Let $H_n$ be the (2n+1)-dimensional Heisenberg group, let p,q be two non-negative integers satisfying p+q=n and let G be the semidirect product of U(p,q) and $H_n$. So $L^2\left(H_n\right)$ has a natural structure of G-module. We obtain a decomposition of $L^2\left(H_n\right)$ as a direct integral of irreducible representations of G. On the other hand, we give an explicit description of the joint spectrum σ(L,iT) in $L^2\left(H_n\right)$ where $L={\sum }_{j=1}^p(X_j^2+Y_j^2)-{\sum }_{j=p+1}^n(X_j^2+Y_j^2)$, and where $X_1,Y_1,...,X_n,Y_n,T$ denotes the standard basis of the Lie algebra of $H_n$. Finally, we obtain a spectral characterization of the...

We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c₀.

Let I = [0, 1] be the compact topological semigroup with max multiplication and usual topology. C(I), $L^p\left(I\right)$, 1 ≤ p ≤ ∞, are the associated Banach algebras. The aim of the paper is to characterise $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$ and their preduals.

Consider I = [0,1] as a compact topological semigroup with max multiplication and usual topology, and let $C\left(I\right),L^p\left(I\right),1\le p\le \infty$, be the associated algebras. The aim of this paper is to study the spaces $Ho{m}_{C\left(I\right)}(L^r\left(I\right),L^p\left(I\right))$, r > p, and their preduals.

We introduce certain spaces of sequences which can be used to characterize spaces of functions of exponential type. We present a generalized version of the sampling theorem and a "nonorthogonal wavelet decomposition" for the elements of these spaces of sequences. In particular, we obtain a discrete version of the so-called φ-transform studied in [6] [8]. We also show how these new spaces and the corresponding decompositions can be used to study multiplier operators on Besov spaces.

Let K be a closed Lie subgroup of the unitary group U(n) acting by automorphisms on the (2n+1)-dimensional Heisenberg group $H_n$. We say that $(K,H_n)$ is a Gelfand pair when the set $L_K^1\left(H_n\right)$ of integrable K-invariant functions on $H_n$ is an abelian convolution algebra. In this case, the Gelfand space (or spectrum) for $L_K^1\left(H_n\right)$ can be identified with the set $\Delta (K,H_n)$ of bounded K-spherical functions on $H_n$. In this paper, we study the natural topology on $\Delta (K,H_n)$ given by uniform convergence on compact subsets in $H_n$. We show that $\Delta (K,H_n)$ is a complete...

We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for $L^1\left(AN\right),$ but also for $L^p\left(AN\right)$, where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.

We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in [CGHM] and [A].

Let G be a Lie group, Xj right invariant vector fields on G, which generate (as a Lie algebra) the Lie algebra of G,L = -Σ Xj2.(...) In this paper we consider L1(G) boundedness of F(L) for (some) metabelian G and a distinguished L on G. Of the main interest is that the group is of exponential growth, and possibly higher rank. Previously positive results about higher rank groups were only about Iwasawa type groups. Also, our groups may be unimodular, so it is the second positive result (after [13])...

Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L=-1/2{\sum }_{j=1}^n[({\partial }_{x_j}+i{y}_j)²+({\partial }_{y_j}-i{x}_j)²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{\lambda }$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $\left|\right|P_{\lambda }{u\left|\right|}_{L^p\left({ℝ}^d\right)}\lesssim {\lambda }^{\varrho \left(p\right)}{\left|\right|u\left|\right|}_{L²\left({ℝ}^d\right)}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p if 2(d+1)/(d-1)...

In this note we define and explore, à la Godement, spectral subspaces of Banach space representations of the Fourier-Eymard algebra of a (nonabelian) locally compact group.

Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty }\left(G\right)$-submodule X̂ of ℬ(L²(G)) (where $V^{\infty }\left(G\right)$ is the weak-* Haagerup tensor product $L^{\infty }\left(G\right){\otimes }_{w*h}{L}^{\infty }\left(G\right)$), define the concept of X̂-operator synthesis and prove that a closed set E...

For locally compact, second countable, type I groups G, we characterize all closed (two-sided) translation invariant subspaces of L²(G). We establish a similar result for K-biinvariant L²-functions (K a fixed maximal compact subgroup) in the context of semisimple Lie groups.