Estimates in Corona Theorems for Some Subalgebras of H∞ Amol Sasane and Sergei Treil

Contractibility of the Maximal Ideal Space of Algebras of Measures in a Half-space

H [n] = {(t1, . . . , tn) ∈ R n \ {0} | ∀j, [tj 6= 0 and t1 = t2 = · · · = tj−1 = 0] ⇒ tj > 0} ∪ {0}. LetM(H) denote the Banach algebra of all complex Borel measures with support contained in H, with the usual addition and scalar multiplication, and with convolution ∗, and the norm being the total variation of μ. It is shown that the maximal ideal space X(M(H)) of M(H), equipped with the Gelfan...

Inertia theorems for operator Lyapunov inequalities

Irrational transfer function classes, coprime factorization and stabilization

Abstract. Classes of irrational function classes, denoted by AS, that lie between the extreme cases of the disk algebra A and the Hardy space H∞(D), are considered. The corona theorem holds for AS, and the following properties are shown: AS is an integral domain, but not a Bézout domain, AS is a Hermite ring with stable rank 1, and the Banach algebra AS has topological stable rank 2. Consequenc...

Analytic Projections, Corona Problem and Geometry of Holomorphic Vector Bundles

The main result of the paper is the theorem giving a sufficient condition for the existence of a bounded analytic projection onto a holomorphic family of (generally infinite-dimensional) subspaces (a holomorphic sub-bundle of a trivial bundle). This sufficient condition is also necessary in the case of finite dimension or codimension of the bundle. A simple lemma of N. Nikolski connects the exi...

Lower Bounds in the Matrix Corona Theorem and the Codimension One Conjecture

Main result of this paper is the following theorem: given δ, 0 < δ < 1/3 and n ∈ N there exists an (n + 1) × n inner matrix function F ∈ H∞ (n+1)×n such that I ≥ F ∗(z)F (z) ≥ δI ∀z ∈ D, but the norm of any left inverse for F is at least [δ/(1−δ)]−n ≥ (2δ). This gives a lower bound for the solution of the Matrix Corona Problem, which is pretty close to the best known upper b bound C · δ−n−1 log...