Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation

For usual Hardy type inequalities the natural “breaking point” (the parameter value where the inequality reverses) is

p

= 1. Recently, J. Oguntuase and L.-E. Persson proved a refined Hardy type inequality with breaking point at

p

= 2. In this paper we show that this refinement is not unique and can be replaced by another refined Hardy type inequality with breaking point at

p

= 2. Moreover, a new refined Hardy type inequality with breaking point at

p

= 3 is obtained. One key idea is to prove some new Jensen type inequalities related to convex or superquadratic funcions, which are also of independent interest.

We study non-negative self-adjoint extensions of a non densely defined non-negative symmetric operator Å with the exit in the rigged Hilbert space constructed by means of the adjoint operator Å

*

(bi-extensions). Criteria of existence and descriptions of such extensions and associated closed forms are obtained. Moreover, we introduce the concept of an extremal nonnegative bi-extension and provide its complete description. After that we state and prove the existence and uniqueness results for extremal non-negative biextensions in terms of the Kreĭn–von Neumann and Friedrichs extensions of a given non-negative symmetric operator. Further, the connections between positive boundary triplets and non-negative self-adjoint bi-extensions are presented

Some important classes of matrices admit a factorization known as bidiagonal decomposition. Bidiagonal decompositions can provide natural parameters to perform computations with high relative accuracy. We prove that corner cutting algorithms provide bidiagonal decompositions with high relative accuracy

This paper presents a survey of recent results, methods, and open problems in the theory of higher-order elliptic boundary value problems on Lipschitz and more general non-smooth domains. The main topics include the maximum principle and pointwise estimates on solutions in arbitrary domains, analogues of the Wiener test governing continuity of solutions and their derivatives at a boundary point, and well-posedness of boundary value problems in domains with Lipschitz boundaries.

Let

X

be a complex Banach space and let

L(X)

be the algebra of all bounded linear operators on

X

. We characterize additive continuous maps from

L(X)

onto itself which preserve the inner local spectral radius at a nonzero fixed vector.

On the real line we consider singular integral operators with a linear Carleman shift and complex conjugation, acting in

$$ \tilde{L}_2(\mathbb{R})$$

, the real space of all Lebesgue measurable complex value functions on ℝ with

p

= 2 power. We show that the original singular integral operator with shift and conjugation is, after extension, equivalent to a singular integral operator without shift and with a 4 × 4 matrix coefficients. By exploiting the properties of the factorization of the symbol of this last operator, it is possible to describe the solution of a generalized Riemann boundary value problem with a Carleman shift.

Extremal vectors were introduced by S. Ansari and P. Enflo in [2], this method produced new and more constructive proofs of existence of invariant subspaces. In this paper, our purpose is to introduce generalized extremal vectors and to study their properties. We firstly check that general properties of extremal vectors also hold for generalized extremal vectors. We give a new useful characterization of generalized extremal vectors. We show that there exist relationships between these vectors and the famous Moore–Penrose pseudo-inverse showing their intrinsic nature. Applications to weighted shift operators are given. In particular, we discuss for quasinilpotent backward weighted shifts the following question: Can the Ansari–Enflo method be used in order to obtain all hyper-invariant subspaces?

Let

ψ

and

$$ \varphi $$

be analytic functions on the open unit disk

$$ (\mathbb{D}) $$

with

$$ \varphi(\mathbb{D})\sqsubseteq\mathbb{D}\, {\rm and\, let }\, 1\leq p<\infty $$

. We characterize the bounded and the compact weighted composition operators

$$ {W_{\psi\varphi}} $$

from the analytic Besov space

B

p

into

BMOA

and into

VMOA

. We also show that there are no isometries among the composition operators.

Given a *-homomorphism

$$\sigma : $$

$$ C(M)\rightarrow\mathcal{L}(H) $$

on a Hilbert space

$$\mathcal{H}$$

for a compact metric space

M

, a projection

P

onto a subspace

$$ P\, {\rm in}\, H$$

is said to be essentially normal relative to

$$\sigma\,{\rm if}[\sigma(\varphi),P]\in\mathcal{K}\,{\rm for} \varphi\in C(M)$$

, where

K

is the ideal of compact operators on

H

. In this note we consider two notions of span for essentially normal projections

P

and

Q

, and investigate when they are also essentially normal. First, we show the representation theorem for two projections, and relate these results to Arveson’s conjecture for the closure of homogenous polynomial ideals on the Drury–Arveson space. Finally, we consider the relation between the relative position of two essentially normal projections and the

K

homology elements defined for them.

Recent work by several authors has revealed the existence of many unexpected classes of normal weighted composition operators. On the other hand, it is known that every normal operator is a complex symmetric operator. We therefore undertake the study of complex symmetric weighted composition operators, identifying several new classes of such operators.

We obtain an analogue of Coburn’s description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.

In the paper some classes of vector differential operators of infinite order are studied and their use for constructing the entire solutions of implicit linear differential equations in a Banach space is considered. In addition, the integral representations of the Cauchy type for vector differential operators of infinite order are obtained.

We recall some results on generalized determinants which support a theory of operator τ -functions in the context of their predeterminants which are operators valued in a Banach–Lie group that are derived from the transition maps of certain Banach bundles. Related to this study is a class of Banach–Lie algebras known as

L

*

-algebras from which several results are obtained in relationship to tau functions. We survey the applicability of this theory to that of Schlesinger systems associated with (operator) equations of Fuschsian type and discuss how meromorphic connections may play a role here.

Tauberian operators have been useful in the study of many different topics of functional analysis. Here we describe some properties and the main applications of tauberian operators, and we point out several concrete problems that remain open.

The aim of this note is to discuss boundedness and compactness of Hankel products and mixed Toeplitz–Hankel products on the Hardy space of the unit sphere in several complex variables. The main adopted tool is an auxiliary pioneering operator involved in an earlier investigation of dual Toeplitz operators on the orthogonal complement of the Hardy space on the unit sphere.

We discuss the direct and inverse scattering theory for the nonlinear Schrödinger equation

$$-\triangle u(x)+h(x,|u(x)|)u(x)=k^{2}u(x),\quad x \epsilon\mathbb{R}^{3}$$

where

h

is a very general and possibly singular combination of potentials. We prove first that the direct scattering problem has a unique bounded solution. We establish also the asymptotic behaviour of scattering solutions. A uniqueness result and a representation formula is proved for the inverse scattering problem with general scattering data. The method of Born approximation is applied for the recovery of jumps in the unknown function from general scattering data and fixed angle data.

Let

p

: ℝ → (1,∞) be a globally log-Hölder continuous variable exponent and

w

: ℝ →[0,∞] be a weight. We prove that the Cauchy singular integral operator

s

is bounded on the weighted variable Lebesgue space

L

p(.)

(ℝ,

w

)= {

f

:

f

w

ϵ

L

p(.)

(ℝ)} if and only if the weight

w

satisfies

$$\mathop{sup}\limits_{-\infty<a<b<\infty}\frac{1}{b-a}\parallel w\chi_{(a,b)}\parallel_{p(.)}\parallel w^{-1}\chi_{(a,b)}\parallel_{p^{\prime}(.)}<\infty\quad (1/p(x)+1/p^{\prime}(x)=1).$$

Let Γ be a non-rectifiable curve on the complex plane ℂ. We extend distributional derivative

$$\bar{\boldsymbol\partial}$$

F

of a function

F

, which is holomorphic in domain

$$\bar{\mathbb{C}}$$

\ Γ, up to continuous functional on the weighted Hölder space and apply this extension for solution of the Riemann boundary value problem on the curve Γ.

For 0 < ϵ < 1, the ϵ-condition spectrum of an element

a

in a complex unital Banach algebra A is defined as,

$$\sigma_{\epsilon}(a)=\{\lambda\,\epsilon\,\mathbb{C}:\lambda-\hbox {a is not invertible or} \parallel\lambda-a\parallel\parallel(\lambda-a)^{-1}\parallel\geq\frac{1}{\epsilon}\}.$$

This is a generalization of the idea of spectrum introduced in [5]. This is expected to be useful in dealing with operator equations. In this paper we prove a mapping theorem for condition spectrum, extending an earlier result in [5]. Let

f

be an analytic function in an open set 9 containing σ

ϵ

(a). We study the relations between the sets

$$\sigma_{\epsilon}(\tilde{f}(a))\,\,\, and \,\,\,f((\sigma)_{\epsilon}(a)).$$

In general these two sets are different. We define functions

$$\phi(\epsilon), \psi(\epsilon)$$

(that take small values for small values of ϵ) and prove that

$$ f (\sigma_{\epsilon}(a))\subseteq \sigma_{\phi}(\epsilon)(\tilde{f}(a))\; \rm and \;\sigma_{\epsilon}(\tilde{f}(a))\subseteq f(\sigma_{\psi}(\epsilon)(a)).$$

The classical Spectral Mapping Theorem is shown as a special case of this result. We give estimates for these functions in some special cases and finally illustrate the results by numerical computations.

In this paper we study Toeplitz operators acting on the super Bergman space on the upper half-plane. We consider the super subgroups of isometries ℝ × S

1

and ℝ

+

× S

1

and we prove that the C

*

-algebras generated by Toeplitz operators whose symbols are invariant under the action of these groups are commutative.

We propose an algorithm for the efficient numerical computation of the periodic Hilbert transform. The function to be transformed is represented in a basis of spline wavelets in Sobolev spaces. The underlying grids have a hierarchical structure which is locally refined during computation according to the behavior of the involved functions. Under appropriate assumptions, we prove that the algorithm can deliver a result with prescribed accuracy. Several test examples illustrate how the method works in practice.

In the earlier papers [16, 19], the B

σ

-function spaces were introduced for the purpose of unifying central Morrey spaces, λ-central mean oscillation spaces and usual Morrey–Campanato spaces.

The purpose of this paper is to establish the B

σ

-Campanato estimates for commutators of Calderón-Zygmund operators on B

σ

-Morrey spaces.

For α >0 and

z

in the unit disk

D

the spaces of fractional Cauchy transforms

F

α

are known as the family of all functions

f

(

z

) such that

f

(

z

)=

$$\int_{T}[K(\overline{x}z)]^{\alpha}d\mu(x)$$

where

K

(

z

)=(1-

z

)

-1

is the Cauchy kernel,

T

is the unit circle and μ ∈

$$\mathcal{M}$$

the set of complex Borel measure on

T

. The Banach space

F

α

may be written as

F

α

=(

F

α

)

a

⊕ (

F

α

)

s

, where (

F

α

)

a

is isomorphic to a closed subspace of

$$\mathcal{M}_a$$

the subset of absolutely continuous measures of

$$\mathcal{M}$$

, and (

F

α

)

s

is isomorphic to

$$\mathcal{M}_s$$

the subspace of

$$\mathcal{M}$$

of singular measures. In this article we show that for α ≥1, the composition operator

C

φ

is compact on

K

α

C

φ

$$C_\varphi[K^{\alpha}(\overline{x}z)]\subset(F_{\alpha})_a$$

and in doing so, extend a result due to [1] who showed that

C

φ

is compact on

F

1

if and only if

C

φ

(

F

1

) ⊂ (

F

1

)

a

.

We study Hankel operators on the weighted Fock spaces F

p

γ

. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu’s characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.

The purpose of this paper is to further exemplify an approach to evolutionary problems originally developed in [3], [4] for a special case and extended to more general evolutionary problems, see [7], compare the survey article [5]. The ideas there are utilized for (1 + 1)-dimensional evolutionary problems, which in a particular case results in a hyperbolic partial differential equation with a Sturm–Liouville type spatial operator constrained by an impedance type boundary condition.

An introduction and survey is given of some recent work on the infinitesimal dynamics of

crystal frameworks

, that is, of translationally periodic discrete bond-node structures in ℝ

d

, for

d

= 2,3,... We discuss the rigidity matrix, a fundamental object from finite bar-joint framework theory, rigidity operators, matrix-function representations and low energy phonons. These phonons in material crystals, such as quartz and zeolites, are known as rigid unit modes, or RUMs, and are associated with the relative motions of rigid units, such as SiO

4

tetrahedra in the tetrahedral polyhedral bondnode model for quartz. We also introduce semi-infinite crystal frameworks, bi-crystal frameworks and associated multi-variable Toeplitz operators.

In this survey we collect and discuss some recent results on the so-called “Furstenberg set problem”, which in its classical form concerns the estimates of the Hausdorff dimension (dim

H

) of the sets in the F

α

-class: for a given α ∈ (0, 1], a set

E

⊆ ℝ

2

is in the F

α

-class if for each

e

∈ ?? there exists a unit line segment

l

e

in the direction of

e

such that dim

H

(

l

∩

E

) ≥ α. For α=1, this problem is essentially equivalent to the “Kakeya needle problem”. Define γ(α)= inf{dim

H

(

E

):

E

∈

F

α

}. The best-known results on γ(α) are the following inequalities:

max {1∕+α;2α}≤γ(α)≤(1+3α)∕2.

In this work we approach this problem from a more general point of view, in terms of a generalized Hausdorff measure

H

h

associated with the dimension function

h

. We define the class

F

h

of Furstenberg sets associated to a given dimension function h. The natural requirement for a set

E

to belong to

F

h

, is that

H

h

(

l

e

∩

E

) > 0 for each direction. We generalize the known results in terms of “logarithmic gaps” and obtain analogues to the estimates given above. Moreover, these analogues allow us to extend our results to the endpoint α = 0. For the upper bounds we exhibit an explicit construction of

F

h

-sets which are small enough. To that end we adapt and prove some results on Diophantine Approximation about the dimension of a set of “wellapproximable numbers”.

We also obtain results about the dimension of Furstenberg sets in the class

F

αβ

, defined analogously to the class

F

α

but only for a fractal set

L

⊂ ?? of directions such that dim

H

(L) ≥ β. We prove analogous inequalities reflecting the interplay between α and β. This problem is also studied in the general scenario of Hausdorff measures.

The boundary value problems for singular degenerate linear and regular degenerate nonlinear differential-operator equations are studied. We prove the well-posedeness of the linear problem and optimal regularity result for the nonlinear problem which occur in fluid mechanics, environmental engineering and in the atmospheric dispersion of pollutants.

We discuss the harmonic spheres conjecture, relating the space of harmonic maps of the Riemann sphere into the loop space of a compact Lie group

G

with the moduli space of Yang–Mills

G

-fields on four-dimensional Euclidean space.

The paper concerns frame multipliers when one of the involved sequences is a Riesz basis. We determine the cases when the multiplier is well defined and invertible, well defined and not invertible, respectively not well defined.

Suppose

K

y

and

K

x

are large sets of observed and reference signals, respectively, each containing

N

signals. Is it possible to construct a filter

F

:

K

y

→

K

x

that requires a priori information only on

few signals

,

p

<<

N

, from

K

x

but

performs better

than the known filters based on a priori information on

every

reference signal from

K

x

? It is shown that the positive answer is achievable under quite unrestrictive assumptions. The device behind the proposed method is based on a special extension of the piecewise linear interpolation technique to the case of random signal sets. The proposed technique provides a single filter to process any signal from the arbitrarily large signal set. The filter is determined in terms of pseudo-inverse matrices so that it always exists.

We consider second and third boundary conditions for the Sturm–Liouville eigenvalue problem with a generalized derivative of Cantor type function as a weight. The property of spectral periodicity of eigenvalues for some class of boundary conditions is established.

We study an abstract integro-differential equations with unbounded operator coefficients in Hilbert space. We obtain the expansion of the strong solutions of such type equations as the exponential series corresponding to the spectra of operator-functions which are the symbols of these equations.