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The structure of shift-invariant subspaces of \(L^2(\mathbb{R}^n)\). (English) Zbl 0986.46018

The author of this note investigates the structure of shift invariant spaces in \(L^2(\mathbb{R}^n)\) under the action of some lattice \(\Gamma= P\mathbb{Z}^n\), where \(P\) is a nonsingular \(n\) by \(n\) real matrix. He treates \(P\) as the unit matrix, since general \(P\) case follows by standard arguments.
The proofs are followed by the idea from H. Helson’s book “Lectures on invariant subspaces”, New York/London (1964; Zbl 0119.11303) and it also can reproduce the former results by R. Ron and Z. Shen of \(L_2(\mathbb{R}^d)\), Can. J. Math. 47, No. 5, 1051-1094 (1995; Zbl 0838.42016).
The typical theorem is in the following:
Theorem. Suppose \(V\subset L^2(\mathbb{R}^n)\) is shift invariant and \(J\) is its range function. For every shift preserving operator \(L: V\to L^2(\mathbb{R}^n)\) there exists a measurable range operator \(R\) on \(J\) such that \[ ({\mathcal T}\circ L) f(x)= R(x)({\mathcal T}f(x))\quad\text{for a.e. }x\in\mathbb{T}^n,\quad f\in V,\tag{\(*\)} \] where \({\mathcal T}: L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n, \ell^2(\mathbb{Z}))\) defined for \(f\in L^2(\mathbb{R}^n)\) by \[ {\mathcal T}f: \mathbb{T}^n\to \ell^2(\mathbb{Z}^n),\quad{\mathcal T}f(x)= (\widehat f(x+ k))_{k\in \mathbb{Z}^n}, \] is an isometric isomorphic between \(L^2(\mathbb{R}^n)\) and \(L^2(\mathbb{T}^n, \ell^2(\mathbb{Z}^n))\). Conversely, given a measurable range operator \(R\) on \(J\) with \(\text{sup ess}_{x\in\mathbb{T}^n}\|R(x)\|< \infty\) there is a bounded shift preserving operator \(L: V\to L^2(\mathbb{R}^n)\) such that \((*)\) holds. The correspondence between \(L\) and \(R\) is one-to-one under the convention that the range operators are identified if they are equal a.e. Moreover, we have \(\|L\|= \text{sup ess}_{x\in\mathbb{T}^n}\|R(x)\|\).

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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