We establish triviality of some holomorphic Banach vector bundles on the maximal ideal space M(H∞) of the Banach algebra H∞ of bounded holomorphic functions on the unit disk D ⊂ C with pointwise multiplication and supremum norm. We apply the result to the study of the Sz.-Nagy operator corona problem. 1. Formulation of Main Results 1.1. We continue our study started in [Br2] of analytic objects...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...

A metric space X is said to be absolutely Lipschitz extendable if every Lipschitz function f from X into any Banach space Z can be extended to any containing space Y ⊇ X , where the loss in the Lipschitz constant in the extension is independent of Y , Z, and f . We show that various classes of natural metric spaces are absolutely Lipschitz extendable. c © 2003 Académie des sciences/Éditions sci...

With this norm, K(W) is a Banach space and even a Banach lattice, i.e., a vector lattice with the property that \g\ ^/, fEk(W) for a measurable g implies gGA(JF) and ||g|| á||/||. Condition (3) ensures [l, Theorem 5] that the norm (4) is continuous with respect to monotone limits. One obtains examples of such functions W as follows. Let (S, B, p.) be a measure space with measure p, which we sha...

We prove that a homogeneous Banach space B on the unit circle T can be embedded as a closed subspace of a dual space ΞB contained in the space of bounded Borel measures on T in such a way that the map B → ΞB defines a bijective correspondence between the class of homogeneous Banach spaces on T and the class of prehomogeneous Banach spaces on T. We apply our results to show that the algebra of a...

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a onedimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be atfinely embedded into the space L0 of all measurable functions.

A survey is given of the work on strong regularity for uniform algebras over the last thirty years, and some new results are proved, including the following. Let A be a uniform algebra on a compact space X and let E be the set of all those points x ∈ X such that A is not strongly regular at x. If E has no non-empty, perfect subsets then A is normal, and X is the character space of A. If X is ei...

Every Banach space X with the Banach-Saks property is reflexive, but the converse is not true (see [4, 5]). Kakutani [6] proved that any uniformly convex Banach space X has the Banach-Saks property. Moreover, he also proved that if X is a reflexive Banach space and θ ∈ (0, 2) such that for every sequence (xn) in S(X) weakly convergent to zero, there exist n1, n2 ∈ N satisfying the Banach-Saks p...

The real geometric properties of spaces of polynomials are discussed in [1, 6]. In particular, it is shown that the symmetric injective tensor product space ⊗̂n,s,εE is not strictly convex if E is a Banach space of dimE ≥ 2 and if n ≥ 2 holds. Let E be a Banach space over a real or complex filed and E is denoted as the Banach dual of E. An element x in the unit sphere SE is called a (real) extre...

‎Let $\Omega_X$ be a bounded‎, ‎circular and strictly convex domain in a complex Banach space $X$‎, ‎and $\mathcal{H}(\Omega_X)$ be the space of all holomorphic functions from $\Omega_X$ to $\mathbb{C}$‎. ‎The growth space $\mathcal{A}^\nu(\Omega_X)$ consists of all $f\in\mathcal{H}(\Omega_X)$‎ ‎such that $$|f(x)|\leqslant C \nu(r_{\Omega_X}(x)),\quad x\in \Omega_X,$$‎ ‎for some constant $C>0$‎...