Multiplier Norm Research Articles

Predual of the Multiplier Algebra of Ap(G) and Amenability

AbstractFor a locally compact group G and 1 < p < ∞, let Ap(G) be the Herz-Figà-Talamanca algebra and let PMp(G) be its dual Banach space. For a Banach Ap(G)-module X of PMp(G), we prove that the multiplier space ℳ(Ap(G); X*) is the dual Banach space of QX, where QX is the norm closure of the linear span Ap(G)X of u f for u 2 Ap(G) and f ∈ X in the dual of ℳ(Ap(G); X*). If p = 2 and PFp(G) ⊆ X, then Ap(G)X is closed in X if and only if G is amenable. In particular, we prove that the multiplier algebra MAp(G) of Ap(G) is the dual of Q, where Q is the completion of L1(G) in the ‖ · ‖M-norm. Q is characterized by the following: f ∈ Q if an only if there are ui ∈ Ap(G) and fi ∈ PFp(G) (i = 1; 2, … ) with such that on MAp(G). It is also proved that if Ap(G) is dense in MAp(G) in the associated w*-topology, then the multiplier norm and ‖ · ‖Ap(G)-norm are equivalent on Ap(G) if and only if G is amenable.

Norms of Multipliers and Best Approximations of Holomorphic Functions of Many Variables

We show that the Lebesgue–Landau constants of linear methods for summation of Taylor series of functions holomorphic in a polydisk and in the unit ball from \(\mathbb{C}^m\) over triangular domains do not depend on the number m. On the basis of this fact, we find a relation between the complete and partial best approximations of holomorphic functions in a polydisk and in the unit ball from \(\mathbb{C}^m\).

Spaces of Lorentz Multipliers

AbstractWe study when the spaces of Lorentz multipliers from Lp,t → Lp,s are distinct. Our main interest is the case when s < t, the Lorentz-improving multipliers. We prove, for example, that the space of multipliers which map Lp,t → Lp,s is different from those mapping Lp,t → Lp,s if either r = p or p′ and 1/s − 1/t ≠ 1/u − 1/v, or r ≠ p or p′. These results are obtained by making careful estimates of the Lorentz multiplier norms of certain linear combinations of Fejer or Dirichlet kernels. For the case when the first indices are different the linear combination we analyze is in the spirit of a Rudin-Shapiro polynomial.

On the 𝐿²→𝐿^{∞} norms of spectral multipliers of “quasi-homogeneous” operators on homogeneous groups

We study the L 2 → L ∞ L^2 \to L^{\infty } norms of spectral projectors and spectral multipliers of left-invariant elliptic and subelliptic second-order differential operators on homogeneous Lie groups. We obtain a precise description of the L 2 → L ∞ L^2 \to L^{\infty } norms of spectral multipliers for some class of operators which we call quasi-homogeneous. As an application we prove a stronger version of Alexopoulos’ spectral multiplier theorem for this class of operators.

The numerical range of products of normal matrices

In an earlier paper, the author developed a formula for the trace class multiplier norm of a matrix of rank at most 2. In this article, applications of this formula are given. In the main result we suppose that f 1,…, f n and g 1,…, g n are given sets of complex numbers. A description is given of the union of the numerical ranges of the product FG as F and G run over all n × n normal matrices with the given sets as eigenvalues.

On the strong type multiplier norms of rational functions in several variables

On the strong type multiplier norms of rational functions in several variables

Trace class multipliers and spectral variation of normal matrices

In this article we show how to estimate the trace multiplier norm of a rank 2 matrix. As an application, an alternative proof of a theorem of Holbrook et al. (Maximal spectral distance, Linear Algebra Appl., 249 (1996) 197–205) on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given.

A lower estimate for the Lebesgue constants of linear means of Laguerre expansions

S. G. Kal’neǐ derived in [5], [6] a quite sharp necessary condition for the multiplier norm of a finite sequence in the setting of Fourier-Jacobi series on L1 with “natural weight” (which ensures a nice convolution structure). In this paper, Kalneǐ’s problem is considered in the setting of Laguerre series on weighted L1-spaces; the admitted scale of weights contains in particular the appropriate “natural weights” occurring in transplantation and convolution.

Projective modules and Hilbert spaces with a Nevanlinna-Pick kernel

In this paper we solve a mapping problem for a particular class of Hilbert modules over an algebra multipliers of a diagonal Nevanlinna-Pick (NP) kernel. In this case, the regular representation provides a multiplier norm which induces the topology on the algebra. In particular, we show that, in an appropriate category, a certain class of Hilbert modules are projective. In addition, we establish a commutant lifting theorem for diagonal NP kernels.

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.

A necessary and sufficient condition for M-matrices and its relation to block LU factorization

We present a necessary and sufficient condition for M-matrices in terms of a special diagonal dominance. Then we use the new result to show that if the block comparison matrix of a block matrix A ̄ is an M-matrix, there exists a block permutation matrix P such that block LU factorization applied to A = P T A ̄ P is stable—i.e., the norms of the block multipliers − A ( k − 1) i, k A ( k − 1) k, k are bounded by 1. We also present a collection of tools in the literature related to the subject matter. We define incomplete M-matrices, prove a necessary and sufficient condition for such matrices, and present their implications for block LU factorization.

Maximal Estimates on Groups, Subgroups, and the Bohr Compactification

Let G be a locally compact abelian group with character group Γ. We study the interplay of boundedness properties for suitably related maximal operators defined by (weak type or strong type) multipliers for G, its subgroups, and its Bohr compactification b( G). These considerations lead to weak type and strong type maximal estimates which generalize fundamental theorems of de Leeuw and Saeki concerning strong type norms of single multipliers. Suppose that 1 ≤ p < ∞, and M is the maximal operator on L p ( G) defined by a sequence {ψ n } ∞ n = 1 of strong type Fourier multipliers which are continuous functions on γ. Our main result establishes that M is of weak type ( p, p) on L p ( G) if and only if the corresponding maximal operator M # on L p ( b( G)) is of weak type ( p, p). This provides a counterpart for locally compact abelian groups of E. M. Stein′s Continuity Principle for compact groups, since the latter characterizes the weak ( p, p) boundedness of M # when 1 ≤ p ≤ 2.

A Characterization of Radial Herz-Schur Multipliers on Free Products of Discrete Groups

A characterization of radial Herz-Schur multipliers on groups which are free products G = ∗ N i=1 G i of subgroups of the same order (finite or infinite) is given. The multiplier norm is computed in terms of trace norms of appropriate trace operators. This applies in particular to free groups F N = ∗ N i=1 Z with block length. This method also applies to characterizing radial multipliers of free product groups of distinct orders, all greater than 2, but in this case we can compute only the multiplier norm up to some constant.

Norms of Schur multipliers

Matrices with extremal ( p, q)-Schur multiplier norms are characterized

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.

Some Banach Algebras without Discontinuous Derivations

It is shown that the completion of $A(G)$ in either the multiplier norm or the completely bounded multiplier norm is a Banach algebra without discontinuous derivations when $G$ is either ${F_2}$ or $\operatorname {SL}(2,\mathbb {R})$.

Some Banach algebras without discontinuous derivations

It is shown that the completion of A ( G ) A(G) in either the multiplier norm or the completely bounded multiplier norm is a Banach algebra without discontinuous derivations when G G is either F 2 {F_2} or SL ⁡ ( 2 , R ) \operatorname {SL}(2,\mathbb {R}) .

Groups acting on trees and approximation properties of the Fourier algebra

Let ▪ be a tree and G a locally compact group acting on ▪ by isometries with respect to the natural metric on ▪. We construct the series of representations of G parametrized by the complex unit disc associated canonically with the distance on ▪ via the matrix coefficients. We apply this series to prove that for any group G acting on a tree in such a way that the stabilizer of a vertex is a compact subgroup of G the Fourier algebra A( G) admits an approximate unit bounded in the multiplier norm on A( G). For the special case of semihomogeneous trees and the full group Aut( ▪) of isometries of ▪ we decompose the constructed representations obtaining finally an analytic continuation of the principal series of Aut( ▪).