Translation Preserving Operators on Locally Compact Abelian Groups
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Translation Preserving Operators on Locally Compact Abelian Groups M. Mortazavizadeh, R. Raisi Tousi
and R. A. Kamyabi Gol
Abstract. We study translation preserving operators, that is operators commuting with translations by a closed subgroup of a locally compact abelian group. We show that there is a one-to-one correspondence between these operators and range operators. Furthermore, we obtain a necessary condition for a translation preserving operator to be Hilbert– Schmidt or of finite trace in terms of its range operator. The last result is proved in the case that the subgroup is discrete or compact. Mathematics Subject Classification. Primary 47A15; Secondary 42B99, 22B99. Keywords. Locally compact abelian group, Multiplication preserving operator, Range function, Translation preserving operator, Range operator.
1. Introduction and Preliminaries For a locally compact abelian (LCA) group G, a translation invariant space is defined to be a closed subspace of L2 (G) that is invariant under translations by elements of a closed subgroup Γ of G. Translation invariant spaces in the case Γ is closed, discrete, and cocompact, called shift invariant spaces, have been studied in [2,3,9,12,13,16], and extended to the case of Γ closed and cocompact (but not necessarily discrete) in [5], see also [10]. Recently, translation invariant spaces have been generalized in [1] to the case when Γ is closed (but not necessarily discrete or cocompact). In [4], Bownik defined shift preserving operators as bounded linear operators on L2 (Rn ) that commute with integer translations and characterized them via range operators. In [14], the authors introduced shift preserving operators on LCA groups, whereas our goal in this paper is to define and study translation preserving operators on LCA groups. We define a translation preserving operator as a bounded linear operator on L2 (G) that commutes with translations by elements of a closed subgroup Γ of G which is not necessarily discrete or cocompact. We 0123456789().: V,-vol
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characterize such operators in terms of range operators and define a multiplication preserving operator on a certain vector valued space. Following an idea of [1], we extend the results in [14] to the general setting when G is an LCA group and Γ is a closed subgroup of G. Indeed, we show that there is a one-to-one correspondence between translation preserving operators on L2 (G) and multiplication preserving operators on the vector valued space. We use this correspondence to get the characterization of translation preserving operators in terms of range operators. We also show that a translation preserving operator has several properties in common with its associated range operator, especially compactness of one implies compactness of the other. We obtain a necessary condition for a translation preserving operator to be Hilbert–Schmidt or of finite trace. This paper is organized as follows. In the rest of this section, we state some required preliminaries and notation relat
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