The Diversity and Beauty of Applied Operator Theory

Frontmatter

Standard versus strict Bounded Real Lemma with infinite-dimensional state space II: The storage function approach

Abstract.

For discrete-time causal linear input/state/output systems, the Bounded Real Lemma explains (under suitable hypotheses) the contractivity of the values of the transfer function over the unit disk for such a system in terms of the existence of a positive-definite solution of a certain Linear Matrix Inequality (the Kalman–Yakubovich–Popov (KYP) inequality). Recent work has extended this result to the setting of infinite-dimensional state space and associated non-rationality of the transfer function, where at least in some cases unbounded solutions of the generalized KYP-inequality are required. This paper is the second installment in a series of papers on the Bounded Real Lemma and the KYP-inequality. We adapt Willems’ storage-function approach to the infinite-dimensional linear setting, and in this way reprove various results presented in the first installment, where they were obtained as applications of infinite-dimensional State-Space-Similarity theorems, rather than via explicit computation of storage functions.

J. A. Ball, G. J. Groenewald, S. ter Horst

Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic

Abstract.

It was shown in a series of recent publications that the eigenvalues of \(n\;\times\;n\) Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of n + 1. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol g(x) = (2 sin(x/2))⁴, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. Here we use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic expansion. This also delivers a counter-example to a conjecture by Ekström, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.

Mauricio Barrera, Albrecht Böttcher, Sergei M. Grudsky, Egor A. Maximenko

Spectral regularity of a C*-algebra generated by two-dimensional singular integral operators

Abstract.

Given a bounded simply connected domain \({U} \subset {\mathbb{C}}\) having a Lyapunov curve as its boundary, let \(\mathcal{L}({L}^{2}(U))\) stand for the \((\mathbb{c}^\ast)\) -algebra of all bounded linear operators acting on the Hilbert space \(\mathcal{L}^{2}(U)\) with Lebesgue area measure. We show that the smallest C^*-subalgebra \(\mathcal{A}\) of \(\mathcal{L}({L}^{2}(U))\) containing the singular integral operator

$$(S_Uf)(z)\;=\;-\frac{1}{\pi}{\int\limits_{U}}\frac{f(w)}{(z-w)^2}dA(w),$$

along with its adjoint

$$(S^*_Uf)(z)\;=\;-\frac{1}{\pi}{\int\limits_{U}}\frac{f(w)}{(z-w)^2}dA(w)$$

all multiplication operators \(aI, a \in\; C(\overline{U})\), and all compact operators on \(\mathcal{L}^{2}(U)\), is spectrally regular. Roughly speaking the latter means the following: if the contour integral of the logarithmic derivative of an analytic \(\mathcal{A}\)-valued function f is vanishing (or is quasi-nilpotent), then f takes invertible values on the inner domain of the contour in question.

Harm Bart, Torsten Ehrhardt, Bernd Silbermann

A spectral shift function for Schröodinger operators with singular interactions

Abstract.

For the pair \(-\Delta,-\Delta-\alpha\delta_\mathcal{C}\) of self-adjoint Schröodinger operators in \(L^2(\mathbb{R}^n)\) a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here δ_C denotes a singular δ-potential which is supported on a smooth compact hypersurface \( { C}\subset(\mathbb{R}^n)\) and δ is a real-valued function on C.

Jussi Behrndt, Fritz Gesztesy, Shu Nakamura

Quantum graph with the Dirac operator and resonance states completeness

Abstract.

Quantum graphs with the Dirac operator at the edges are considered. Resonances (quasi-eigenvalues) and resonance states are found for certain star-like graphs and graphs with loops. Completeness of the resonance states on finite subgraphs is studied. Due to use of a functional model, the problem reduces to factorization of the characteristic matrixfunction. The result is compared with the corresponding completeness theorem for the Schrödinger quantum graph.

Irina V. Blinova, Igor Y. Popov

Robert Sheckley’s Answerer for two orthogonal projections

Abstract.

The meta theorem of this paper is that Halmos’ two projections theorem is something like Robert Sheckley’s Answerer: no question about the W^*- and C^*-algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supersymmetry equality introduced by Avron, Seiler, and Simon. A noteworthy insight of the paper reveals that the supersymmetric approach is nothing but Halmos in different language and hence an equivalent Answerer.

Albrecht Böttcher, Ilya M. Spitkovsky

Toeplitz kernels and model spaces

Abstract.

We review some classical and more recent results concerning kernels of Toeplitz operators and their relations with model spaces, which are themselves Toeplitz kernels of a special kind. We highlight the fundamental role played by the existence of maximal vectors for every nontrivial Toeplitz kernel.

M. Cristina Câamara, Jonathan R. Partington

Frames, operator representations, and open problems

Abstract.

A frame in a Hilbert space H is a countable collection of elements in H that allows each \(f\epsilon\mathcal {H}\) to be expanded as an (infinite) linear combination of the frame elements. Frames generalize the wellknown orthonormal bases, but provide much more exibility and can often be constructed with properties that are not possible for orthonormal bases. We will present the basic facts in frame theory with focus on their operator theoretical characterizations and discuss open problems concerning representations of frames in terms of iterations of a fixed operator. These problems come up in the context of dynamical sampling, a topic that has recently attracted considerably interest within harmonic analysis. The goal of the paper is twofold, namely, that experts in operator theory will explore the potential of frames, and that frame theory will benefit from insight provided by the operator theory community.

Ole Christensen, Marzieh Hasannasab

A survey on solvable sesquilinear forms

Abstract.

The aim of this paper is to present a unified theory of many Kato type representation theorems in terms of solvable forms on a Hilbert space \((H,\langle\cdot,\cdot\rangle)\) In particular, for some sesquilinear forms Ω on a dense domain \(D\subseteq\mathcal {H}\) one looks for a representation \(\Omega(\xi,\eta)= \langle T\xi,\eta\rangle\) \((\xi\epsilon\mathcal{D}\mathcal(T),\eta\epsilon D)\) where T is a densely defined closed operator with domain \(D(\mathcal{T})\subseteq \mathcal{D}\). There are two characteristic aspects of a solvable form on H. One is that the domain of the form can be turned into a reexive Banach space that need not be a Hilbert space. The second one is that representation theorems hold after perturbing the form by a bounded form that is not necessarily a multiple of the inner product of H.

Rosario Corso

An application of limiting interpolation to Fourier series theory

Abstract.

The limiting real interpolation method is applied to describe the behavior of the Fourier coefficients of functions that belong to spaces which are “very close” to L₂. The Fourier coefficients are taken with respect to bounded orthonormal systems.

Leo R. Ya. Doktorski

Isomorphisms of AC(σ) spaces for countable sets

Abstract.

It is known that the classical Banach–Stone theorem does not extend to the class of AC(σ) spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if σ is restricted to the set of compact polygons, then all the corresponding AC(σ) spaces are isomorphic (as algebras). In this paper we examine the case where σ is the spectrum of a compact operator, and show that in this case one can obtain an infinite family of homeomorphic sets for which the corresponding function spaces are not isomorphic.

Ian Doust, Shaymaa Al-shakarchi

Restricted inversion of split-Bezoutians

Abstract.

The main aim of the present paper is to compute inverses of split-Bezoutians considered as linear operators restricted to subspaces of symmetric or skewsymmetric vectors. Such results are important, e.g., for the inversion of nonsingular, centrosymmetric or centroskewsymmetric Toeplitz-plus-Hankel Bezoutians B of order n. To realize this inversion we present algorithms with O(n²) computational complexity, which involves an explicit representation of B^–1 as a sum of a Toeplitz and a Hankel matrix. Based on different ideas such inversion formulas have already been proved in previous papers by the authors. Here we focus on the occurring splitting parts since they are of interest also in a more general context. The main key is the solution of the converse problem: the inversion of Toeplitz-plus-Hankel matrices. An advantage of this approach is that all appearing special cases can be dealt with in the same, relatively straightforward way without any additional assumptions.

Torsten Ehrhardt, Karla Rost

Generalized backward shift operators on the ring ℤ[[x]], Cramer’s rule for infinite linear systems, and p-adic integers

Abstract.

Let A be a generalized backward shift operator on ℤ[[x]] and f(x) be a formal power series with integer coefficients. A criterion for the existence of a solution of the linear equation (Ay)(x) + f(x) = y(x) in ℤ[[x]] is obtained. An explicit formula for its unique solution in ℤ[[x]] is found as well. The main results are based on using the p-adic topology on ℤ and on using a formal version of Cramer’s rule for solving infinite linear systems.

Sergey Gefter, Anna Goncharuk

Feynman path integral regularization using Fourier Integral Operator ζ-functions

Abstract.

We will have a closer look at a regularized path integral definition based on Fourier Integral Operator ς-functions and the generalized Kontsevich-Vishik trace, as well as physical examples. Using Feynman's path integral formulation of quantum mechanics, it is possible to formally write partition functions and expectations of observables in terms of operator traces. More precisely, Let U be the wave propagator (a Fourier Integral Operator of order 0) and Ω an observable (a pseudo-differential operator), then the expectation 〈Ω〉 can formally be expressed as \( \langle{\Omega}\rangle = \frac {{\rm {tr}}({U}\Omega)} {{\rm{tr}} U}\). Unfortunately, the operators U and UΩ are not of trace-class in general. Hence, “regularizing the path integral” can be understood as “defining these traces.” In particular, the traces should extend the classical trace on trace-class operators. We therefore consider the generalized Kontsevich-Vishik trace (i.e., Fourier Integral Operator ς-functions) since its restriction to pseudo- differential operators (obtained through Wick rotations if they are possible) is the unique extension of the classical trace. Applying the construction of the generalized Kontsevich-Vishik trace yields a new definition of the Feynman path integral whose predictions coincide with a number of well-known physical examples.

Tobias Hartung

Improving Monte Carlo integration by symmetrization

Abstract.

The error scaling for Markov chain Monte Carlo (MCMC) techniques with N samples behaves like 1/√N. This scaling makes it often very time intensive to reduce the error of calculated observables, in particular for applications in 4-dimensional lattice quantum chromodynamics as our theory of the interaction between quarks and gluons. Even more, for certain cases, where the infamous sign problem appears, MCMC methods fail to provide results with a reliable error estimate. It is therefore highly desirable to have alternative methods at hand which show an improved error scaling and have the potential to overcome the sign problem. One candidate for such an alternative integration technique we used is based on a new class of polynomially exact integration rules on U(N) and SU(N) which are derived from polynomially exact rules on spheres. We applied these rules successfully to a non-trivial, zero-dimensional model with a sign problem and obtained arbitrary precision results. In this article we test a possible way to apply the integration rules for spheres to the case of a one-dimensional U(1) model, the topological rotor, which already leads to a problem of very high dimensionality.

Tobias Hartung, Karl Jansen, Hernan Leövey, Julia Volmer

More on the density of analytic polynomials in abstract Hardy spaces

Abstract.

Let \(\left\{F_n\right\}\) be the sequence of the Fejér kernels on the unit circle \(\mathbb{T}\).The First author recently proved that if X is a separable Banach function space on \(\mathbb{T}\) such that the Hardy–Littlewood maximal operator M is bounded on its associate space \(X^\prime\), then \(\| f * F_n - f \|_X \to 0\) for every \(f \in X\; \mathrm{as}\; n \to \infty\). This implies that the set of analytic polynomials \(\mathcal{P}_A\) is dense in the abstract Hardy space \(H \left[X \right]\) built upon a separable Banach function space X such that M is bounded on \(X^\prime\). In this note we show that there exists a separable weighted L¹ space X such that the sequence \(f * F_n\) does not always converge to \(f \in X\) in the norm of X. On the other hand, we prove that the set \(\mathcal{P}_A\) is dense in \(H \left[X \right]\) under the assumption that X is merely separable.

Alexei Karlovich, Eugene Shargorodsky

Pseudodifferential operators with compound non-regular symbols

Abstract.

The boundedness and compactness of Fourier pseudodifferential operators with compound symbols in subclasses of \(L^\infty\left(\mathbb{R}^2, L^{1}\left(\mathbb{R}\right)\right)\) is studied on weighted Lebesgue spaces \(L^p\left(\mathbb{R}, w\right)\) with \(p\;\in\;\left(1,\;\infty\right)\) and Muckenhoupt weights \(w\;\in\;A_p\left(\mathbb{R}\right)\) by applying the techniques of oscillatory integrals. The boundedness and compactness conditions are also obtained for Mellin pseudodifferential operators with compound symbols in subclasses of \(L^\infty\left(\mathbb{R}^2_{+}, L^{1}\left(\mathbb{R}\right)\right),\) which act on the spaces \(L^p\left(\mathbb{R}_{+}, d\mu\right),\) where \(d\mu \left(t\right)\;=\;dt/t\; \mathrm{for}\;t\in \mathbb{R}_{+}.\) The latter results allow one to reduce the smoothness of slowly oscillating Carleson curves Γ and slowly oscillating Muckenhoupt weights w in the Fredholm study of singular integral operators with shifts on weighted Lebesgue spaces \(L^p\left(\Gamma, w\right)\).

Yuri I. Karlovich

Asymptotically sharp inequalities for polynomials involving mixed Hermite norms

Abstract.

The paper concerns best constants in Markov-type inequalities between the norm of a higher derivative of a polynomial and the norm of the polynomial itself. The norm of the polynomial and its derivative is taken in L² on the real axis with the weight |t|^2α e ^–t2 and |t|^2β e ^–t2, respectively. We determine the leading term of the asymptotics of the constants as the degree of the polynomial goes to infinity.

Holger Langenau

A two-parameter eigenvalue problem for a class of block-operator matrices

Abstract.

We consider a symmetric block operator spectral problem with two spectral parameters. Under some reasonable restrictions, we state localisation theorems for the pair-eigenvalues and discuss relations to a class of non-self-adjoint spectral problems.

Michael Levitin, Hasen Mekki Öztürk

Finite sections of the Fibonacci Hamiltonian

Abstract.

We study finite but growing principal square submatrices A_n of the one- or two-sided infinite Fibonacci Hamiltonian A. Our results show that such a sequence (A_n), no matter how the points of truncation are chosen, is always stable – implying that A_n is invertible for sufficiently large n and A^–1 _n → A^–1 pointwise.

Marko Lindner, Hagen Söding

Spectral asymptotics for Toeplitz operators and an application to banded matrices

Abstract.

We consider a class of compact Toeplitz operators on the Bergman space on the unit disc. The symbols of the operators in our class are assumed to have a sufficiently regular power-like behaviour near the boundary of the disc. We compute the asymptotics of the singular values of Toeplitz operators in this class. We use this result to obtain the asymptotics of the singular values for a class of compact banded matrices.

Alexander Pushnitski

Beyond fractality: piecewise fractal and quasifractal algebras

Abstract.

Fractality is a property of C*-algebras of approximation sequences with several useful consequences: for example, if (A_n) is a sequence in a fractal algebra, then the pseudospectra of the An converge in the Hausdorff metric. The fractality of a separable algebra of approximation sequences can always be forced by a suitable restriction. This observation leads to the question to describe the possible fractal restrictions of a given algebra. In this connection we define two classes of algebras beyond the class of fractal algebras (piecewise fractal and quasifractal algebras), give examples for algebras with these properties, and present some first results on the structure of quasifractal algebras (being continuous fields over the set of their fractal restrictions).

Steffen Roch

Unbounded operators on Hilbert C*-modules and C*-algebras

Abstract.

Hilbert C*-modules are generalizations of Hilbert spaces equipped with scalar products taking values in C*-algebras. The failure of the projection theorem leads to new difficulties for the operator theory on Hilbert C*-modules compared to the Hilbert space setting. In this paper we discuss two classes of unbounded operators (regular operators, graph regular operators) on Hilbert C*-modules and C*-algebras.

Konrad Schmüdgen

A characterization of positive normal functionals on the full operator algebra

Abstract.

Using the recent theory of Krein-von Neumann extensions for positive functionals we present several simple criteria to decide whether a given positive functional on the full operator algebra B(H) is normal. We also characterize those functionals defined on the left ideal of finite rank operators that have a normal extension.

Zoltán Sebestyén, Zsigmond Tarcsay, Tamás Titkos

The linearised Korteweg–deVries equation on general metric graphs

Abstract.

We consider the linearised Korteweg–deVries equation, sometimes called Airy equation, on general metric graphs with edge lengths bounded away from zero. We show that properties of the induced dynamics can be obtained by studying boundary operators in the corresponding boundary space induced by the vertices of the graph. In particular, we characterise unitary dynamics and contractive dynamics. We demonstrate our results on various special graphs, including those recently treated in the literature.

Christian Seifert

Bounded multiplicative Toeplitz operators on sequence spaces

Abstract.

In this paper, we study the linear mapping which sends the sequence \(x=\left(x_n\right)_{n\in\mathbb{N}}\;\mathrm{to}\;y=\left(y_n\right)_{n\in\mathbb{N}}\;\mathrm{where}\;y_n\;=\;\sum\nolimits^\infty_{k=1}f\left(n/k\right)x_k \;\mathrm{for}\;f:\mathbb{Q}^{+}\to\;\mathbb{C}.\) This operator is the multiplicative analogue of the classical Toeplitz operator, and as such we denote the mapping by\(\mathcal{M}_f\). We show that for \(1\leq p\leq q\leq\infty,\;\mathrm{if}\;f\;\in l^r\left(\mathbb{Q}^{+}\right),\;\mathrm{then}\;\mathcal{M}_f\;:\;l^p\;\to\;l^q\) is bounded where \(\frac{1}{r}\;=\;1-\frac{1}{p}\;+\;\frac{1}{q}.\) Moreover, for the cases when p=1 with any \(q,\;p\;=\;q, \mathrm{and}\;q\;=\;\infty\) with any p, we Find that the operator norm is given by \(\|\mathcal{M}_f\|_{p,q}\;=\;\|f\|_{r,\mathbb{Q}^{+}}\;\mathrm{When}\;f\geq 0.\) Finding a necessary condition and the operator norm for the remaining cases highlights an interesting connection between the operator norm of \(\mathcal{M}_f\) and elements in l^p that have a multiplicative structure, when considering \(f\;:\;\mathbb{N}\;\to\;\mathbb{C}.\) We also provide an argument suggesting that \(f\;\in\;l^{r}\) may not be a necessary condition for boundedness when \(1<p<q<\infty\).

Nicola Thorn

On higher index differential-algebraic equations in infinite dimensions

Abstract.

We consider initial value problems for differential-algebraic equations in a possibly infinite-dimensional Hilbert space. Assuming a growth condition for the associated operator pencil, we prove existence and uniqueness of solutions for arbitrary initial values in a distributional sense. Moreover, we construct a nested sequence of subspaces for initial values in order to obtain classical solutions.

Sascha Trostorff, Marcus Waurick

Characterizations of centrality by local convexity of certain functions on C*-algebras

Abstract.

We provide a quite large function class which is useful to distinguish central and non-central elements of a C*-algebra in the following sense: for each element f of this function class, a self-adjoint element a of a C*-algebra is central if and only if the function f is locally convex at a.

Dáaniel Virosztek

Double-scaling limits of Toeplitz determinants and Fisher–Hartwig singularities

Abstract.

Double-scaling limits of Toeplitz determinants D_n(f_t) generated by a set of functions f_t ∈ L¹ are discussed as both n → ∞ and t → 0 simultaneously, which is currently of great importance in mathematics and in physics. The main focus is on the cases where the number of Fisher–Hartwig singularities changes as t → 0. All the results on double-scaling limits are discussed in the context of applications in random matrix theory and in mathematical physics.

Jani A. Virtanen