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Resolvent estimates in Beurling-Sobolev algebras. (English. Russian original) Zbl 0938.46047

St. Petersbg. Math. J. 10, No. 6, 901-964 (1999); translation from Algebra Anal. 10, No. 6, 1-92 (1998).
Let \(A\) be a commutative unital Banach algebra continuously embedded in the space \(C(X)\) of all continuous functions on a Hausdorff topological space \(X\), \(A\subset C(X)\). Hence \(X\) can be indentified with a subset of the maximal ideal space \({\mathcal M} = {\mathcal M}( A)\) and clos \(X\) may be regarded as the visible part of \(\mathcal M\). The spectrum of \(A\) is said to be \(n\)-visible (or \(n\)-visible from \(X\)), \(n=0,1,\dots \), if \(\widehat f(\mathcal {M})= \operatorname{clos}(f(X))\) for all \(f=(f_1,\dots,f_n)\in A^n\), where \(\widehat f\) denotes the Gelfand transform of \(f\in A\). Let \(0<\delta \leq 1\). The spectrum of \(A\) is said to be \((\delta , n)\)-visible if there exists a constant \(c_n\) such that any Bezout equation \(\sum _{k=1}^n g_kf_k = e\) with data \(f=(f_1,\dots, f_n)\in A^n\) that satisfy \(\inf_{x\in X}\sum^n_{k=1} |f_k(x)|^2 > 0\) and the normalization condition \(\|f\|^2 = \sum^n_{k=1}\|f\|^2 \leq 1\) has a solution \(g\in A^n\) with \(\|g\|\leq c_n\). Let \(c_n(\delta,A)\) be the smallest number for which the constant \(c_n=c_n(\delta, A) +\varepsilon\) meets the requirements of this definition (for every \(\varepsilon > 0\)). This definition is a quantitative version of the preceding one, adapted to the case of the norm controlled invertibility instead of mere invertibility. In particular, \(c_1(\delta, A)=\sup \{\|f^{-1}\|: f\in A, \|f\|\leq 1\), \(|f(s)|\geq \delta \) for \( s\in X\}.\) For any integer \(n\geq 1\) there exists a critical constant \(0\leq \delta_{n}(A) \leq 1\), such that for \(\delta_n(A)< \delta \leq 1\) the spectrum of \(A\) is \((\delta, n)\)-visible, and for \(0<\delta<\delta_n(A)\) it is not.
The authors study the norm controlled invertibility in the general concept of the quantitative visibility of the spectrum. The main goal of this paper is to estimate and compute the critical constant \(\delta_1(A)\) and the majorants \(c_1(\delta, A)\) for some Banach algebras \(A\) and visible parts \(X\) of their spectra. More precisely, upper estimates for inverses are sought in the weighted Beurling-Sobolev algebras \(A={\mathcal F}l^{p}({\mathbb Z},w)\) and \(A={\mathcal F}l^{p}({\mathbb Z_{+}},w)\) of positive spectral radius. It is shown that under some weight regularity conditions, the classical Wiener algebra of absolutely convergent Taylor (Fourier) series (i.e., the analytic algebra \(A={\mathcal F}l^1({\mathbb Z_{+}})\), or the symmetric algebra \({\mathcal F}l^1({\mathbb Z})\)) is the only exceptional case in which the norm control \(\|f^{-1}\|\leq c_1(\delta, A)\), where \(\delta = \inf |\widehat f |\) and \(\|f \|\leq 1\), fails for some \(\delta > 0\). For the analytic Wiener algebra the critical constant \(\delta_1(A)\) and the sharp resolvent growth rate are computed, which sheds new light on the well-known Wiener-Pitt phenomen for measure algebras. The same method provides a lower estimate for the critical constant for the measure algebra \(M(G)\) on any infinite locally compact Abelian group \(G\). For the other Beurling-Sobolev algebras \(A\), the growth rate of \(c_1(\delta, A)\) as \(\delta \to 0\) is estimated in terms of the weight function \(w\). To this end, the Green formula space \(\mathcal D A\) and its multiplier space mult\((\mathcal D A)\) are introduced; it is proved that there is a compact embedding \(a\subset\) mult\((\mathcal D A)\). The best polynomial approximations related to this embedding are estimated. This is the key point in obtaining the upper bounds in question for \(c_1(\delta, A)\).

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A10 Measure algebras on groups, semigroups, etc.
43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
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