Riesz Projection Research Articles

The invariant subspace problem for bi-expansive operators in Krein spaces

The invariant subspace problem is one of the most classical questions in operator theory and has attracted research studies for several years. The solutions for this question exist for compact operators. Recently, this question has been solved for bi-ontractive operators defined on a Krein space. The authors established that every bi-contractive operators defined on a Krein space, has maximal semi-definite invariant sub-spaces. In this paper, we address this question for bi-expansive operators defined on a Krein space using Riesz projection and fixed fundamental decomposition. We established that every bi-expansive operator defined on a Krein space has maximal invariant sub-spaces.

On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces

Let X be a Banach function space over the unit circle such that the Riesz projection P is bounded on X and let H[X] be the abstract Hardy space built upon X. We show that the essential norm of the Toeplitz operator T(a):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(a):H[X]\\rightarrow H[X]$$\\end{document} coincides with ‖a‖L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert a\\Vert _{L^\\infty }$$\\end{document} for every a∈C+H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in C+H^\\infty $$\\end{document} if and only if the essential norm of the backward shift operator T(e-1):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(\ extbf{e}_{-1}):H[X]\\rightarrow H[X]$$\\end{document} is equal to one, where e-1(z)=z-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{e}_{-1}(z)=z^{-1}$$\\end{document}. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.

Around Discontinuity for the Riesz Projection in the Uniform Metric

Around Discontinuity for the Riesz Projection in the Uniform Metric

When are the norms of the Riesz projection and the backward shift operator equal to one?

The lower estimate by Gohberg and Krupnik (1968) and the upper estimate by Hollenbeck and Verbitsky (2000) for the norm of the Riesz projection P on the Lebesgue space Lp lead to ‖P‖Lp→Lp=1/sin⁡(π/p) for every p∈(1,∞). Hence L2 is the only space among all Lebesgue spaces Lp for which the norm of the Riesz projection P is equal to one. Banach function spaces X are far-reaching generalisations of Lebesgue spaces Lp. We prove that the norm of P is equal to one on the space X if and only if X coincides with L2 and there exists a constant C∈(0,∞) such that ‖f‖X=C‖f‖L2 for all functions f∈X. Independently from this, we also show that the norm of P on X is equal to one if and only if the norm of the backward shift operator S on the abstract Hardy space H[X] built upon X is equal to one.

Resonance Expansion of Quadratic Quantities with Regularized Quasinormal Modes

Resonance expansions are an intuitive approach to capture the interaction of an optical resonator with light. Herein, a quasinormal mode expansion approach for quadratic observables is presented. The approach exploits the rigorous Riesz projection method. A numerical implementation of a state‐of‐the‐art quantum light source demonstrates the applicability with emphasis on modal expansions outside the underlying nanophotonic resonator.

Contractive inequalities between Dirichlet and Hardy spaces

We prove a conjecture of Brevig, Ortega-Cerdà, Seip and Zhao about contractive inequalities between Dirichlet and Hardy spaces and discuss its consequent connection with the Riesz projection.

The Coburn Lemma and the Hartman–Wintner–Simonenko Theorem for Toeplitz Operators on Abstract Hardy Spaces

Let X be a Banach function space on the unit circle {mathbb {T}}, let X' be its associate space, and let H[X] and H[X'] be the abstract Hardy spaces built upon X and X', respectively. Suppose that the Riesz projection P is bounded on X and ain L^infty {setminus }{0}. We show that P is bounded on X'. So, we can consider the Toeplitz operators T(a)f=P(af) and T({overline{a}})g=P({overline{a}}g) on H[X] and H[X'], respectively. In our previous paper, we have shown that if X is not separable, then one cannot rephrase Coburn’s lemma as in the case of classical Hardy spaces H^p, 1<p<infty , and guarantee that T(a) has a trivial kernel or a dense range on H[X]. The first main result of the present paper is the following extension of Coburn’s lemma: the kernel of T(a) or the kernel of T({overline{a}}) is trivial. The second main result is a generalisation of the Hartman–Wintner–Simonenko theorem saying that if T(a) is normally solvable on the space H[X], then 1/ain L^infty .

Riesz Projection and Essential S-spectrum in Quaternionic Setting

This paper is devoted to the investigation of the Weyl and the essential S-spectra of a bounded right quaternionic linear operator in a right quaternionic Hilbert space. Using the quaternionic Riesz projection, the S-eigenvalue of finite type is both introduced and studied. In particular, we have shown that the Weyl and the essential S-spectra do not contain eigenvalues of finite type. We have also described the boundary of the Weyl S-spectrum and the particular case of the spectral theorem of the essential S-spectrum.

Computation of eigenfrequency sensitivities using Riesz projections for efficient optimization of nanophotonic resonators

Resonances are omnipresent in physics and essential for the description of wave phenomena. We present an approach for computing eigenfrequency sensitivities of resonances. The theory is based on Riesz projections and the approach can be applied to compute partial derivatives of the complex eigenfrequencies of any resonance problem. Here, the method is derived for Maxwell’s equations. Its numerical realization essentially relies on direct differentiation of scattering problems. We use a numerical implementation to demonstrate the performance of the approach compared to differentiation using finite differences. The method is applied for the efficient optimization of the quality factor of a nanophotonic resonator.

Riesz projection and bounded mean oscillation for Dirichlet series

We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.

Lumped Mass Finite Element Method of the Nonlinear BBM Equation

In this paper, the full-discrete approximation scheme of the lumped mass nonconforming finite element method for BBM equation is discussed. Without the Riesz projection used in the traditional finite element analysis, the optimal error estimations are derived based on interpolation technique and special properties of element.

Computing Edge States without Hard Truncation

We present a numerical method which accurately computes the discrete spectrum and associated bound states of Hamiltonians which model electronic "edge" states localized at boundaries of one and two-dimensional crystalline materials. The problem is non-trivial since arbitrarily large finite "hard" truncations of the Hamiltonian in the infinite bulk direction tend to produce spurious bound states partially supported at the truncation. Our method, which overcomes this difficulty, is to compute the Green's function of the Hamiltonian by imposing an appropriate boundary condition in the bulk direction; then, the spectral data is recovered via Riesz projection. We demonstrate our method's effectiveness by studies of edge states at a graphene zig-zag edge in the presence of defects modeled both by a discrete tight-binding model and a continuum PDE model under finite difference discretization. Our method may also be used to study states localized at domain wall-type edges in one and two-dimensional materials where the edge Hamiltonian is infinite in both directions; we demonstrate this for the case of a tight-binding model of distinct honeycomb structures joined along a zig-zag edge.

Quasi-normal mode expansion as a tool for the design of nanophotonic devices

Many nanophotonic devices rely on the excitation of photonic resonances to enhance light-matter interaction. The understanding of the resonances is therefore of a key importance to facilitate the design of such devices. These resonances may be analyzed by use of the quasi-normal mode (QNM) theory. Here, we illustrate how QNM analysis may help study and design resonant nanophotonic devices. We will in particular use the QNM expansion of far-field quantities based on Riesz projection to design optical antennas.

Some results on eigenvalues of finite type, resolvents and Riesz projections

The present paper considers a separable Hilbert space H and a bounded linear operator A:H↦H that has an eigenvalue of finite type at λ0∈C. Using the image and the kernel of certain operators that are defined recursively starting from λ0−A, a construction of the local Smith form and extended canonical system of root functions of λ−A at λ0 is provided. This allows to express the partial multiplicities, the algebraic multiplicity, the Jordan structure and Jordan chains of A at λ0 in terms of the quantities defined by the recursion. Similarly, formulas for the rank, image and kernel of the operators in the principal part of the resolvent (λ−A)−1 at λ0, and thus also of the Riesz projection, are given in terms of the quantities defined by the recursion.

Spectral properties for some extensions of isometric operators

In this paper, we show that the spectrum, Weyl spectrum, and Browder spectrum are continuous on the set of all 2-isometric operators. We also prove that if T, \(S^{*}\) are 2-isometric and invertible isometric operators respectively and X is a Hilbert–Schmidt operator such that \(TX=XS\), then \(T^{*}X=XS^{*}\). Moreover, we show that every Riesz projection E with respect to a non-zero isolated spectral point \(\lambda \) of a 2-isometric operator T is self-adjoint and satisfies \({\mathcal {R}}(E) = {\mathcal {N}}(T-\lambda ) = {\mathcal {N}}(T- \lambda )^{*}\). Further, we show that quasi-2-isometric operator satisfies Bishop’s property (\(\beta \)). Finally, we prove Weyl type theorems for \(f(d_{TS})\), where \( d_{TS}\) denote the generalized derivation or the elementary operator with quasi-2-isometric operator entries T and \(S^{*}\) and \(f \in H(\sigma (d_{TS}))\), the set of analytic functions which are defined on an open neighborhood of \(\sigma (d_{TS})\).

Weyl’s theorem for paranormal closed operators

In this article, we discuss a few spectral properties of paranormal closed operator (not necessarily bounded) defined in a Hilbert space. This class contains closed symmetric operators. First, we show that the spectrum of such an operator is non-empty and give a characterization of closed range operators in terms of the spectrum. Using these results, we prove the Weyl’s theorem: if T is a densely defined closed paranormal operator, then \(\sigma (T)\setminus \omega (T)=\pi _{00}(T)\), where \(\sigma (T),\, \omega (T)\) and \(\pi _{00}(T)\) denote the spectrum, the Weyl spectrum and the set of all isolated eigenvalues with finite multiplicities, respectively. Finally, we prove that the Riesz projection \(E_\lambda \) with respect to any non-zero isolated spectral value \(\lambda \) of T is self-adjoint and satisfies \(R(E_\lambda )=N(T-\lambda I)=N(T-\lambda I)^*\).

Preservation of the Unconditional Basis Property under Non-Self-Adjoint Perturbations of Self-Adjoint Operators

Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$ \alpha_{2k+1}-\alpha_{2k} \geq b|\alpha_{2k+1}+\alpha_{2k}|^p\quad \text{ for some }\,b>0,\,p\in[0,1). $$ Suppose that a linear operator $B$ in $H$ is $p$-subordinated to $T$, i.e. $\mathcal D(B) \supset\mathcal D(T)$ and $\|Bx\| \leq b'\,\|Tx\|^p\|x\|^{1-p} +M\|x\| \text{\, for all } x\in \mathcal D(T)$, with some $b'>0$ and $M\geq 0$. Then the spectrum of the perturbed operator $A=T+B$ lies in the union of a rectangle in $\mathbb{C}$ and double parabola $P_{p,h} = \bigl\{\lambda \in \mathbb{C}\,\bigl|\,|\mathop{\rm Im} \lambda|\leq h|\mathop{\rm Re} \lambda|^p\bigr\}$, provided that $h>b'$. The vertical strips $\Omega_k =\{\lambda\in\mathbb{C}|\,|r_k-{\rm Re}\,\lambda|\leq \delta r_k^p\}$, $r_k =(\alpha_{2k}+\alpha_{2k+1})/2$, belong to the resolvent set of $T$, provided that $\delta <b -b'$ and $ |k|\geq N$ for $N$ large enough. For $|k|\ge N+1$, denote by $\Pi_k$ the curvilinear trapezoid formed by the lines ${\rm Re}\,\lambda = r_{k-1}$, ${\rm Re}\,\lambda = r_{k}$, and the boundary of the parabola $P_{p,h}$. Assume that $Q_0$ is the Riesz projection corresponding to the (bounded) part of the spectrum of $T$ that lies outside $\bigcup_{|k|\geq N+1}\Pi_k$. And let $Q_k$, $|k|\ge N+1$, be the Riesz projection for the part of the spectrum of $T$ confined within $\Pi_k$. Main result of the work consists in proving that the system of the invariant subspaces $Q_k(H)$, $|k|\geq N+1$, together with the invariant subspace $Q_0(H)$ forms an unconditional basis of subspaces in the space $H$.

Peierls’ substitution via minimal coupling and magnetic pseudo-differential calculus

We revisit the celebrated Peierls–Onsager substitution for weak magnetic fields with no spatial decay conditions. We assume that the non-magnetic [Formula: see text]-periodic Hamiltonian has an isolated spectral band whose Riesz projection has a range which admits a basis generated by [Formula: see text] exponentially localized composite Wannier functions. Then we show that the effective magnetic band Hamiltonian is unitarily equivalent to a Hofstadter-like magnetic matrix living in [Formula: see text]. In addition, if the magnetic field perturbation is slowly variable in space, then the perturbed spectral island is close (in the Hausdorff distance) to the spectrum of a Weyl quantized minimally coupled symbol. This symbol only depends on [Formula: see text] and is [Formula: see text]-periodic; if [Formula: see text], the symbol equals the Bloch eigenvalue itself. In particular, this rigorously formulates a result from 1951 by J. M. Luttinger.

Linear functions and duality on the infinite polytorus

We consider the following question: Are there exponents $2<p<q$ such that the Riesz projection is bounded from $L^q$ to $L^p$ on the infinite polytorus? We are unable to answer the question, but our counter-example improves a result of Marzo and Seip by demonstrating that the Riesz projection is unbounded from $L^\infty$ to $L^p$ if $p\geq 3.31138$. A similar result can be extracted for any $q>2$. Our approach is based on duality arguments and a detailed study of linear functions. Some related results are also presented.

The Brown–Halmos theorem for a pair of abstract Hardy spaces

Let H [ X ] and H [ Y ] be abstract Hardy spaces built upon Banach function spaces X and Y over the unit circle T . We prove an analogue of the Brown–Halmos theorem for Toeplitz operators T a acting from H [ X ] to H [ Y ] under the only assumption that the space X is separable and the Riesz projection P is bounded on the space Y . We specify our results to the case of variable Lebesgue spaces X = L p ( ⋅ ) and Y = L q ( ⋅ ) and to the case of Lorentz spaces X = Y = L p , q ( w ) , 1 < p < ∞ , 1 ≤ q < ∞ with Muckenhoupt weights w ∈ A p ( T ) .