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Summer School Lecture Notes


Subalgebras of Graph C*-algebras

I give a self-contained introduction to two novel classes of nonselfadjoint operator algebras, namely the generalised analytic Toeplitz algebras L G , associated with the “Fock space” of a graph G, and subalgebras of graph C*-algebras. These two topics are somewhat independent but in both cases I shall focus on fundamental techniques and problems related to classifying isomorphism types and to the recovery of underlying foundational structures, be they graphs or groupoids.
Stephen C. Power

C*-algebras and Asymptotic Spectral Theory

The presented material is a slightly polished and extended version of lectures given at Lisbon, WOAT 2006. Three basic topics of numerical functional analysis are discussed: stability, fractality, and Fredholmness. It is further shown that these notions are corner stones in order to understand a few topics in asymptotic spectral theory: asymptotic behavior of singular values, ε-pseudospectra, norms. Four important examples are discussed: Finite sections of quasidiagonal operators, Toeplitz operators, band-dominated operators with almost periodic coefficients, and general band-dominated operators. The elementary theory of C*-algebras serves as the natural background of these topics.
Bernd Silbermann

Toeplitz Operator Algebras and Complex Analysis

The aim of this survey article is to present the recent work concerning Hilbert spaces of holomorphic functions on hermitian symmetric domains of arbitrary rank and dimension, in relation to operator theory (Toeplitz C*-algebras and their representations), harmonic analysis (discrete series of semi-simple Lie groups) and quantization (covariant functional calculi and Berezin transformation).
Harald Upmeier

Workshop Contributed Articles


Rotation Algebras and Continued Fractions

This paper discusses two problems related with the approximation of rotation algebras: (i) estimating the norm of almost Mathieu operators and (ii) studying a certain AF algebra associated with the continued fraction algorithm. The Effros-Shen AF algebras naturally arise as primitive quotients of this algebra.
Florin P. Boca

On the Fredholm Index of Matrix Wiener-Hopf plus/minus Hankel Operators with Semi-almost Periodic Symbols

Conditions for the Fredholm property of Wiener-Hopf plus/minus Hankel operators with semi-almost periodic Fourier matrix symbols are exhibited. Under such conditions, a formula for the sum of the Fredholm indices of these Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators is derived. Concrete examples are worked out in view of the computation of the Fredholm indices.
Giorgi Bogveradze, Luís P. Castro

Diffraction by a Strip and by a Half-plane with Variable Face Impedances

A study is presented for boundary value problems arising from the wave diffraction theory and involving variable impedance conditions. Two different geometrical situations are considered: the diffraction by a strip and by a half-plane. In the first case, both situations of real and complex wave numbers are analyzed, and in the second case only the complex wave number case is considered. At the end, conditions are founded for the well-posedness of the problems in Bessel potential space settings. These conditions depend on the wave numbers and the impedance properties.
Luís P. Castro, David Kapanadze

Factorization Algorithm for Some Special Matrix Functions

We will see that it is possible to construct an algorithm that allows us to determine an effective factorization of some matrix functions. For those matrix functions it is shown that its explicit factorization can be obtained through the solutions of two non-homogeneous equations.
Ana C. Conceição, Viktor G. Kravchenko

On a Radon Transform

In this article a special type of Radon transform (Kipriyanov-Radon transform K γ ) is considered and some properties of this transform are proved. The main results of this work are the inversion formulas of K γ , which were obtained with a help of general B-hypersingular integrals.
Ekaterina Gots, Lev Lyakhov

Extensions of σ-C* -algebras

Let A be a σ-C*-algebra. The bounded part b(A) of A introduced by Konrad Schmüdgen in [4] is a C*-algebra for some C*-norm. We shall show that if A is a split extension of a σ-C*-algebra B by a closed two-sided ideal I then b(A) will be a split extension of the C*-algebra b(B) by the closed two-sided b(I). A number of results concerning the bounded part of a σ-C*-algebra are established.
Rachid El Harti

Higher-order Asymptotic Formulas for Toeplitz Matrices with Symbols in Generalized Hölder Spaces

We prove higher-order asymptotic formulas for determinants and traces of finite block Toeplitz matrices generated by matrix functions belonging to generalized Hölder spaces with characteristic functions from the Bari-Stechkin class. We follow the approach of Böttcher and Silbermann and generalize their results for symbols in standard Hölder spaces.
Alexei Yu. Karlovich

Nonlocal Singular Integral Operators with Slowly Oscillating Data

The paper is devoted to studying the Fredholmness of (nonlocal) singular integral operators with shifts N = (Σ a g + V g )P + + (Σa g V g )P on weighted Lebesgue spaces L p (Γ,w) where 1 < p < ∞, Γ is an unbounded slowly oscillating Carleson curve, w is a slowly oscillating Muckenhoupt weight, the operators P ± = 1/2 (I ± S Γ) are related to the Cauchy singular integral operator SΓ, a g ± are slowly oscillating coefficients, V g are shift operators given by V g f = f o g, and g are slowly oscillating shifts in a finite subset of a subexponential group G acting topologically freely on Γ. The Fredholm criterion for N consists of two parts: of an invertibility criterion for polynomial functional operators A ± = Σa g ± V g in terms of invertibility of corresponding discrete operators on the space l p (G), and of a condition of local Fredholmness of N at the endpoints of Γ established by applying Mellin pseudodifferential operators with compound slowly oscillating V (ℝ)-valued symbols where V (ℝ) is the Banach algebra of absolutely continuous functions of bounded total variation on ℝ.
Yuri I. Karlovich

Poly-Bergman Projections and Orthogonal Decompositions of L 2-spaces Over Bounded Domains

The paper is devoted to obtaining explicit representations of poly-Bergman and anti-poly-Bergman projections in terms of the singular integral operators S D and S D * on the unit disk D, studying relations between different true poly-Bergman and true anti-poly-Bergman spaces on the unit disk that are realized by the operators S D and S D * , establishing two new orthogonal decompositions of the space L 2(U, dA) (in terms of poly-Bergman and anti-poly-Bergman spaces) for an arbitrary bounded open set U ⊂ ℂ with the Lebesgue area measure dA, considering violation of Dzhuraev’s formulas and establishing explicit forms of the Bergman and anti-Bergman projections for several open sectors.
Yuri I. Karlovich, Luís V. Pessoa

Vekua’s Generalized Singular Integral on Carleson Curves in Weighted Variable Lebesgue Spaces

For a Carleson curve Γ we establish the boundedness, in weighted Lebesgue spaces L p(·)(Γ, ϱ) with variable exponent p(·), of the generalized singular integral operator which arises in the theory of I.N.Vekua generalized analytic functions. The obtained result is an extension of the known results even in the case of constant p. We also show that Vekua’s generalized singular integral exists a.e. for fL 1(Γ) on an arbitrary rectifiable curve.
Vakhtang Kokilashvili, Stefan Samko

On Homotopical Non-invertibility of C*-extensions

We have presented recently an example of a C*-extension, which is not invertible in the semigroup of homotopy classes of C*-extensions. Here we reveal the cause for existence of homotopy non-invertible C*-extensions: it is related to non-exact C*-algebras and to possibility to distinguish different tensor C*-norms by K-theory. We construct a special C*-algebra, K-theory of which hosts an obstruction for homotopical non-invertibility, and show that this obstruction for our example does not vanish.
Vladimir Manuilov

Galois-fixed Points and K-theory for GL(n)

Let F be a nonarchimedean local field and let G = GL(n) = GL(n, F). Let E/F be a finite Galois extension. We use the Hasse-Herbrand function ψ E/F to identify the K-theory groups of the reduced C*-algebra C* r GL(n, F) with the Galois-fixed points of the K-theory groups of the reduced C*-algebra C* r GL(n, E).
Sérgio Mendes

Spectral Factorization, Unstable Canonical Factorization, and Open Factorization Problems in Control Theory

The spectral or canonical factorization of matrix- or operator-valued function F defined on the imaginary axis is defined as F = Y *X, where Y ±1,X ±1 are H (bounded and holomorphic on Rez > 0), or, more generally, Y ±1,X ±1 belong to some weighted strong H2 space.
It is well known that the invertibility of the corresponding Toeplitz operator P + FP 0+ is necessary for this factorization to exist, where P + : L2 → H2 is the orthogonal projection. When F is positive, this condition is also sufficient for the factors to be H. In the general (indefinite) case, this is not so. However, if F is smooth enough, then the H canonical factorization does exist even in the indefinite case; we give a solution assuming that F is the Fourier transform of a measure with no singular continuous part.
If the (Popov function determined by the) transfer function of a control system has a canonical factorization, then a well-posed optimal state feedback exists for the corresponding control problem. Conversely, a well-posed optimal state feedback determines a canonical factorization of the transfer function. We generalize this to unstable systems, i.e., to transfer functions that are holomorphic and bounded on some right half-plane | Rez > r.
Then we show that if the generalized Popov Toeplitz operator is uniformly positive, then the canonical factorization exists (the stable case is well known). However, the results on the regularity of the factors and in the nonpositive case remain very few — we explain them and the remaining open problems.
Kalle M. Mikkola, Ilya M. Spitkovsky

Compact Linear Operators Between Probabilistic Normed Spaces

A pair (X, N) is said to be a probabilistic normed space if X is a real vector space, N is a mapping from X into the set of all distribution functions (for xX, the distribution function N(x) is denoted by N x , and N x (t) is the value N x at t ∈ ℝ) satisfying the following conditions:
N x (0) = 0
N x (t) = 1 for all t > 0 iff x = 0
N αx (t) = N x \( (\frac{t} {{|\alpha |}})\) for all α ∈ ℝ∖0,
N x+y (s + t) ≥ min N x (s), N y (t) for all x, yX, and s, t ∈ ℝ0 +.
In this article, we study compact linear operators between probabilistic normed spaces.
Kourosh Nourouzi

Essential Spectra of Pseudodifferential Operators and Exponential Decay of Their Solutions. Applications to Schrödinger Operators

The aim of this paper is to study relations between the location of the essential spectrum and the exponential decay of eigenfunctions of pseudodifferential operators on L p (ℝ n ) perturbed by singular potentials.
Our approach to this problem is via the limit operators method. This method associates with each band-dominated operator A a family op(A) of so-called limit operators which reflect the properties of A at infinity. Consider the compactification of ℝn by the “infinitely distant” sphere S n−1. Then the set op(A) can be written as the union of its components op ηω (A) where ω runs through the points of S n−1 and where op ηω (A) collects all limit operators of A which reflect the properties of A if one tends to infinity “in the direction of ω”. Set \( sp_{n_\omega } A: = \cup _{A_h \in op_{\eta \omega } (A)} spA_h \).
We show that the distance of an eigenvalue λsp ess A to sp ηω A determines the exponential decay of the λ-eigenfunctions of A in the direction of ω. We apply these results to estimate the exponential decay of eigenfunctions of electro-magnetic Schrödinger operators for a large class of electric potentials, in particular, for multiparticle Schrödinger operators and periodic Schrödinger operators perturbed by slowly oscillating at infinity potentials.
Vladimir S. Rabinovich, Steffen Roch

On Finite Sections of Band-dominated Operators

In an earlier paper we showed that the sequence of the finite sections P n AP n of a band-dominated operator A on l p (ℤ) is stable if and only if the operator A is invertible, every limit operator of the sequence (P n AP n ) is invertible, and if the norms of the inverses of the limit operators are uniformly bounded. The purpose of this short note is to show that the uniform boundedness condition is redundant.
Vladimir S. Rabinovich, Steffen Roch, Bernd Silbermann

Characterization of the Range of One-dimensional Fractional Integration in the Space with Variable Exponent

Within the frameworks of weighted Lebesgue spaces with variable exponent, we give a characterization of the range of the one-dimensional Riemann-Liouville fractional integral operator in terms of convergence of the corresponding hypersingular integrals. We also show that this range coincides with the weighted Sobolev-type space L α, p(·)[(a, b)ϱ].
Humberto Rafeiro, Stefan Samko

Orbit Representations and Circle Maps

We yield C*-algebras representations on the orbit spaces from the family of interval maps f(x) = βx+α (mod 1) lifted to circle maps, in which case β ∈ N.
Each orbit will encode an unitary equivalence class of an irreducible representation of: a Cuntz algebra O β if = 0 and β > 1; an irrational rotation algebra A β if α ∉ ℚ and β = 1; and a Cuntz-Krieger O Aα,β whenever β > 1 and the critical point is periodic, where A α,β is the underlying Markov transition matrix of f.
Carlos Correia Ramos, Nuno Martins, Paulo R. Pinto

On Generalized Spherical Fractional Integration Operators in Weighted Generalized Hölder Spaces on the Unit Sphere

For spherical convolution operators with a given power type asymptotic of their Fourier-Laplace multiplier we prove a statement on the boundedness within the framework of weighted generalized Hölder spaces on the unit sphere. The result obtained explicitly shows how spherical convolution operators under consideration improve the behavior of the continuity modulus of functions.
Natasha Samko, Boris Vakulov
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