Non-selfadjoint Operators Research Articles

Pseudospectra and eigenvalue asymptotics for disordered non-selfadjoint operators in the semiclassical limit

The purpose of this note is to review certain recent results concerning the pseudospectra and the eigenvalues asymptotics of non-selfadjoint semiclassical pseudo-differential operators subject to small random perturbations.

Morita invariance of unbounded bivariant K-theory

We introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov’s bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a refinement of the usual interior Kasparov product. We illustrate our results by proving C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^1$$\\end{document}-versions of well-known C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebraic Morita equivalences in the context of hereditary subalgebras, conformal equivalences and crossed products by discrete groups.

Investigation of the Spectrum of Nonself-Adjoint Discontinuous Sturm-Liouville Operator

In this paper, we study nonself-adjoint Sturm-Liouville operator containing both the discontinuous coefficient and discontinuity conditions at some point on the positive half-line. The eigenvalues and the spectral singularities of this problem are examined and it is proved that this problem has a finite number of spectral singularities and eigenvalues with finite multiplicities under two different additional conditions. Furthermore, the principal functions corresponding to the eigenvalues and the spectral singularities of this operator are determined.

Spectral theory and self-similar blowup in wave equations

This is an expository article that describes the spectral-theoretic aspects in the study of the stability of self-similar blowup for nonlinear wave equations. The linearization near a self-similar solution leads to a genuinely nonself-adjoint operator which is difficult to analyze. The main goal of this article is to provide an accessible account of the only known method that is capable of providing sufficient spectral information to complete the stability analysis. The exposition is based on a mini course given at the Summer School on Geometric Dispersive PDEs in Obergurgl, Austria, in September 2022.

The pseudospectra of black holes in AdS

We study the stability of quasinormal modes (QNMs) in electrically charged black brane spacetimes that asymptote to AdS by means of the pseudospectrum. Methodologically, we adopt ingoing Eddington-Finkelstein coordinates to cast QNMs in terms of a generalised eigenvalue problem involving a non-selfadjoint operator; this simplifies the computation significantly in comparison with previous results in the literature. Our analysis reveals spectral instability for (neutral) scalar as well as gravitoelectric perturbations. This indicates that the equilibration process of perturbed black branes is sensitive to external perturbations. Particular attention is given on the hydrodynamic modes, which are found to be the least unstable. In contrast with computations in hyperboloidal coordinates, we find that the pseudospectral contour lines cross to the upper half plane. This indicates the existence of pseudo-resonances as well as the possibility of transient instabilities. We also investigate the asymptotic structure of pseudospectral contour levels and we find remarkable universality across all sectors, persistent in the extremal limit.

Wold decomposition and C*-envelopes of self-similar semigroup actions on graphs

We study the Wold decomposition for representations of a self-similar semigroup P action on a directed graph E. We then apply this decomposition to the case where P=N to study the C*-envelope of the associated universal non-selfadjoint operator algebra AN,E by carefully constructing explicit non-trivial dilations for non-boundary representations. In particular, it is shown that the C*-envelope of AN,E coincides with the self-similar graph C*-algebra OZ,E.

On the spectrum of non-selfadjoint Dirac operators with two-point boundary conditions

On the spectrum of non-selfadjoint Dirac operators with two-point boundary conditions

On the Spectrum of Nonself-Adjoint Dirac Operators with Two-Point Boundary Conditions

On the Spectrum of Nonself-Adjoint Dirac Operators with Two-Point Boundary Conditions

Gains of integrability and local smoothing effects for quadratic evolution equations

We characterize geometrically the semigroups generated by non-selfadjoint quadratic differential operators (e−tqw)t≥0 enjoying local smoothing effects and providing gains of integrability. More precisely, we prove that the evolution operators e−tqw map Lp on Lq∩C∞, for all 1≤p≤q≤∞, if and only if the singular space of the quadratic operator qw is included in the graph of a linear map. We also provide quantitative estimates for the associated operator norms in the short-time asymptotics 0<t≪1.

Maximal sectorial operators and invariant operator ranges

For unbounded maximal sectorial operators we establish necessary and sufficient conditions for the domain equality domA=domA⁎ and for the equality ReA=AR of operator real part ReA and form real part AR. Here ReA=12(A+A⁎) is half of the operator sum defined on domA∩domA⁎, whereas AR=12(A+˙A⁎) is the self-adjoint operator given by half of the form-sum of A and A⁎ so that, in general, ReA⊆AR. The natural question posed in [6], whether for a maximal sectorial operator A the equalitydomA=domA⁎implies the equalityReA=AR, is answered negatively in this paper. We construct families of unbounded coercive m-sectorial operators A such that domA=domA⁎ for which ReA is a closed symmetric non-selfadjoint operator or a non-closed essentially selfadjoint operator. Moreover, we show that the domain equalities domA=domA⁎ and domReA=domAR are equivalent to problems of invariant operator ranges of bounded selfadjoint or unitary operators as well as to the existence of bounded operators with specific operator range properties.

Negatif Yoğunluk Fonksiyonuna Sahip Kendine Eşlenik Olmayan Schrödinger Operatörü Üzerine Bir Çalışma

This study focuses on the spectral features of the non-selfadjoint singular operator with an out-of-the-ordinary type weight function. Take into consideration the one-dimensional time-dependent Schrödinger type differential equation&#x0D; -y^''+q(x)y=μ^2 ρ(x)y,x∈[0,∞),&#x0D; holding the initial condition&#x0D; y(0)=0,&#x0D; and the density function defined with a completely negative value as&#x0D; ρ(x)=-1.&#x0D; There is an enormous number of the papers considering the positive values of ρ(x) for both continuous and discontinuous cases. The structure of the density function affects the analytical properties and representations of the solutions of the equation. Unlike the classical literature, we use the hyperbolic type representations of the equation’s fundamental solutions to obtain the operator’s spectrum. Additionally, the requirements for finiteness of eigenvalues and spectral singularities are addressed. Hence, Naimark’s and Pavlov’s conditions are adopted for the negative density function case.

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects

We study semigroups generated by accretive non-selfadjoint quadratic differential operators. We give a description of the polar decomposition of the associated evolution operators as products of a selfadjoint operator and a unitary operator. The selfadjoint parts turn out to be also evolution operators generated by time-dependent real-valued quadratic forms that are studied in details. As a byproduct of this decomposition, we give a geometric description of the regularizing properties of semigroups generated by accretive non-selfadjoint quadratic operators. Finally, by using the interpolation theory, we take advantage of this smoothing effect to establish subelliptic estimates enjoyed by quadratic operators.

Spectral Expansion for Nonself-Adjoint Differential Operators with Periodic Matrix Coefficients

Spectral Expansion for Nonself-Adjoint Differential Operators with Periodic Matrix Coefficients

Numerical solution of Korteweg–de Vries equation using discrete least squares meshless method

In this study, a meshless approach called Discrete Least Squares Meshless is used to solve the third-order nonlinear Korteweg–de Vries equation numerically. In shallow water, the Korteweg–de Vries equation illustrates the unidirectional propagation of waves. For function approximation, the suggested technique uses moving least squares shape functions. Furthermore, the least squares functional constructed on certain auxiliary points known as sampling points is minimized in order to generate the system of algebraic equations. This approach may be considered meshless since it does not need a mesh for field variable approximation or system of equations creation; it also gives a symmetric and positive definite matrix even for non-self adjoint differential operators. To demonstrate the applicability of the current meshless approach, the propagation of a single soliton and multi-solitons is explored. The approach may be applied directly to the Korteweg–de Vries equation because to the high degrees of continuity of the moving least squares shape function when an exponential weight function is utilized. Several benchmark numerical instances are explored and the resulting results are compared to earlier research to test the capabilities of the suggested approach for the solution of Korteweg–de Vries equation.

Nonselfadjoint impedance in Generalized Optimized Schwarz Methods

Abstract We present a convergence theory for Optimized Schwarz Methods that rely on a nonlocal exchange operator and covers the case of coercive possibly nonselfadjoint impedance operators. This analysis also naturally deals with the presence of cross-points in subdomain partitions of arbitrary shape. In the particular case of hermitian positive definite impedance, we recover the theory proposed in Claeys &amp; Parolin (2021).

Pseudospectrum and binary black hole merger transients

The merger phase of binary black hole coalescences is a transient between an initial oscillating regime (inspiral) and a late exponentially damped phase (ringdown). In spite of the non-linear character of Einstein equations, the merger dynamics presents a surprisingly simple behaviour consistent with effective linearity. On the other hand, energy loss through the event horizon and by scattering to infinity renders the system non-conservative. Hence, the infinitesimal generator of the (effective) linear dynamics is a non-selfadjoint operator. Qualitative features of transients in linear dynamics driven by non-selfadjoint (in general, non-normal) operators are captured by the pseudospectrum of the time generator. We propose the pseudospectrum as a unifying framework to thread together the phases of binary black hole coalescences, from the inspiral-merger transition up to the late quasinormal mode ringdown.

Construction of quasimodes for non-selfadjoint operators via propagation of Hagedorn wave-packets

We construct quasimodes for some non-selfadjoint semiclassical operators at the boundary of the pseudo-spectrum using propagation of Hagedorn wave-packets. Assuming that the imaginary part of the principal symbol of the operator is non-negative and vanishes on certain points of the phase-space satisfying a subelliptic finite-type condition, we construct quasimodes that concentrate on these non-damped points. More generally, we apply this technique to construct quasimodes for non-selfadjoint semiclassical perturbations of the harmonic oscillator that concentrate on non-damped periodic orbits or invariant tori satisfying a weakgeometric- control condition.

Abstract Evolution Equations with an Operator Function in the Second Term

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the second term of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator.

Quaternionic triangular linear operators

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S‐spectrum. This allow us to obtain conditions when a non‐selfadjoint operator admits a triangular representation.