Spectral Properties of Some Linear Matrix Differential Operators in Lp-spaces on $${\mathbb{R}}$$

A detailed study is made of matrix-valued, ordinary linear differential operators T in for 1 < p < ∞, which arise as the perturbation of a constant coefficient differential operator of order n ≥ 1 by a lower order differential operator which has a factorisation S = AB for suitable operators A and B. Via techniques from L ^p -harmonic analysis, perturbation theory and local spectral theory, it is shown that T satisfies certain local resolvent estimates, which imply the existence of local functional calculi and decomposability properties of T.      

Certain invariants of extension algebras of C({\mathbb{T}}^{2})

In this paper, we compute certain invariants of extension algebras of the torus algebra by , where is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space H. These extension algebras are also constructed up to isomorphism. Received: 5 July 2007, Revised: 14 February 2008     

On nonatomic submeasures on {\mathbb{N}}

A submeasure μ defined on the subsets of is nonatomic if for every ℓ ≥ 1 there exists a partition of into a finite number of parts on which μ is bounded from above by 1/ℓ. In this paper we answer several natural questions concerning nonatomic submeasures d _F that are determined (like the standard density) by a family F of finite subsets of . We first show that if the number of n-element sets in F grows at most exponentially with n, then d _F is nonatomic; but if this growth condition fails, then d _F need not be nonatomic in general. We next prove that, for a nonatomic submeasure d _F , the minimal number of sets in a 1/ℓ-small partition of can grow arbitrarily fast with ℓ. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type d _F . The second author acknowledges a generous support of the Foundation for Polish Science.     

A Spectrally Minimal Realization Formula for H^{\infty}({\mathbb{D}})

In this article we prove a representation theorem for functions, such that the realization formula is spectrally minimal in the following sense: the spectrum of the main operator in the realization intersects the unit circle precisely at those points where the given function has no holomorphic extension. We also extend this result to operator-valued H ^∞ functions. Received: March 12, 2007. Revised: April 12, 2007. Accepted: May 7, 2007.     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem

We survey various increasingly more general operator-theoretic formulations of generalized left-tangential Nevanlinna-Pick interpolation for Schur multipliers on the Drury-Arveson space. An adaptation of the methods of Potapov and Dym leads to a chain-matrix linear-fractional parametrization for the set of all solutions for all but the last of the formulations for the case where the Pick operator is invertible. The last formulation is a multivariable analogue of the Abstract Interpolation Problem formulated by Katsnelson, Kheifets and Yuditskii for the single-variable case; we obtain a Redheffer-type linear-fractional parametrization for the set of all solutions (including in degenerate cases) via an adaptation of ideas of Arov and Grossman.      

$${\mathcal{C}}_{0}$$ ( X)-algebras,stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras

We study permanence properties of the classes of stable and so-called -stable -algebras, respectively. More precisely, we show that a (X)-algebra A is stable if all its fibres are, provided that the underlying compact metrizable space X has finite covering dimension or that the Cuntz semigroup of A is almost unperforated (a condition which is automatically satisfied for -algebras absorbing the Jiang–Su algebra tensorially). Furthermore, we prove that if is a K ₁-injective strongly self-absorbing -algebra, then A absorbs tensorially if and only if all its fibres do, again provided that X is finite-dimensional. This latter statement generalizes results of Blanchard and Kirchberg. We also show that the condition on the dimension of X cannot be dropped. Along the way, we obtain a useful characterization of when a -algebra with weakly unperforated Cuntz semigroup is stable, which allows us to show that stability passes to extensions of -absorbing -algebras. Research supported by: Deutsche Forschungsgemeinschaft (through the SFB 478), by the EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280), and by the Center for Advanced Studies in Mathematics at Ben-Gurion University     

Complemented ideals in A({\mathbb{R}}^{d}) of algebraic curves

It is shown that for an algebraic curve the ideal of real analytic functions vanishing on X is complemented in if and only if in every a ∈ X every irreducible component of the germ X_a is either regular or a point. Received: 5 January 2009     

Multiplicity one theorem for ({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))

Let F be either or . Consider the standard embedding and the action of GL_n(F) on GL_n+1(F) by conjugation. We show that any GL_n(F)-invariant distribution on GL_n+1(F) is invariant with respect to transposition. We prove that this implies that for any irreducible admissible smooth Fréchet representations π of GL_n+1(F) and of GL_n(F),

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

The absolute Galois groups of finite extensions of $${\mathbb{R}}(t)$$

Let R be a real closed field and L be a finite extension of R(t). We prove that Gal(L) ≅ Gal(R(t)) if L is formally real and Gal(L) is the free profinite group of rank card (R) if L is not formally real. Received: 3 April 2007     

A duality property of an uncountable product of $${\mathbb{Z}}$$

We construct a quotient group of an uncountable product of integers which is not Pontryagin reflexive.     

The Fredholm Index of Locally Compact Band-dominated Operators on L^p \,(\user2{\mathbb{R}})

We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator A on and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of A. The results are applied to operators of convolution type with almost periodic symbol.     

On nonatomic submeasures on {\mathbb{N}}. III

In our earlier paper (Arch. Math. 91 (2008), 76–85), we proved that if F is a sequence of finite nonempty subsets of such that a certain quantity t(F) is finite, then the associated submeasure d_F on is nonatomic. In the present note, we give two curious characterizations of the set of such sequences F. The second author is partially supported by the Foundation for Polish Science.     

Syzygies of Multi-Variable Higher Spin Dirac Operators on $$\mathbb{R}^{3}$$

This paper is a short report on the generalization of some results of our previous paper [12] to the case of spin j/2 Dirac operators in real dimension three for arbitrary odd integer j. We use an explicit formula for the local expression of such operators to study their algebraic properties, construct the compatibility conditions of the overdetermined system associated to the operator in several spatial variables, and we prove that its associated algebraic complex, dual do the BGG sequence coming from representation theory, has substantially the same pattern as the Cauchy-Fueter complex. The author is a member of the Eduard Čech Center and his research is supported by the relative grants.     

A note on weak convergence in L^{1}_{\rm loc}({\mathbb{R}})

Let be a sequence of Borel measurable functions satisfying, for a function the inequalities

Integral Representations for Generalized Difference Kernels Having a Finite Number of Negative Squares

An integral representation is derived for matrix-valued generalized difference kernels which have a finite number of negative squares. The representation is used to extend such kernels to the real line with a bound on the number of negative squares. The main results are obtained by means of an operator interpolation theorem. The nondegenerate case is assumed.      

An L p -estimate for surfaces of constant mean curvature in {\mathbb{S}}^3

In this paper, we prove a one end theorem for complete noncompact Riemannian manifolds and apply it to complete noncompact stable minimal hypersurfaces. Received: 10 September 2007