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Palle E. T. Jorgensen

Publications and source records attributed to Palle E. T. Jorgensen.

At least 73 records · Page 4Linked to original sources
arXiv2012

Scalar spectral measures associated with an Operator-Fractal

We examine the operator $U_5$ defined on $L^2(μ_{\frac14})$ where $μ_{\frac14}$ is the 1/4 Cantor measure. The operator $U_5$ scales the elements of the canonical exponential spectrum for $L^2(μ_{\frac14})$ by 5 --- that is, $Ue_γ = e_{5γ}$ where $e_γ(t) = e^{2πi γt}$. It is known that $U_5$ has a self-similar structure, which makes its spectrum, which is currently unknown, of particular interest. In order to better understand the spectrum of $U_5$, we demonstrate a decomposition of the projection valued measures and scalar spectral measures associated with $U_5$. We are also able to compute associated Radon-Nikodym derivatives between the scalar measures. Our decomposition utilizes a system of operators which form a representation of the Cuntz algebra $\mathcal{O}_2$.

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arXiv2012

Self-adjoint extensions of network Laplacians and applications to resistance metrics

Let $(G,c)$ be an infinite network, and let $\mathcal{E}$ be the canonical energy form. Let $Δ_2$ be the Laplace operator with dense domain in $\ell^2(G)$ and let $Δ_{\mathcal{E}}$ be the Laplace operator with dense domain in the Hilbert space $\mathcal{H}_\mathcal{E}$ of finite energy functions on $G$. It is known that $Δ_2$ is essentially self-adjoint, but that $Δ_{\mathcal{E}}$ is \emph{not}. In this paper, we characterize the Friedrichs extension of $Δ_{\mathcal{E}}$ in terms of $Δ_2$ and show that the spectral measures of the two operators are mutually absolutely continuous with Radon-Nikodym derivative $λ$ (the spectral parameter), in the complement of $λ=0$. We also give applications to the effective resistance on $(G,c)$. For transient networks, the Dirac measure at $λ= 0$ contributes to the spectral resolution of the Friedrichs extension of $Δ_{\mathcal{E}}$ but not to that of the self-adjoint $\ell^2$ Laplacian.

math.SP↗
arXiv2011

An Operator-Fractal

Certain Bernoulli convolution measures (μ) are known to be spectral. Recently, much work has concentrated on determining conditions under which orthonormal Fourier bases (i.e. spectral bases) exist. For a fixed measure known to be spectral, the orthonormal basis need not be unique; indeed, there are often families of such spectral bases. Let λ= 1/(2n) for a natural number n and consider the Bernoulli measure (μ) with scale factor λ. It is known that L^2(μ) has a Fourier basis. We first show that there are Cuntz operators acting on this Hilbert space which create an orthogonal decomposition, thereby offering powerful algorithms for computations for Fourier expansions. When L^2(μ) has more than one Fourier basis, there are natural unitary operators U, indexed by a subset of odd scaling factors p; each U is defined by mapping one ONB to another. We show that the unitary operator U can also be orthogonally decomposed according to the Cuntz relations. Moreover, this operator-fractal U exhibits its own self-similarity.

math.OA↗
arXiv2011

Decomposition of wavelet representations and Martin boundaries

We study a decomposition problem for a class of unitary representations associated with wavelet analysis, wavelet representations, but our framework is wider and has applications to multi-scale expansions arising in dynamical systems theory for non-invertible endomorphisms. Our main results offer a direct integral decomposition for the general wavelet representation, and we solve a question posed by Judith Packer. This entails a direct integral decomposition of the general wavelet representation. We further give a detailed analysis of the measures contributing to the decomposition into irreducible representations. We prove results for associated Martin boundaries, relevant for the understanding of wavelet filters and induced random-walks, as well as classes of harmonic functions. Our setting entails representations built from certain finite-to-one endomorphisms $r$ in compact metric spaces $X$, and we study their dilations to automorphisms in induced solenoids. Our wavelet representations are covariant systems formed from the dilated automorphisms. They depend on assigned measures $μ$ on $X$. It is known that when the data $(X, r, μ)$ are given the associated wavelet representation is typically reducible. By introducing wavelet filters associated to $(X, r)$ we build random walks in $X$, path-space measures, harmonic functions, and an associated Martin boundary. We construct measures on the solenoid $(X_\infty, r_\infty)$, built from $(X, r)$. We show that $r_\infty$ induces unitary operators $U$ on Hilbert space $\H$ and representations $π$ of the algebra $L^\infty(X)$ such that the pair $(U, r_\infty)$, together with the corresponding representation $π$ forms a cross-product in the sense of $C^*$-algebras. We note that the traditional wavelet representations fall within this wider framework of $(\H, U, π)$ covariant crossed products.

math.FA↗
arXiv2011

Momentum Operators in Two Intervals: Spectra and Phase Transition

We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs. Compared to earlier studies, it yields a more direct link between spectrum and geometry.

math.SP↗
arXiv2011

Fourier duality for fractal measures with affine scales

For a family of fractal measures, we find an explicit Fourier duality. The measures in the pair have compact support in $\br^d$, and they both have the same matrix scaling. But the two use different translation vectors, one by a subset $B$ in $\br^d$, and the other by a related subset $L$. Among other things, we show that there is then a pair of infinite discrete sets $Γ(L)$ and $Γ(B)$ in $\br^d$ such that the $Γ(L)$-Fourier exponentials are orthogonal in $L^2(μ_B)$, and the $Γ(B)$-Fourier exponentials are orthogonal in $L^2(μ_L)$. These sets of orthogonal "frequencies" are typically lacunary, and they will be obtained by scaling in the large. The nature of our duality is explored below both in higher dimensions and for examples on the real line. Our duality pairs do not always yield orthonormal Fourier bases in the respective $L^2(μ)$-Hilbert spaces, but depending on the geometry of certain finite orbits, we show that they do in some cases. We further show that there are new and surprising scaling symmetries of relevance for the ergodic theory of these affine fractal measures.

math.FA↗
arXiv2011

Isospectral measures

In recent papers a number of authors have considered Borel probability measures $μ$ in $\br^d$ such that the Hilbert space $L^2(μ)$ has a Fourier basis (orthogonal) of complex exponentials. If $μ$ satisfies this property, the set of frequencies in this set are called a spectrum for $μ$. Here we fix a spectrum, say $Γ$, and we study the possibilities for measures $μ$ having $Γ$ as spectrum.

math.FA↗
arXiv2011

Spectral reciprocity and matrix representations of unbounded operators

Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical conformal Laplacians and Hamiltonians from statistical mechanics. For an infinite discrete set $X$, we consider operators acting on Hilbert spaces of functions on $X$, and their representations as infinite matrices; the focus is on $\ell^2(X)$, and the energy space $\mathcal{H}_{\mathcal E}$. In particular, we prove that these operators are always essentially self-adjoint on $\ell^2(X)$, but may fail to be essentially self-adjoint on $\mathcal{H}_{\mathcal E}$. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the $\mathcal{H}_{\mathcal E}$ operators with the use of a new approximation scheme.

math.FA↗
arXiv2010

Affine fractals as boundaries and their harmonic analysis

We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we establish boundary representations for these fractals. We prove that as sets these fractals arise as boundaries of functions in closed subspaces of the Hardy space $H^2$. By this we mean that there are lacunary subsets $Γ$ of the non-negative integers, and associated closed $Γ$-subspace in the Hardy space $H^2(\bd)$, $\bd$ denoting the disk, such that for every function $f$ in in $H^2(Γ)$, and for every point $z$ in $\bd$, $f(z)$ admits a boundary integral represented by an associated measure $μ$, with integration over $\suppμ$ placed as a Cantor subset on the circle $\bt := \{bd}(\bd)$. We study families of pairs: measures $μ$ and sets $Γ$ of lacunary form, admitting lacunary Fourier series in $L^2(μ)$; i.e., configurations $Γ$ arranged with a geometric progression of empty spacing, or missing parts, gaps. Given $Γ$, we find corresponding generalized Szeg\" o kernels $G_Γ$, and we compare them to the classical Szeg\" o kernel for $\bd$. Rather than the more traditional approach of starting with $μ$, and then asking for possibilities for sets $Γ$, such that we get Fourier series representations, we turn the problem upside down; now starting instead with a countably infinite discrete subset $Γ$, and, within a new duality framework, we study the possibilities for choices of measures $μ$.

math.FA↗
arXiv2010

Multiplication operators on the energy space

This paper studies the "energy space" $\mathcal{H}_{\mathcal{E}}$ (the Hilbert space of functions of finite energy, aka the Dirichlet-finite functions) on an infinite network (weighted connected graph), from the point of view of the multiplication operators $M_f$ associated to functions $f$ on the network. We show that the multiplication operators $M_f$ are not Hermitian unless $f$ is constant, and compute the adjoint $M_f^\star$ in terms of a reproducing kernel for $\mathcal{H}_{\mathcal{E}}$. A characterization of the bounded multiplication operators is given in terms of positive semidefinite functions, and we give some conditions on $f$ which ensure $M_f$ is bounded. Examples show that it is not sufficient that $f$ be bounded or have finite energy. Conditions for the boundedness of $M_f$ are also expressed in terms of the behavior of the simple random walk on the network. We also consider the bounded elements of $\mathcal{H}_{\mathcal{E}}$ and the (possibly unbounded) multiplication operators corresponding to them. In a previous paper, the authors used functional integration to construct a type of boundary for infinite networks. The boundary is described here in terms of a certain subalgebra of these multiplication operators, and is shown to embed into the Gel'fand space of that subalgebra. In the case when the only harmonic functions of finite energy are constant, we show that the Gel'fand space is the 1-point compactification of the underlying network.

math.OA↗
arXiv2010

Operators Induced by Graphs

In this paper, we consider certain elements in von Neumann algebras generated by graph groupoids. In particular, we are interested in finitely supported elements, called graph operators. We study the characterizations for self-adjointness, the unitary property, hyponormality and normality of graph operators.

math.RT↗
arXiv2010

Functional Analysis

Notes from a course taught by Palle Jorgensen in the fall semester of 2009. The course covered central themes in functional analysis and operator theory, with an emphasis on topics of special relevance to such applications as representation theory, harmonic analysis, mathematical physics, and stochastic integration.

math.FA↗
arXiv2010

A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walks

A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship between the natural Dirichlet form $\mathcal E$ and the discrete Laplace operator $Δ$ on a finite network is given by $\mathcal E(u,v) = \la u, \Lap v\ra_2$, where the latter is the usual $\ell^2$ inner product. We extend this formula to infinite networks, where a new (boundary) term appears. The Laplace operator is typically unbounded in this context; we construct a reproducing kernel for the space of functions of finite energy which allows us to specify a dense domain for $Δ$ and give several criteria for the transience of the random walk on the network. The extended Gauss-Green identity and the reproducing kernel are the foundation for a boundary integral representation for harmonic functions of finite energy, akin to that of Martin boundary theory.

math.FA↗
arXiv2010

Spectral measures and Cuntz algebras

We consider a family of measures $μ$ supported in $\br^d$ and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in $L^2(μ)$ consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset $Γ$ in $\br^d$. Here we offer two computational devices for understanding the interplay between the possibilities for such sets $Γ$ (spectrum) and the measures $μ$ themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz $C^*$-algebras $\mathcal O_N$.

math.FA↗
arXiv2010

Resistance boundaries of infinite networks

A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a Hilbert space structure ${\mathcal H}_{\mathcal E}$ on the space of functions of finite energy. The relationship between the natural Dirichlet form $\mathcal E$ and the discrete Laplace operator $Δ$ on a finite network is given by $\mathcal E(u,v) = \la u, \Lap v\ra_2$, where the latter is the usual $\ell^2$ inner product. We describe a reproducing kernel $\{v_x\}$ for $\mathcal E$ and used it to extends the discrete Gauss-Green identity to infinite networks: \[{\mathcal E}(u,v) = \sum_{G} u Δv + \sum_{\operatorname{bd}G} u \tfrac{\partial}{\partial \mathbf{n}} v,\] where the latter sum is understood in a limiting sense, analogous to a Riemann sum. This formula immediately yields a boundary sum representation for the harmonic functions of finite energy. Techniques from stochastic integration allow one to make the boundary $\operatorname{bd}G$ precise as a measure space, and give a boundary integral representation (in a sense analogous to that of Poisson or Martin boundary theory). This is done in terms of a Gel'fand triple $S \ci {\mathcal H}_{\mathcal E} \ci S'$ and gives a probability measure $\mathbb{P}$ and an isometric embedding of ${\mathcal H}_{\mathcal E}$ into $L^2(S',\mathbb{P})$, and yields a concrete representation of the boundary as a set of linear functionals on $S$.

math.FA↗
arXiv2009

Operator theory of electrical resistance networks

A resistance network is a weighted graph $(G,c)$ with intrinsic (resistance) metric $R$. We embed the resistance network into the Hilbert space ${\mathcal H}_{\mathcal E}$ of functions of finite energy. We use the resistance metric to study ${\mathcal H}_{\mathcal E}$, and vice versa and show that the embedded images of the vertices $\{v_x\}$ form a reproducing kernel for this Hilbert space. We also obtain a discrete version of the Gauss-Green formula for resistance networks and show that resistance networks which support nonconstant harmonic functions of finite energy have a certain type of \emph{boundary}. We obtain an analytic boundary representation for the harmonic functions of finite energy in a sense analogous to the Poisson or Martin boundary representations, but with different hypotheses, and for a different class of functions. In the process, we construct a dense space of "smooth" functions of finite energy and obtain a Gel'fand triple for ${\mathcal H}_{\mathcal E}$. This allows us to represent the resistance network as a system of Gaussian random variables indexed by vertices. We also study the spectral representation for $Δ$ on ${\mathcal H}_{\mathcal E}$ and show how nonzero defect entails a nontrivial boundary. All of the above are are detected by the operator theory of ${\mathcal H}_{\mathcal E}$ but not $\ell^2$. Our results apply to the Heisenberg model for the isotropic ferromagnet, improving earlier results of R. T. Powers on the problem of long-range order (in reference to KMS states on the $C^\ast$-algebra of the model).

math.OA↗
arXiv2009

A Hilbert space approach to effective resistance metric

A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form $\mathcal E$ produces a Hilbert space structure (which we call the energy space ${\mathcal H}_{\mathcal E}$) on the space of functions of finite energy. We use the reproducing kernel $\{v_x\}$ constructed in \cite{DGG} to analyze the effective resistance $R$, which is a natural metric for such a network. It is known that when $(G,c)$ supports nonconstant harmonic functions of finite energy, the effective resistance metric is not unique. The two most natural choices for $R(x,y)$ are the ``free resistance'' $R^F$, and the ``wired resistance'' $R^W$. We define $R^F$ and $R^W$ in terms of the functions $v_x$ (and certain projections of them). This provides a way to express $R^F$ and $R^W$ as norms of certain operators, and explain $R^F \neq R^W$ in terms of Neumann vs. Dirichlet boundary conditions. We show that the metric space $(G,R^F)$ embeds isometrically into ${\mathcal H}_{\mathcal E}$, and the metric space $(G,R^W)$ embeds isometrically into the closure of the space of finitely supported functions; a subspace of ${\mathcal H}_{\mathcal E}$. Typically, $R^F$ and $R^W$ are computed as limits of restrictions to finite subnetworks. A third formulation $R^{tr}$ is given in terms of the trace of the Dirichlet form $\mathcal E$ to finite subnetworks. A probabilistic approach shows that in the limit, $R^{tr}$ coincides with $R^F$. This suggests a comparison between the probabilistic interpretations of $R^F$ vs. $R^W$.

math.FA↗