On Convolution Dominated Operators

Abstract

Abstract. For a locally compact group G we consider the algebra CD(G)In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators CDreg(G). For amenable G which is rigidly symmetric as a discrete group we show that any element of CDreg(G) is invertible in CDreg(G) if it is invertible as a bounded operator on L 2 (G). We give an example of a symmetric group E for which the convolution dominated operators are not inverse-closed in the bounded operators on L 2 (E).Mathematics Subject Classification. Primary 47B35; Secondary 43A20.

Convolution dominated operators on compact extensions of abelian groups

“…In case that the operator T is in CD ∞ , we can take the dominating f in (L ∞ , l 1 ). The argument in Remark 2.4 of [9] shows that the infimum in ((2.3)) is attained. Actually the function i∈H ess sup…”

“…From Proposition 2.3 of [9] we know that any convolution dominated operator is an integral operator with respect to a kernel. Calling kernels equivalent if they coincide l.a.e on G × G, we have a linear bijection between the convolution dominated operators and the equivalence classes of kernels satisfying ((2.5)).…”

“…Note that (1) covers nilpotent Lie groups that admit a rational structure. The real "ax+b" group (for this group the convolution dominated operators are not inverse closed in B(L 2 (G)) [9]) shows that the compactness condition in (2) is needed. Conditions (1) and ( 2) cover all known examples where CD(G) is inverse-closed in B(L 2 (G)).…”

Inverse-Closedness of Subalgebras of Integral Operators with Almost Periodic Kernels

“…Clearly, N 1 ⊂ N b . Kernels of the class N 1 and operators induced by them were considered in [10,9,20,24,36,37,38].…”

“…The history of full subalgebras have its origin in Wiener's theorem on absolutely convergent Fourier series [57]. For further results related to inverse closed classes, see [3,4,5,6,10,11,20,22,23,24,25,26,27,28,29,30,32,34,38,41,47,54,55] and references therein.…”