pinecone/core-2020-05-10-deduplication huggingface dataset:
Eigenvalues for perturbed periodic Jacobi matrices by the Wigner-von
  Neumann approach

The Wigner-von Neumann method, which was previously used for perturbing
continuous Schr\"{o}dinger operators, is here applied to their discrete
counterparts. In particular, we consider perturbations of arbitrary
$T$-periodic Jacobi matrices. The asymptotic behaviour of the subordinate
solutions is investigated, as too are their initial components, together giving
a general technique for embedding eigenvalues, $\lambda$, into the operator's
absolutely continuous spectrum. Introducing a new rational function,
$C(\lambda;T)$, related to the periodic Jacobi matrices, we describe the
elements of the a.c. spectrum for which this construction does not work (zeros
of $C(\lambda;T)$); in particular showing that there are only finitely many of
them
The norm of the non-self-adjoint harmonic oscillator semigroup

We identify the norm of the semigroup generated by the non-self-adjoint
harmonic oscillator acting on $L^2(\Bbb{R})$, for all complex times where it is
bounded. We relate this problem to embeddings between Gaussian-weighted spaces
of holomorphic functions, and we show that the same technique applies, in any
dimension, to the semigroup $e^{-tQ}$ generated by an elliptic quadratic
operator acting on $L^2(\Bbb{R}^n)$. The method used --- identifying the
exponents of sharp products of Mehler formulas --- is elementary and is
inspired by more general works of L. H\"ormander, A. Melin, and J. Sj\"ostrand.Comment: 22 pages, 1 figure. Significant revision following referee's
  comments. To appear in Integral Equations and Operator Theory; published
  version may diffe
Dirac-Krein systems on star graphs

We study the spectrum of a self-adjoint Dirac-Krein operator with potential
on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This
operator is defined by a Dirac-Krein differential expression with summable
matrix potentials on each edge, by self-adjoint boundary conditions at the
outer vertices, and by a self-adjoint matching condition at the common central
vertex of $\mathcal G$. Special attention is paid to Robin matching conditions
with parameter $\tau \in\mathbb R\cup\{\infty\}$. Choosing the decoupled
operator with Dirichlet condition at the central vertex as a reference
operator, we derive Krein's resolvent formula, introduce corresponding
Weyl-Titchmarsh functions, study the multiplicities, dependence on $\tau$, and
interlacing properties of the eigenvalues, and prove a trace formula. Moreover,
we show that, asymptotically for $R\to \infty$, the difference of the number of
eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer
$\kappa_0$, which we call dislocation index, at most by $n+2$.Comment: Accepted for publication in IEO
The Cesaro operator in growth Banach spaces of analytic functions

The Cesaro operator $\mathsf{C}$, when acting in the classical growth Banach
spaces $A^{-\gamma}$ and $A_0^{-\gamma}$, for $\gamma > 0 $, of analytic
functions on $\mathbb{D}$, is investigated. Based on a detailed knowledge of
their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the
norms of these operators precisely. It is then possible to characterize the
mean ergodic and related properties of $\mathsf{C}$ acting in these spaces. In
addition, we determine the largest Banach space of analytic functions on
$\mathbb{D}$ which $\mathsf{C}$ maps into $A^{-\gamma}$ (resp. into
$A_0^{-\gamma}$); this optimal domain space always contains $A^{-\gamma}$
(resp. $A_0^{-\gamma}$) as a proper subspace.Comment: 17 page
Weak product spaces of Dirichlet series

Let $\mathscr{H}^2$ denote the space of ordinary Dirichlet series with square
summable coefficients, and let $\mathscr{H}^2_0$ denote its subspace consisting
of series vanishing at $+\infty$. We investigate the weak product spaces
$\mathscr{H}^2\odot\mathscr{H}^2$ and $\mathscr{H}^2_0\odot\mathscr{H}^2_0$,
finding that several pertinent problems are more tractable for the latter
space. This surprising phenomenon is related to the fact that
$\mathscr{H}^2_0\odot\mathscr{H}^2_0$ does not contain the infinite-dimensional
subspace of $\mathscr{H}^2$ of series which lift to linear functions on the
infinite polydisc.
  The problems considered stem from questions about the dual spaces of these
weak product spaces, and are therefore naturally phrased in terms of
multiplicative Hankel forms. We show that there are bounded, even Schatten
class, multiplicative Hankel forms on $\mathscr{H}^2_0 \times \mathscr{H}^2_0$
whose analytic symbols are not in $\mathscr{H}^2$. Based on this result we
examine Nehari's theorem for such Hankel forms. We define also the skew product
spaces associated with $\mathscr{H}^2\odot\mathscr{H}^2$ and
$\mathscr{H}^2_0\odot\mathscr{H}^2_0$, with respect to both half-plane and
polydisc differentiation, the latter arising from Bohr's point of view. In the
process we supply square function characterizations of the Hardy spaces
$\mathscr{H}^p$, for $0 < p < \infty$, from the viewpoints of both types of
differentiation. Finally we compare the skew product spaces to the weak product
spaces, leading naturally to an interesting Schur multiplier problem.Comment: This paper has been accepted for publication in Integral Equations
  and Operator Theor
Intrinsic operators from holomorphic function spaces to growth spaces

We determine the boundedness and compactness of a large class of operators,
mapping from general Banach spaces of holomorphic functions into a particular
type of spaces of functions determined by the growth of the functions, or the
growth of the functions derivatives. The results show that the boundedness and
compactness of such intrinsic operators depends only on the behaviour on the
point evaluation functionals. They also generalize previous similar results
about several specific classes of operators, such as the multiplication,
composition and integral operators.Comment: 22 pages; the final publication is available at Springer via
  http://dx.doi.org/10.1007/s00020-017-2361-
Kadison's Pythagorean Theorem and essential codimension

Kadison's Pythagorean theorem (2002) provides a characterization of the
diagonals of projections with a subtle integrality condition. Arveson (2007),
Kaftal, Ng, Zhang (2009), and Argerami (2015) all provide different proofs of
that integrality condition. In this paper we interpret the integrality
condition in terms of the essential codimension of a pair of projections
introduced by Brown, Douglas and Fillmore (1973), or, equivalently of the index
of a Fredholm pair of projections introduced by Avron, Seiler, and Simon
(1994). The same techniques explain the integer occurring in the
characterization of diagonals of selfadjoint operators with finite spectrum by
Bownik and Jasper (2015).Comment: 15 page
Nonlinear bang-bang eigenproblems and optimization of resonances in
  layered cavities

Quasi-normal-eigenvalue optimization is studied under constraints $b_1(x) \le
B(x) \le b_2 (x)$ on structure functions $B$ of 2-side open optical or
mechanical resonators. We prove existence of various optimizers and provide an
example when different structures generate the same optimal
quasi-(normal-)eigenvalue. To show that quasi-eigenvalues locally optimal in
various senses are in the spectrum $\Sigma^{nl}$ of the bang-bang eigenproblem
$y" = - \omega^2 y [ b_1 + (b_2 - b_1) \chi_{\mathbb{C}_+} (y^2 ) ]$, where
$\chi_{\mathbb{C}_+} (\cdot)$ is the indicator function of the upper complex
half-plane $\mathbb{C}_+$, we obtain a variational characterization of the
nonlinear spectrum $\Sigma^{nl}$ in terms of quasi-eigenvalue perturbations. To
address the minimization of the decay rate $| \mathrm{Im} \ \omega |$, we study
the bang-bang equation and explain how it excludes an unknown optimal $B$ from
the optimization process. Computing one of minimal decay structures for 1-side
open settings, we show that it resembles gradually size-modulated 1-D stack
cavities introduced recently in Optical Engineering. In 2-side open symmetric
settings, our example has an additional centered defect. Nonexistence of global
decay rate minimizers is discussed.Comment: 39 pages, 2 figures, 2 tables. The results of the paper were
  partially reported at the International Congress of Mathematicians (Seoul ICM
  2014) and in several seminar and workshop talk
Operator representations of frames: boundedness, duality, and stability

The purpose of the paper is to analyze frames $\{f_k\}_{k\in \mathbf Z}$
having the form $\{T^kf_0\}_{k\in\mathbf Z}$ for some linear operator $T:
\mbox{span} \{f_k\}_{k\in \mathbf Z} \to \mbox{span}\{f_k\}_{k\in \mathbf Z}$.
A key result characterizes boundedness of the operator $T$ in terms of
shift-invariance of a certain sequence space. One of the consequences is a
characterization of the case where the representation $\{f_k\}_{k\in \mathbf
Z}=\{T^kf_0\}_{k\in\mathbf Z}$ can be achieved for an operator $T$ that has an
extension to a bounded bijective operator $\widetilde{T}: \cal H \to \cal H.$
In this case we also characterize all the dual frames that are representable in
terms of iterations of an operator $V;$ in particular we prove that the only
possible operator is $V=(\widetilde{T}^*)^{-1}.$ Finally, we consider stability
of the representation $\{T^kf_0\}_{k\in\mathbf Z};$ rather surprisingly, it
turns out that the possibility to represent a frame on this form is sensitive
towards some of the classical perturbation conditions in frame theory. Various
ways of avoiding this problem will be discussed. Throughout the paper the
results will be connected with the operators and function systems appearing in
applied harmonic analysis, as well as with general group representations
Converging expansions for Lipschitz self-similar perforations of a plane
  sector

In contrast with the well-known methods of matching asymptotics and
multiscale (or compound) asymptotics, the " functional analytic approach " of
Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of
expansions around interior small holes of size $\epsilon$ for solutions of
elliptic boundary value problems. Using the method of layer potentials, the
asymptotic behavior of the solution as $\epsilon$ tends to zero is described
not only by asymptotic series in powers of $\epsilon$, but by convergent power
series. Here we use this method to investigate the Dirichlet problem for the
Laplace operator where holes are collapsing at a polygonal corner of opening
$\omega$. Then in addition to the scale $\epsilon$ there appears the scale
$\eta = \epsilon^{\pi/\omega}$. We prove that when $\pi/\omega$ is irrational,
the solution of the Dirichlet problem is given by convergent series in powers
of these two small parameters. Due to interference of the two scales, this
convergence is obtained, in full generality, by grouping together integer
powers of the two scales that are very close to each other. Nevertheless, there
exists a dense subset of openings $\omega$ (characterized by Diophantine
approximation properties), for which real analyticity in the two variables
$\epsilon$ and $\eta$ holds and the power series converge unconditionally. When
$\pi/\omega$ is rational, the series are unconditionally convergent, but
contain terms in log $\epsilon$.Comment: Integral Equations and Operator Theory, Springer Verlag, 201
Effects of Rayleigh waves on the essential spectrum in perturbed doubly
  periodic elliptic problems

We give an example of a scalar second order differential operator in the
plane with double periodic coefficients and describe its modification, which
causes an additional spectral band in the essential spectrum. The modified
operator is obtained by applying to the coefficients a mirror reflection with
respect to a vertical or horizontal line. This change gives rise to Rayleigh
type waves localized near the line. The results are proven using asymptotic
analysis, and they are based on high contrast of the coefficient functions.Comment: 14 pages, 3 figure
Unitary Correlation Sets

The unitary correlation sets defined by the first author in conjunction with
tensor products of $\mathcal{U}_{nc}(n)$ are further studied. We show that
Connes' embedding problem is equivalent to deciding whether or not two smaller
versions of the unitary correlation sets are equal. Moreover, we obtain the
result that Connes' embedding problem is equivalent to deciding whether or not
two cross norms on $M_n \otimes M_n$ are equal for all $n \geq 2$.Comment: 24 pages, fixed a small error in the proof of Theorem 5.3 (v2
Hamiltonian systems and Sturm-Liouville equations: Darboux
  transformation and applications

We introduce GBDT version of Darboux transformation for symplectic and
Hamiltonian systems as well as for Shin-Zettl systems and Sturm-Liouville
equations. These are the first results on Darboux transformation for
general-type Hamiltonian and for Shin-Zettl systems. The obtained results are
applied to the corresponding transformations of the Weyl-Titchmarsh functions
and to the construction of explicit solutions of dynamical symplectic systems,
of two-way diffusion equations and of indefinite Sturm-Liouville equations. The
energy of the explicit solutions of dynamical systems is expressed (in a quite
simple form) in terms of the parameter matrices of GBDT.Comment: Section 7 of this paper is related to our recent paper
  arXiv:1603.0870
Sharp estimates for the Szeg\H{o} projection on the distinguished
  boundary of model worm domains

In this paper we study the regularity of the Szeg\H{o} projection on Lebesgue
and Sobolev spaces on the distinguished boundary of the unbounded model worm
domain $D_\beta$.
  We denote by $d_b(D_\beta)$ the distinguished boundary of $D_\beta$ and
define the corresponding Hardy space $\mathscr{H}^2(D_\beta)$. This can be
identified with a closed subspace of $L^2(d_b(D_\beta),d\sigma)$, that we
denote by $\mathscr{H}^2(d_b(D_\beta))$, where $d\sigma$ is the naturally
induced measure on $d_b(D_\beta)$.
  The orthogonal Hilbert space projection $\mathscr{P}:
L^2(d_b(D_\beta),d\sigma)\to \mathscr{H}^2(d_b(D_\beta))$ is called the
Szeg\H{o} projection on the distinguished boundary.
  We prove that $\mathscr{P}$, initially defined on the dense subspace
$L^2(d_b( D_\beta),d\sigma)\cap L^p(d_b(D_\beta), d\sigma)$ extends to a
bounded operator $\mathscr{P}: L^p(d_b(D_\beta), d\sigma)\to L^p(d_b(D_\beta),
d\sigma)$ if and only if
$\textstyle{\frac{2}{1+\nu_\beta}}<p<\textstyle{\frac{2}{1-\nu_\beta}}$ where
$\nu_\beta=\textstyle{\frac{\pi}{2\beta-\pi}},\beta>\pi$. Furthermore, we also
prove that $\mathscr{P}$ defines a bounded operator $\mathscr{P}:
W^{s,2}(d_b(D_\beta),d\sigma)\to W^{s,2}(d_b(D_\beta), d\sigma)$ if and only if
$0\leq s<\textstyle{\frac{\nu_\beta}{2}}$ where $W^{s.2}(d_b( D_\beta),
d\sigma)$ denotes the Sobolev space of order $s$ and underlying $L^2$-norm.
  Finally, we prove a necessary condition for the boundedness of $\mathscr{P}$
on $W^{s,p}(d_b(D_\beta), d\sigma)$, $p\in(1,\infty)$, the Sobolev space of
order $s$ and underlying $L^p$-norm.Comment: 27 page
A Hankel matrix acting on spaces of analytic functions

If $\mu $ is a positive Borel measure on the interval $[0, 1)$ we let
$\mathcal H_\mu $ be the Hankel matrix $\mathcal H_\mu =(\mu _{n, k})_{n,k\ge
0}$ with entries $\mu _{n, k}=\mu _{n+k}$, where, for $n\,=\,0, 1, 2, \dots $,
$\mu_n$ denotes the moment of order $n$ of $\mu $. This matrix induces formally
the operator $$\mathcal{H}_\mu (f)(z)=
\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n,k}{a_k}\right)z^n$$ on the
space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit
disc $\mathbb D $. This is a natural generalization of the classical Hilbert
operator. In this paper we improve the results obtained in some recent papers
concerning the action of the operators $H_\mu $ on Hardy spaces and on M\"obius
invariant spaces.Comment: arXiv admin note: text overlap with arXiv:1612.0830
The Classical Moment Problem and Generalized Indefinite Strings

We show that the classical Hamburger moment problem can be included in the
spectral theory of generalized indefinite strings. Namely, we introduce the
class of Krein-Langer strings and show that there is a bijective correspondence
between moment sequences and this class of generalized indefinite strings. This
result can be viewed as a complement to the classical results of M. G. Krein on
the connection between the Stieltjes moment problem and Krein-Stieltjes strings
and I. S. Kac on the connection between the Hamburger moment problem and 2x2
canonical systems with Hamburger Hamiltonians.Comment: 25 page
Moment problem for symmetric algebras of locally convex spaces

It is explained how a locally convex (lc) topology $\tau$ on a real vector
space $V$ extends to a locally multiplicatively convex (lmc) topology
$\overline{\tau}$ on the symmetric algebra $S(V)$. This allows the application
of the results on lmc topological algebras obtained by Ghasemi, Kuhlmann and
Marshall to obtain representations of $\overline{\tau}$-continuous linear
functionals $L: S(V)\rightarrow \mathbb{R}$ satisfying $L(\sum S(V)^{2d})
\subseteq [0,\infty)$ (more generally, $L(M) \subseteq [0,\infty)$ for some
$2d$-power module $M$ of $S(V)$) as integrals with respect to uniquely
determined Radon measures $\mu$ supported by special sorts of closed balls in
the dual space of $V$. The result is simultaneously more general and less
general than the corresponding result of Berezansky, Kondratiev and \v Sifrin.
It is more general because $V$ can be any lc topological space (not just a
separable nuclear space), the result holds for arbitrary $2d$-powers (not just
squares), and no assumptions of quasi-analyticity are required. It is less
general because it is necessary to assume that $L : S(V) \rightarrow
\mathbb{R}$ is $\overline{\tau}$-continuous (not just continuous on each
homogeneous part of $S(V)$).Comment: 19 pages, revised according to referee's comments, updated
  references, to appear in Integral Equations and Operator Theor
On an integral formula for Fredholm determinants related to pairs of
  spectral projections

We consider Fredholm determinants of the form identity minus product of
spectral projections corresponding to isolated parts of the spectrum of a pair
of self-adjoint operators. We show an identity relating such determinants to an
integral over the spectral shift function in the case of a rank-one
perturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1}
-\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big)
\right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\,
\frac{\xi(x)\xi(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes the
spectral projection of a self-adjoint operator on a set $J\in
\text{Borel}(\mathbb R)$. The operators $A$ and $B$ are self-adjoint, bounded
from below and differ by a rank-one perturbation and $\xi$ denotes the
corresponding spectral shift function. The set $I$ is a union of intervals on
the real line such that its boundary lies in the resolvent set of $A$ and $B$
and such that the spectral shift function vanishes there i.e. $I$ contains
isolated parts of the spectrum of $A$ and $B$. We apply this formula to the
subspace perturbation problem.Comment: version as publishe
Aluthge transforms of 2-variable weighted shifts

We introduce two natural notions of multivariable Aluthge transforms (toral
and spherical), and study their basic properties. In the case of 2-variable
weighted shifts, we first prove that the toral Aluthge transform does not
preserve (joint) hyponormality, in sharp contrast with the 1-variable case.
Second, we identify a large class of 2-variable weighted shifts for which
hyponormality is preserved under both transforms. Third, we consider whether
these Aluthge transforms are norm-continuous. Fourth, we study how the Taylor
and Taylor essential spectra of 2-variable weighted shifts behave under the
toral and spherical Aluthge transforms; as a special case, we consider the
Aluthge transforms of the Drury-Arveson 2-shift. Finally, we briefly discuss
the class of spherically quasinormal 2-variable weighted shifts, which are the
fixed points for the spherical Aluthge transform
Dirichlet spectrum of the Fichera layer

We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a
prototypal infinite polyhedral layer, that is formed by three perpendicular
quarter-plane walls of constant width joining each other. Alternatively, this
domain can be viewed as an octant from which another "parallel" octant is
removed. It contains six edges (three convex and three non-convex) and two
corners (one convex and one non-convex). It is a canonical example of
non-smooth conical layer. We name it after Fichera because near its non-convex
corner, it coincides with the famous Fichera cube that illustrates the
interaction between edge and corner singularities. This domain could also be
called an octant layer.
  We show that the essential spectrum of the Laplacian on such a domain is a
half-line and we characterize its minimum as the first eigenvalue of the
two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type
strategy, we also prove that its discrete spectrum is finite and that a lower
bound is given by the ground state of a special Sturm-Liouville operator. By
finite element computations taking singularities into account, we exhibit
exactly one eigenvalue under the essential spectrum threshold leaving a
relative gap of 3%. We extend these results to a variant of the Fichera layer
with rounded edges (for which we find a very small relative gap of 0.5%), and
to a three-dimensional cross where the three walls are full thickened planes.Comment: 33 page
Multimodal Standing Gravity Waves: a Completely Resonant System

International audienceThe standing gravity wave problem on an infinitely deep fluid layer is considered under the form of a nonlinear non local scalar PDE of second order as in [6]. Nonreso-nance at quadratic order of the infinite dimensional bifurcation equation, allows to give the explicit form of the quadratic change of variables able to suppress quadratic terms in the nonlinear equation. We state precisely the equivalence between formulations in showing that the above unbounded change of variable is invertible. The infinite set of solutions which can be expanded in powers of amplitude ε is then given up to order ε 2
Reversible Bifurcation of Homoclinic Solutions in Presence of an Essential Spectrum

International audienceWe consider bifurcations of a class of infinite dimensional reversible dynamical systems which possess a family of symmetric equilibria near the origin. We also assume that the linearized operator at the origin $% L_{\varepsilon }$ has an essential spectrum filling the entire real line, in addition to the simple eigenvalue at 0. Moreover, for parameter values $% \varepsilon <0$ there is a pair of imaginary eigenvalues which meet in 0 for $\varepsilon =0$, and which disappear for $\varepsilon >0$. The above situation occurs for example when one looks for travelling waves in a system of superposed perfect fluid layers, one being infinitely deep. We give quite general assumptions which apply in such physical examples, under which one obtains a family of bifurcating solutions homoclinic to every equilibrium near the origin. These homoclinics are symmetric and decay algebraically at infinity, being approximated at main order by the Benjamin - Ono homoclinic. For the water wave example, this corresponds to a family of solitary waves, such that at infinity the upper layer slides with a uniform velocity, over the bottom layer (at rest)
Existence of Multimodal Standing Gravity Waves

International audienceWe consider two-dimensional standing gravity waves on the surface of an infinitely deep perfect fluid, the flow being potential. It is known that the linearized problem is completely resonant. Following the method described in [4], we prove the existence of an infinity of multimodal standing gravity waves, corresponding to any choice of asymptotic expansion in powers of the amplitude ε, indicated in [2] and [3]. Each one of these solutions exist for a set of values of ε being dense in 0