Let A be a uniform algebra, and let $\phi$ be a self-map of the spectrum $M_A$ of A that induces a composition operator $C_{\phi}$, on A. It is shown that the image of $M_A$ under some iterate ${\phi}^n$ of \phi is hyperbolically bounded if and only if \phi has a finite number of attracting cycles to which the iterates of $\phi$ converge. On the other hand, the image of the spectrum of A under $\phi$ is not hyperbolically bounded if and only if there is a subspace of $A^{**}$ "almost" isometric to ${\ell}_{\infty}$ on which ${C_{\phi}}^{**}$ "almost" an isometry. A corollary of these characterizations is that if $C_{\phi}$ is weakly compact, and if the spectrum of A is connected, then $\phi$ has a unique fixed point, to which the iterates of $\phi$ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].

We prove that a Banach space that is convex-transitive and such that for some element u in the unit sphere, and for every subspace Μ containing u, it happens that the subset of norm attaining functionals on Μ is second Baire category in $M^{*}$ is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to $\ell$$_1$, where the norm is the restriction of a Luxembourg norm on $L_1$. In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.m.

In [25] Taskinen shows that if $\{E_n\}_n\;and\;\{F_n\}_n$ are two sequences of Frechet spaces such that ($E_m,\;F_n$) has the BB-property for all m and n then (${\Pi}_m\;E_m,\;{\Pi}_n\;F_n$) also has the ΒΒ-property. Here we investigate when this result extends to (i) arbitrary products of Frechet spaces, (ii) countable products of DFN spaces, (iii) countable direct sums of Frechet nuclear spaces. We also look at topologies properties of ($H(U),\;\tau$) for U balanced open in a product of Frechet spaces and $\tau\;=\;{\tau}_o,\;{\tau}_{\omega}\;or\;{\tau}_{\delta}$.

Q-reflexive locally convex spaces are spaces where $\widehat{\bigotimes}_{n,s,\pi}\;{E_e}^{"}\;and\;\overline{{(P(^nE),\;\taub)_i}^'}$ are isomorphic in a canonical way for every n. We investigate properties and find examples of such spaces.

An entire function $f\;{\in}\;H(\mathbb{C})$ is called universal with respect to translations if for any $g\;{\in}\;H(\mathbb{C}),\;R\;>\;0,\;and\;{\epsilon}\;>\;0$, there is $n\;{\in}\;{\mathbb{N}}$ such that $$\mid$f(z\;+\;n)\;-\;g(z)$\mid$\;<\;{\epsilon}$ whenever $$\mid$z$\mid$\;{\leq}\;R$. Similarly, it is universal with respect to differentiation if for any g, R, and $\epsilon$, there is n such that $$\mid$f^{(n)}(z)\;-\;g(z)$\mid$\;<\;{\epsilon}\;for\;$\mid$z$\mid$\;{\leq}\;R$. In this note, we review G. MacLane's proof of the existence of universal functions with respect to differentiation, and we give a simplified proof of G. D. Birkhoff's theorem showing the existence of universal functions with respect to translation. We also discuss Godefroy and Shapiro's extension of these results to convolution operators as well as some new, related results and problems.

This paper is an expository presentation of a large part of the results, about zeros of real polynomials on Banach spaces, that have been obtained in recent years. Also new results, for or-thogonally additive polynomials on $L_p$ spaces, are given.

In this paper we will consider the weighted composition operators W = $uC_{\tau}$ between $L^{p}$$(X,\sum,\mu$) spaces and Orlicz spaces $L^{\phi}$$(X,\sum,\mu$) generated by measurable and non-singular transformations $\tau$ from X into itself and measurable functions u on X. We characterize the functions u and transformations $\tau$ that induce weighted composition operators between $L^{p}$ -spaces by using some properties of conditional expectation operator, pair (u,${\gamma}$) and the measure space $(X,\sum,\mu$). Also, some other properties of these types of operators will be investigated.

A $JB^{*}-triple$ is a Banach space A on which the group Aut(B) of biholomorphic automorphisms acts transitively on the open unit ball B of A. In this case, a triple product {$\cdots$} from $A\;\times\;A\;\times\;A\;to\;A$ can be defined in a canonical way. If A is also the dual of some Banach space $A_{*}$, then A is said to be a JBW triple. A projection R on A is said to be structural if the identity {Ra, b, Rc} = R{a, Rb, c, }holds. On $JBW^{*}-triples$, structural projections being algebraic objects by definition have also some interesting metric properties, and it is possible to give a full characterization of structural projections in terms of the norm of the predual $A_{*}$ of A. It is shown, that the class of structural projections on A coincides with the class of the adjoints of neutral GL-projections on $A_{*}$. Furthermore, the class of GL-projections on $A_{*}$ is naturally ordered and is completely ortho-additive with respect to L-orthogonality.

Let Ε and F be locally convex spaces over C. We assume that Ε is a nuclear space and F is a Banach space. Let f be a holomorphic mapping from Ε into F. Then we show that f is of uniformly bounded type if and only if, for an arbitrary locally convex space G containing Ε as a closed subspace, f can be extended to a holomorphic mapping from G into F.

Let $D_1,\;D_2$ be convex domains in complex normed spaces $E_1,\;E_2$ respectively. When a mapping $f\;:\;D_1{\rightarrow}D_2$ is holomorphic with f(0) = 0, we obtain some results like the Schwarz lemma. Furthermore, we discuss a condition whereby f is linear or injective or isometry.

In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical $L^{p}(\mu)$ spaces. In the case $1\;{\leq}\;p\;{\leq}\;2$ we obtain optimal results and for finite dimensional complex Banach spaces, in a special case, we have improved an early result by K. Ball [3]. By using these results, in some cases we are able to find best possible lower bounds for the norms of homogeneous polynomials which are products of linear forms. In particular, we give an estimate in the case of a real Hilbert space which seems to be a difficult problem. We have also obtained some results on the so-called n-th (linear) polarization constant of a Banach space which is an isometric property of the space. Finally, known polynomial inequalities have been derived as simple consequences of various results related to plank problems.

This talk discusses conditions on the numerical range of a holomorphic function defined on a bounded convex domain in a complex Banach space that imply that the function has a unique fixed point. In particular, extensions of the Earle-Hamilton Theorem are given for such domains. The theorems are applied to obtain a quantitative version of the inverse function theorem for holomorphic functions and a distortion form of Cartan's unique-ness theorem.

In this expository article we survey some recent developments on the construction of a holomorphic functional calculus for an infinite number of elements in a commutative unital Banach algebra.

Let $\Sigma_{$\mid$\alpha$\mid$=m}\;s_{\alpha}z^{\alpha},\;z\;{\in}\;{\mathbb{C}}^n$ be a unimodular m-homogeneous polynomial in n variables (i.e. $$\mid$s_{\alpha}$\mid$\;=\;1$ for all multi indices $\alpha$), and let $R\;{\subset}\;{\mathbb{C}}^n$ be a (bounded complete) Reinhardt domain. We give lower bounds for the maximum modules $sup_{z\;{\in}\;R\;$\mid$\Sigma_{$\mid$\alpha$\mid$=m}\;s_{\alpha}z^{\alpha}$\mid$$, and upper estimates for the average of these maximum moduli taken over all possible m-homogeneous Bernoulli polynomials (i.e. $s_{\alpha}\;=\;{\pm}1$ for all multi indices $\alpha$). Examples show that for a fixed degree m our estimates, for rather large classes of domains R, are asymptotically optimal in the dimension n.

Let $A_b(B_E)$ be the Banach algebra of all complex valued bounded continuous functions on the closed unit ball $B_E$ of a complex Banach space E, and holomorphic in the interior of $B_E$, endowed with the sup norm. We present some sufficient conditions for a set to be a boundary for $A_b(B_E)$ in case E belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in [6]. We also prove the non-existence of the Shilov boundary for $A_b(B_E)$ and give some examples of boundaries.