Hadamard Multipliers Research Articles

Cesaro summability of Taylor series in higher order weighted Dirichlet-type spaces

For a positive integer \(m\) and a finite non-negative Borel measure \(\mu\) on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces \(\mathcal H_{\mu, m}\). We show that if \(\alpha\gt\frac{1}{2}\), then for any \(f\) in \(\mathcal H_{\mu, m}\) the sequence of generalized Cesaro sums \(\{\sigma_n^{\alpha}[f]\}\) converges to \(f\). We further show that if \(\alpha=\frac{1}{2}\) then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer \(m\).

Hadamard Multipliers on Weighted Dirichlet Spaces

The Hadamard product of two power series is obtained by multiplying them coefficientwise. In this paper we characterize those power series that act as Hadamard multipliers on all weighted Dirichlet spaces on the disk with superharmonic weights, and we obtain sharp estimates on the corresponding multiplier norms. Applications include an analogue of Fej\'er's theorem in these spaces, and a new estimate for the weighted Dirichlet integrals of dilates.

Semigroups of Hadamard multipliers on the space of real analytic functions

An operator $M$ acting on the space of real analytic functions is called a multiplier if every monomial is its eigenvector. In this paper we state some results concerning the problem of generating strongly continuous semigroups by multipliers. In particular we show when the Euler differential operator of finite order is a generator and when it is not.

Hadamard Multipliers on Spaces of Holomorphic Functions

Several representation theorems of multipliers are derived: in terms of analytic functionals, and germs of holomorphic functions. The co-induced topology via the representation theorems is discussed. As an application of the representation theorems the density of non-invertible multipliers is proved. Moreover, Euler differential operators are distinguished among all multipliers.

Interpolation of holomorphic functions and surjectivity of Taylor coefficient multipliers

We prove criteria for global analytic solvability for some Euler type partial differential equations — a class of linear partial differential equations with variable (polynomial) coefficients. This is based on a very general Mellin type theory developed in the present paper which is valid for analytic functionals (in particular, distributions of compact support) substituting the classical Mellin transform. As a consequence, we get a characterization of moment sequences of analytic functionals by interpolation of holomorphic functions satisfying some restrictive growth conditions. This allows to study the so-called Hadamard type multipliers on real analytic functions of several variables, i.e., operators for which monomials are eigenvectors. We characterize multiplier sequences (i.e., sequences of eigenvalues of multipliers) by interpolation and describe the image of some multipliers, especially, the image of the principal examples of Hadamard multipliers: Euler type partial differential operators. We also get some results showing that the image remains unchanged after some perturbation of the original Hadamard type operator.

Hadamard Multipliers and Abel Dual of Hardy Spaces

The paper is devoted to the study of Hadamard multipliers of functions from the abstract Hardy classes generated by rearrangement invariant spaces. In particular the relation between the existence of such multiplier and the boundedness of the appropriate convolution operator on spaces of measurable functions is presented. As an application, the description of Hadamard multipliers intoH∞is given and the Abel type theorem for mentioned Hardy spaces is proved.

New Pólya–Schoenberg type theorems

In this paper the theory of Hadamard product multipliers is extended from the unit disk in the complex plane to arbitrary so-called disk-like domains, i.e. such domains which are the union of disks or half-planes, all containing the origin. In such a domain, say Ω, we define (the class R α d ( Ω ) of) generalized prestarlike functions of order α ⩽ 1 and ask for Hadamard multipliers g analytic at z = 0 for which f ∈ R α d ( Ω ) implies g ∗ f ∈ R α d ( Ω ) . We prove that such a multiplier necessarily has to be analytic in Ω ∗ : = { u v : u ∈ Ω , v ∈ C ∖ Ω } . In many cases (we prove this for all proper disks containing the origin) we actually find that R α d ( Ω ∗ ) is the precise description of the set of all such multipliers. For these disks, Ω γ say, the domains Ω γ ∗ turn out to be bounded by the outer loops of certain Limaçons of Pascal. The parameter γ is related to the characteristic q ( Ω γ ) = ( 1 − γ ) / ( 1 + γ ) : = r / s of the disk, where r is the shortest distance of the origin to the boundary of that disk, and s the largest. Large subclasses of R α d ( Ω ∗ ) are being explicitly determined. For the case γ = 0 , i.e. Ω γ = Ω γ ∗ = D , this result coincides with an old one by Ruscheweyh and Sheil-Small, previously conjectured by G. Pólya and I.J. Schoenberg. The notion of the characteristic of a disk (containing the origin) is then extended to general disk-like domains, and some multipliers are identified for those general classes R α d ( Ω ) . The previously determined class of ‘universally prestarlike functions’, defined in the slit-domain C ∖ [ 1 , ∞ ] , is identified as the class of ‘universal multipliers’ for R α d ( Ω ) in any disk-like domain Ω.

Universally prestarlike functions as convolution multipliers

Universally prestarlike functions (of order α ≤ 1) in the slit domain $${\Lambda:=\mathbb{C}{\setminus}[1,\infty]}$$ have recently been introduced in Ruscheweyh et al. (Israel J Math, to appear). This notation generalizes the corresponding one for functions in the unit disk $${\mathbb{D}}$$ (and other circular domains in $${\mathbb{C}}$$ ). In this paper we study the behaviour of universally prestarlike functions under the Hadamard product. In particular it is shown that these function classes (with α fixed), are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.

Maximal spectral distance

The exact upper bound for the norm distance between two normal matrices is given as a function of their eigenvalues. Among the applications, we obtain an inequality related to the Cartesian decomposition of a matrix and the evaluation of the Hadamard multiplier norms for matrices in a certain broad class.

Finding norms of Hadamard multipliers

The norm of a matrix B as a Hadamard multiplier is the norm of the map X at X · B, where is the Hadamard or entrywise product of matrices. Watson proposed an algorithm for finding lower bounds for the Hadamard multiplier norm of a matrix. It is shown how Watson's algorithm can be used to give upper bounds as well, which, in many cases, yield the Hadamard multiplier norm to any desired accuracy. A sharp form of Wittstock's decomposition theorem is proved for the special case of Hadamard multiplication.

Geometry and the norms of Hadamard multipliers

The norm of a matrix B as a Hadamard multiplier is the norm of the map X → X • B where • is the Hadamard or entrywise product of matrices. We give a geometric interpretation to a criterion of Haagerup and use this interpretation to find an explicit formula for the Hadamard multiplier norms of real 2 × 2 matrices. For Hermitian matrices with one positive eigenvalue, we give necessary and sufficient conditions for the norm to be the magnitude of the largest entry.

Norms of Hadamard Multipliers

For B a fixed matrix, the authors consider the problem of finding the norm of the map $X \mapsto X \bullet B$, where $ \bullet $ is the Hadamard or entrywise product of matrices and the norm of a matrix is its spectral norm. Using techniques from the theory of Kren spaces, the problem is rewritten for Hermitian matrices as a minimization problem whose solution, for small matrices, can be obtained from standard optimization software. The Hadamard multiplier norm for an arbitrary matrix is given in terms of a Hermitian extension. The results are applied to refute a conjecture of R. V. McEachin concerning the value of a constant in an operator inequality.

Triangular truncation and finding the norm of a Hadamard multiplier

For B a fixed matrix, we study the problem of using a criterion of Haagerup to find the norm of the map A ↦ A · B , where · is the Hadamard or entrywise product of matrices. The techniques developed are applied to triangular truncation, and it is proved that if K n is the norm of triangular truncation of n × n matrices, then K n log n →π −1 .

An application of Hadamard multiplication to operators on weighted Hardy spaces

The monomials z n form an orthogonal basis for the weighted Hardy Hilbert spaces. It is shown that versions of some natural operators defined on different weighted Hardy spaces are related (in the corresponding orthonormal bases) by a Hadamard product map. Cases in which the Hadamard multiplier is positive are established, and the results used to derive new norm and spectral information for operators on the weighted spaces from well-known information about operators on the usual (unweighted) Hardy space.