scalar Aronszajn-Donoghue type results, we find that real numbers can only be eigenvalues for two extensions of Sturm-Liouville operator when the boundary conditions are restricted to corresponding to affine lines in the space from which the perturbation parameter is taken. Furthermore, we determine much of the spectral representation of those Sturm-Liouville operators that can be reached by perturbation theory.

math.SP↗
arXiv2022

Singular Boundary Conditions for Sturm--Liouville Operators via Perturbation Theory

We show that all self-adjoint extensions of semi-bounded Sturm--Liouville operators with general limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in\{1,2\}$. This characterization generalizes the well-known analog for semi-bounded Sturm--Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as \begin{align*} \boldsymbol{A}_Θ=\boldsymbol{A}_0+{\bf B}Θ{\bf B}^*, \end{align*} where $\boldsymbol{A}_0$ is a distinguished self-adjoint extension and $Θ$ is a self-adjoint linear relation in $\mathbb{C}^d$. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to $\boldsymbol{A}_0$, i.e. it belongs to $\mathcal{H}_{-1}(\boldsymbol{A}_0)$. The construction of a boundary triple and compatible boundary pair for the symmetric operator ensure that the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations $Θ$. As an example, self-adjoint extensions of the classical symmetric Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

math.SP↗
arXiv2022

Spectral Measures for Derivative Powers via Matrix-Valued Clark Theory

The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the derivative operator, providing a full account from self-adjoint boundary conditions to computing aspects of the operators' matrix-valued spectral measures. In particular, the support and weights of the Clark (spectral) measures are computed via the connection between matrix-valued contractive analytic functions and matrix-valued nonnegative measures through the Herglotz Representation Theorem. For operators associated with several powers of the derivative, explicit formulae for these measures are included. While eigenfunctions and eigenvalues for these operators with fixed boundary conditions can often be computed using direct methods from ordinary differential equations, this approach provides a more complete picture of the spectral information.

math.SP↗
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↗
arXiv2020

Perspectives on General Left-Definite Theory

In 2002, Littlejohn and Wellman developed a celebrated general left-definite theory for semi-bounded self-adjoint operators with many applications to differential operators. The theory starts with a semi-bounded self-adjoint operator and constructs a continuum of related Hilbert spaces and self-adjoint operators that are intimately related with powers of the initial operator. The development spurred a flurry of activity in the field that is still ongoing today. The main goal of this expository (with the exception of Proposition 1) manuscript is to compare and contrast the complementary theories of general left-definite theory, the Birman--Krein--Vishik (BKV) theory of self-adjoint extensions and singular perturbation theory. In this way, we hope to encourage interest in left-definite theory as well as point out directions of potential growth where the fields are interconnected. We include several related open questions to further these goals.

math.FA↗
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↗
arXiv2019

Properties and Decompositions of Domains for Powers of the Jacobi Differential Operator

We set out to build a framework for self-adjoint extension theory for powers of the Jacobi differential operator that does not make use of classical deficiency elements. Instead, we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics. This new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains. The maximal domain for $n$-th composition of the Jacobi operator is characterized in terms of a smoothness condition for each derivative, and the endpoint behavior of functions in the underlying Hilbert space can then be classified, for $j\in\mathbb{N}_0$, by $(1-x)^j$, $(1+x)^j$, $(1-x)^{-α+j}$ and $(1+x)^{β+j}$. Most of these behaviors can only occur when functions are in the associated minimal domain, and this leads to a formulation of the defect spaces with a convenient basis. Self-adjoint extensions, including the important left-definite domain, are then given in terms of the new basis functions for the defect spaces using GKN theory. Comments are made for the Laguerre operator as well.

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↗
arXiv2019

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.

math.SP↗
arXiv2019

Rank one perturbations and Anderson-type Hamiltonians

Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts, began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators $$H_ω= H + V_ω$$ on a separable Hilbert space $\mathcal{H}$, where the perturbation is given by $$V_ω= \sum_n ω_n (\cdot, φ_n)φ_n$$ with a sequence $\{φ_n\}\subset\mathcal{H}$ and independent identically distributed random variables $ω_n$. We show that the the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank one perturbation. This result connects one of the least trackable perturbation problem (with almost surely non-compact perturbations) with one where the perturbation is `only' of rank one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has non-zero Lebesgue measure.

math.FA↗
arXiv2017

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 $Γ$; 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 $Γ$ the resulting perturbed operator $T_Γ$ 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 $T_Γ$ (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.

math.FA↗
arXiv2017

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 $φ$ is given by $T_φf=P(φf)$, 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 $φ$ 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.

math.CV↗
arXiv2017

Boundary Conditions associated with the General Left-Definite Theory for Differential Operators

In the early 2000's, Littlejohn and Wellman developed a general left-definite theory for certain self-adjoint operators by fully determining their domains and spectral properties. The description of these domains do not feature explicit boundary conditions. We present characterizations of these domains given by the left-definite theory for all operators which possess a complete system of orthogonal eigenfunctions, in terms of classical boundary conditions.

math.CA↗
arXiv2017

Moment Representations of Type I X2 Exceptional Laguerre Polynomials

The $X_m$ exceptional orthogonal polynomials (XOP) form a complete set of eigenpolynomials to a differential equation. Despite being complete, the XOP set does not contain polynomials of every degree. Thereby, the XOP escape the Bochner classification theorem. In literature two ways to obtain XOP have been presented. When m=1, Gram-Schmidt orthogonalization of a so-called "flag" was used. For general m, the Darboux transform was applied. Here, we present a possible flag for the X_m exceptional Laguerre polynomials. We can write more about this. We only want to make specific picks when we also derive determinantal representations. There is a large degree of freedom in doing so. Further, we derive determinantal representations of the X_2 exceptional Laguerre polynomials involving certain adjusted moments of the exceptional weights. We find a recursion formula for these adjusted moments. The particular canonical flag we pick keeps both the determinantal representation and the moment recursion manageable.

math.CA↗