On the numerical radius of operators in Lebesgue spaces

We show that the absolute numerical index of the space Lp(μ) is (where ). In other words, we prove that

     

Composition operators on Bergman spaces over the punctured plane

We study the spaces of holomorphic functions on the punctured plane which are square integrable with respect to a weight. We determine when these spaces are finite-dimensional and which are composition operators acting on them, studying their cyclicity and hypercyclicity properties.     

Weighted geometric discrepancies and numerical integration on reproducing kernel Hilbert spaces

We extend the notion of L₂-B-discrepancy introduced in [E. Novak, H. Wo?niakowski, L₂ discrepancy and multivariate integration, in: W.W.L. Chen, W.T. Gowers, H. Halberstam, W.M. Schmidt, and R.C. Vaughan (Eds.), Analytic Number Theory. Essays in Honour of Klaus Roth, Cambridge University Press, Cambridge, 2009, pp. 359-388] to what we shall call weighted geometric L₂-discrepancy. This extension enables us to consider weights in order to moderate the importance of different groups of variables, as well as to consider volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of Euclidean spaces.We relate the weighted geometric L₂-discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wo?niakowski.Furthermore, we prove an upper bound for the numerical integration error for cubature formulas that use admissible sample points. The set of admissible sample points may actually be a subset of the integration domain of measure zero. We illustrate that particularly in infinite-dimensional numerical integration it is crucial to distinguish between the whole integration domain and the set of those sample points that actually can be used by the algorithms.     

Multicyclicity of unbounded normal operators and polynomial approximation in C

A remarkable and much cited result of Bram [J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75-94] shows that a star-cyclic bounded normal operator in a separable Hilbert space has a cyclic vector. If, in addition, the operator is multiplication by the variable in a space L²(m) (not only unitarily equivalent to it), then it has a cyclic vector in L^∞(m). We extend Bram's result to the case of a general unbounded normal operator, implying by this that the (classical) multiplicity and the multicyclicity of the operator (cf. [N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002]) coincide. It follows that if m is a sigma-finite Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L²(C,τ) is the norm-closure of the polynomials in z.     

Some estimates for Bochner-Riesz operators on the weighted Herz-type Hardy spaces

In this paper, by using the atomic decomposition and molecular characterization of the homogeneous and non-homogeneous weighted Herz-type Hardy spaces , we obtain some weighted boundedness properties of the Bochner-Riesz operator and the maximal Bochner-Riesz operator on these spaces for α=n(1/p−1/q), 0<p?1 and 1<q<∞.     

On binormality in non-separable Banach spaces

We study binormality, a separation property of the norm and weak topologies of a Banach space. We show that every Banach space which belongs to a P-class is binormal. We also show that the asplundness of a Banach space is equivalent to a related separation property of its dual space.     

On certain spaces of vector measures of bounded variation

If (Σ,X) is a measurable space and X a Banach space we investigate the X-inheritance of copies of ?_∞ in certain subspaces Δ(Σ,X) of bvca(Σ,X), the Banach space of all X-valued countable additive measures of bounded variation equipped with the variation norm. Among the consequences of our main theorem we get a theorem of J. Mendoza on the X-inheritance of copies of ?_∞ in the Bochner space L₁(μ,X) and other of the author on the X-inheritance of copies of ?_∞ in bvca(Σ,X).     

Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures

Under the assumption that μ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón-Zygmund operators or fractional integrals with RBMO(μ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(μ) functions, of the Hardy space H¹(μ) of Tolsa.     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

On k-networks and weak bases for spaces

In this paper, we answer some problems on k-networks and weak bases posed by Liu and Tanaka (2007) [19].     

The inclusion relation between Sobolev and modulation spaces

The inclusion relations between the Lp-Sobolev spaces and the modulation spaces is determined explicitly. As an application, mapping properties of unimodular Fourier multiplier eiα|D| between Lp-Sobolev spaces and modulation spaces are discussed.     

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.     

Spectral properties and their applications of frequency response operators in linear continuous-time periodic systems

Spectral properties related to the frequency response operators of finite-dimensional linear continuous-time periodic (FDLCP) systems are examined rigorously and thoroughly in the paper. As applications of the spectral features, positive realness of FDLCP systems is scrutinized and then a harmonic Hamiltonian criterion is derived for the H_∞ norm of FDLCP systems. In particular, positive realness of FDLCP systems is interpreted in term of Toeplitz operators for the first time in this study, together with a testing algorithm. Deriving the harmonic Hamiltonian criterion with a frequency-domain approach bridges the spectra of the frequency response operators in FDLCP systems with their time-domain behaviors.     

Hardy spaces for the strip

In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H²(S)) basis of polynomials.     

On the stability of the linear mapping in Banach modules

We prove the generalized Hyers-Ulam-Rassias stability of the linear mapping in Banach modules over a unital Banach algebra.     

Workshop lecture on products of Fréchet spaces

The general question, “When is the product of Fréchet spaces Fréchet?” really depends on the questions of when a product of α₄ Fréchet spaces (also known as strongly Fréchet or countably bisequential spaces) is α₄, and when it is Fréchet. Two subclasses of the class of strongly Fréchet spaces shed much light on these questions. These are the class of α₃ Fréchet spaces and its subclass of ℵ₀-bisequential spaces. The latter is closed under countable products, the former not even under finite products. A number of fundamental results and open problems are recalled, some further highlighting the difference between being α₃ and Fréchet and being ℵ₀-bisequential.     

On the perturbations of spectra of upper triangular operator matrices

In this paper we investigate perturbations of the regular spectrum of an upper triangular operator matrix such as acting on a Hilbert space H⊕K.     

Recursive approximation of the high dimensional max function

This paper proposes a smoothing method for the general n-dimensional max function, based on a recursive extension of smoothing functions for the two-dimensional max function. A theoretical framework is introduced, and some applications are discussed. Finally, a numerical comparison with a well-known smoothing method is presented.     

On the normalized eigenvalue problems for nonlinear elliptic operators II

This paper continues our previous research on the following form of normalized eigenvalue problem