Related papers: Solvability of the B\'ezout Equation for Banach Al…

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H^\infty of bounded holomorphic functions on the unit disk D\subset C with pointwise multiplication and supremum norm. In…

We establish triviality of some holomorphic Banach vector bundles on the maximal ideal space of the Banach algebra of bounded holomorphic functions on the unit disk with pointwise multiplication and supremum norm. We apply the result to the…

We study the differential equation $\frac{\partial G}{\partial\bar z}=g$ with an unbounded Banach-valued Bochner measurable function $g$ on the open unit disk $\mathbb D\subset\mathbb C$. We prove that under some conditions on the growth…

Let $\mathbb D^n\subset\mathbb C^n$ be the open unit polydisk, $K\subset\mathbb D^n$ be an $n$-ary Cartesian product of planar sets, and $\hat U\subset \mathfrak M^n$ be an open neighbourhood of the closure $\bar K$ of $K$ in $\mathfrak…

Let $E$ be a Banach lattice on $\mathbb Z$ having order continuous norm. We show that for any function $f = \{f_j\}_{j \in \mathbb Z}$ from the Hardy space $H_\infty (E)$ such that $\delta \leqslant \|f (z)\|_E \leqslant 1$ for all $z$ from…

We study the corona problem on the unit ball and the unit polydisc in $\CC^n$. We provide affirmative solutions to both problems.

Let ${\mathbb{D}}=\{z\in \mathbb{C}:|z|<1\}$ and for an integer $d\geq 1$, let $S_d$ denote the symmetric group, consisting of of all permutations of the set $\{1,\cdots, d\}$. A function $f:{\mathbb{D}}^d\rightarrow \mathbb{C}$ is…

This paper addresses the Corona problem for slice hyperholomorphic functions for a single quaternionic variable. While the Corona problem is well-understood in the context of one complex variable, it remains highly challenging in the case…

We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed…

In this paper, we obtain estimates for the solutions to the classical B{\'e}zout equation that are analogous to Carleson's solution to the corona theorem for the bounded analytic functions on the open unit disk. As an application, we extend…

The corona problem was motivated by the question of the density of the open unit disc in the maximal ideal space of the algebra of bounded holomorphic functions on the unit disc. The corona problem connects operator theory, function theory,…

We establish an equivalency of the Corona problem (1962) and Gleason problem (1964) in the theory of several complex variables. As an application, we give an affirmative solution of the Corona problem for certain bounded pseudoconvex…

R.M. Aron et al. proved that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra $\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelin showed that…

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in $\mathbb{C}^{n}$, and for the algebra of bounded analytic functions on certain strictly pseudoconvex…

Let $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the open unit ball $B_X$ of a complex Banach space $X$. Considering the Gelfand transform image $\widehat{\mathcal{A}}$ of the Banach algebra $\mathcal{A}$, which is…

We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect…

The space of Bloch functions on bounded symmetric domains is extended by considering Bloch functions $f$ on the unit ball $B_E$ of finite and infinite dimensional complex Banach spaces in two different ways: by extending the classical Bloch…

The algebra $H^\infty(D)$ of bounded holomorphic functions on $D\subset\mathbb C$ is projective free for a wide class of infinitely connected domains. In particular, for such $D$ every rectangular left-invertible matrix with entries in…

We reduce the polynomial cluster value problem for the algebra of bounded analytic functions, $H^{\infty}$, on the ball of Banach spaces $X$ to the same polynomial cluster value problem for $H^{\infty}$ but on the ball of those spaces which…

Let $H^\infty(\Delta)$ be the uniform algebra of bounded analytic functions on the open unit disc $\Delta$, and let $\mathfrak{M}(H^\infty)$ be the maximal ideal space of $H^\infty(\Delta)$. By regarding $\Delta$ as an open subset of…