The problem of efficient inversions and Bezout equations (Q2772777)

The spectrum (maximal ideal space) \({\mathfrak M}(A)\) of a commutative unital Banach algebra \(A\) is said to be \(n\)-visible form \(X\in{\mathfrak M}(A)\), if \(\{(f_1(x),\dots, f_n(x))\in \mathbb{C}^n: x\in X\}\) is dense in \(\{(f_1(x),\dots, f_n(x))\in \mathbb{C}^n: x\in{\mathfrak M}(A)\}\), for all \(n\)-tuples \(f= (f_1,\dots, f_n)\in A^n\), and is said to be completely visible if it is \(n\)-visible for all \(n\). The spectrum \({\mathfrak M}(A)\) is \(n\)-visible iff for every \(f\in A^n\) satisfying NEWLINE\[NEWLINE\delta^2:= \inf \Bigl\{\sum|f_k(x)|^2> 0:x\in X\Bigr\},\tag{\(*\)}NEWLINE\]NEWLINE there is a \(g\in A^n\) with \(\sum f_kg_k= e\) -- the unity of \(A\). The spectrum of \(A\) is said to be \((\delta,n)\)-visible (from \(X\)), \(0<\delta\leq 1\), if there is a constant \(c_n\) such that whenever \(f\) satisfies \((*)\) and is normalized so that \(\|f\|^2= \sum\|f_i\|^2\leq 1\), then \(\|g\|< c_n\). It is said to be completely \(\delta\)-visible if it is \((\delta-n)\)-visible for all \(n\) and \(\sup_n c_n< \infty\). Denote by \(c_n(\delta)= c_n(\delta, A,X)\) the best possible constants, i.e., the above is satisfied for all normalized \(n\)-tuples \(f\) and all \(c_n= c_n(\delta)+\varepsilon\) for all \(\varepsilon> 0\).NEWLINENEWLINENEWLINEThe goal of the theory presented in this survey is to estimate from above and from below (and compute, if possible) the majorants \(c_n(\delta, A,X)\) and the critical constants \(\delta_n(A,X)= \inf\{\delta: c_n(\delta, A,X)<\infty\}\). In particular, the author discusses the case when \(A= M(G)\) is the measure algebra of an LCA group \(G\) and \(X=\widehat G\) is the dual group. The author also considers the measure algebras of a large class of topological Abelian semigroups, and their subalgebras -- the (semi)group algebra of an LCA (semi)group, the algebra of almost periodic functions, the algebra of absolutely convergent Dirichlet series, as well as the weighted Beurling-Sobolev algebras, \(H^\infty\)-quotient algebras and some finite-dimensional algebras.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00019].