On The Exact Norm Value Op Some Bmoa Functions

In this paper we construct two large classes of BMOA functions with BMOA norm equal to the H² norm, respecting equal to the H^∞ norm. We also determine all functions in the disc algebra with BMOA norm equal to the Bloch norm.     

The Szeg? Metric Associated to Hardy Spaces of Clifford Algebra Valued Functions and Some Geometric Properties

In analogy to complex function theory we introduce a Szeg? metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in \mathbbR^m+1{\mathbb{R}^{m+1}} . These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szeg? metric turns out to have a pseudo-invariance under M?bius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.     

Uniform Approximation Properties for Spaces of Analytic Functions

We prove that the Banach space of bounded analytic functions and the disc algebra have the uniform bounded approximation property of order p for 1 < p < ∞.     

Norm Estimate of the Cauchy Transform on L P (Ω)

We give two sided estimates of the norm of Cauchy transform on L^p(Ω)(1 ≤ p < ∞) in the case when Ω is the unit disc or the bounded simply connected domain with piecewise C¹ boundary. As a consequence we get the better constant in Poincare inequality. Also, we conjecture the exact value norm of Cauchy transform on L^p(D), where D is the unit disc.     

The topological center of the spectrum of some distal algebras

The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions , is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map , to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra W _k , the norm closure of the algebra generated by the set of functions , will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.     

Best simultaneous approximation in L∞(μ, X)

Let Y be a reflexive subspace of the Banach space X, let (Ω, Σ, μ) be a finite measure space, and let L_∞(μ, X) be the Banach space of all essentially bounded μ ‐Bochner integrable functions on Ω with values in X, endowed with its usual norm. Let us suppose that Σ₀ is a sub‐σ ‐algebra of Σ, and let μ₀ be the restriction of μ to Σ₀. Given a natural number n, let N be a monotonous norm in ?ⁿ . We prove that L_∞(μ, Y) is N ‐simultaneously proximinal in L_∞(μ,X), and that if X is reflexive then L_∞(μ₀, X) is N ‐simultaneously proximinal in L_∞(μ, X) in the sense of Fathi, Hussein, and Khalil [3]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on Z ^N with symbols which are bounded on Z ^N ×T ^N together with their derivatives with respect to the second variable. In the same way as partial differential operators on R ^N are included in an algebra of pseudodifferential operators, difference operators on Z ^N are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces l _w ^p (Z ^N ) and to Phragmen–Lindelöf type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrödinger operators and the decay of their eigenfunctions at infinity.     

Norms of composition operators on weighted Hardy spaces

The norm of a bounded composition operator induced by a disc automorphism is estimated on weighted Hardy spaces H ²(β) in which the classical Hardy space is continuously embedded. The estimate obtained is accurate in the sense that it provides the exact norm for particular instances of the sequence β. As a by-product of our results, an estimate for the norm of any bounded composition operator on H ²(β) is obtained.     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA ²(C ^N),N≧1.     

On multicyclic operators and the Vasjunin-Nikol'ski206-1206-1206-1discotheca

If A is a bounded linear multicyclic operator acting on a complex Banach spaceX, then thedisc of A is defined by: disc A = sup(R ∈ Cyc A) min{dimR′: R′ ? R, R′ ∈ Cyc A}, where Cyc A denotes the family of all finite dimensional subspacesR ofX such that X = (R+AR+A ² R+?)^?. It is shown that if the set {λ ∈ ?: dim ker (λ-A)^* ≥ n} has nonempty interior (in particular, if A is a Fredholm operator of index -n), then disc A ≥ n+1. This result affirmatively answers a question of V.I. Vasjunin and N.K. Nikol'skiï. In the case whenX is a Hilbert space, it is shown that the set of all operators A such that A is n-multicyclic, but disc A =∞, is dense in the set of all n-multicyclic operators. If M_λ = "multiplication by λ" acting on the disk algebra (and many other spaces of continuous and/or analytic functions), then M_λ is cyclic, but disc M_λ = ∞. However, the analogous result is false if the disk algebra is replaced by the algebra of functions analytic on the disk and smooth on the boundary, or algebras of Lipschitz functions. If T is a multicyclic unicellular operator, then T is cyclic and disc T=1.     

Bounds on Supremum Norms for Hecke Eigenfunctions of Quantized Cat Maps

We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are ≪_ε N^ε for all ε > 0 in the case of N square free. Submitted: March 6, 2006; Accepted: April 30, 2006     

Homogeneous polynomials on the ball and polynomial bases

It is shown that the spaces of homogenous polynomials on the complex 2-ball are uniformly isomorphic tol ^∞-spaces. The argument is based on explicit constructions and the decomposition method. A new construction is given of bases in the spaceA _N of monomials 1,z,z ², ...,z ^N−1 on the disc (due to Bochkarev [Boc]). Also using decomposition methods, the existence of a base in the ball algebra is obtained.     

Coherence of some rings of functions

To say that a commutative ring $\text{R}$ with unit is coherent amounts to saying, in case $\text{R}$ has no divisors of zero, that the intersection of two finitely generated ideals in $\text{R}$ is finitely generated. We prove that the ring H^∞ of bounded analytic functions in the unit disc is coherent, while the disc algebra A is not coherent. For any positive measure μ, L^∞(μ) is coherent.     

On the Generalized Cesáro Summability of Trigonometric Fourier Series

The class of generalized bounded variation is considered. For functions from this class, the deviation of the generalized Cesáro means of negative order in the norm of L^r, (1 ≤ r ≤ ∞) is estimated.

     

Singularities in derived categories

Let A be a finite dimensional algebra over an algebraically closed field k and let M and N be two complexes in the bounded derived category D^b(A) of finitely generated A-modules. Together with Alexander Zimmermann we have defined a notion of degeneration for derived categories. We say that M degenerates to N if there is a complex Z and an exact triangle N→M⊕Z→Z→N[1]. In this paper we define and study the type of singularity at every degeneration in the bounded derived categrory.     

The inclusion of the Schur algebra in B(?2) is not inverse-closed

The Schur algebra is the algebra of operators which are bounded on ℓ ¹ and on ℓ ^∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ℓ ², and whose inverse is not bounded on ℓ ¹ nor on ℓ ^∞. In particular, this shows that the Schur algebra is not inverse-closed.     

Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C^∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C^∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier.     

Pre-dual of the Function Algebra A ?��(D) and Representation of Functions in Dirichlet Series

In this paper we describe, via the Laplace transformation of analytic functionals, a pre-dual to the function algebra A ^−∞(D) (D being either a bounded C ²-smooth convex domain in ${\mathbb{C}^N (N > 1)}${\mathbb{C}^N (N > 1)} , or a bounded convex domain in \mathbbC{\mathbb{C}}) as a space of entire functions with certain growth. A possibility of representation of functions from the pre-dual space in a form of Dirichlet series with frequencies from D, is also studied.     

Sufficient conditions for the projective freeness of Banach algebras

Let R be a unital semi-simple commutative complex Banach algebra, and let M(R) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M(R) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H^∞(X) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.     

One class of C*-algebras generated by a family of partial isometries and multiplicators

We consider a C*-subalgebra of the algebra of all bounded operators on the Hilbert space of square-summable functions defined on some countable set. The algebra under consideration is generated by a family of partial isometries and the multiplier algebra isomorphic to the algebra of all bounded functions defined on the mentioned set. The partial isometry operators satisfy correlations defined by a prescribed map on the set. We show that the considered algebra is ?-graduated. After that we construct the conditional expectation from the latter onto the subalgebra responding to zero. Using this conditional expectation, we prove that the algebra under consideration is nuclear.