The local analysis of convergence for Newton’s method has been extensively studied by numerous researchers under a plethora of sufficient conditions. However, the complexity of extending the convergence domain requires very general conditions such as the ones depending on the majorant principle in order to include as large classes of operators as possible. In the present article, such an analysis is developed under the weak majorant condition. The new results extend earlier ones using similar information. Finally, the numerical examples complement the theory.

Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z| \lt 1\}$ into a complex Banach space X with $\left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q \lt \infty$, we denote

\begin{equation*}R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \left\lVert x_{0}\right\rVert^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \left\lVert x_{n}\right\rVert\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\}\end{equation*}

\begin{equation*}R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1\right\}.\end{equation*}

$R_{p,q,\phi}(\Omega,X)$

$X=\mathcal{B}(\mathcal{H})$

$\mathcal{H}$

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher.

We define positively expansive semigroups of linear operators on Banach spaces. We characterize these semigroups in terms of the point spectrum of the infinitesimal generator. In particular, we prove that a positively expansive semigroup is neither uniformly bounded nor equicontinuous. We apply our results to the Lasota equation.

We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.

We extend our study of variability regions, Ali et al. [‘An application of Schur algorithm to variability regions of certain analytic functions–I’, Comput. Methods Funct. Theory, to appear] from convex domains to starlike domains. Let $\mathcal {CV}(\Omega )$ be the class of analytic functions f in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in {\Omega }$ . As an application of the main result, we determine the variability region of $\log f'(z_0)$ when f ranges over $\mathcal {CV}(\Omega )$ . By choosing a particular $\Omega $ , we obtain the precise variability regions of $\log f'(z_0)$ for some well-known subclasses of analytic and univalent functions.

We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space ℝN. We show that given a weighted Lp-space $L_w^p({\mathbb {R}}^N)$ with 1 ⩽ p < ∞ and a fast growing weight w, there is a Schauder basis $(e_n)_{n=1}^\infty$ in $L_w^p({\mathbb {R}}^N)$ with the following property: given an arbitrary positive integer m there exists nm > 0 such that, if the initial data f belongs to the closed linear span of en with n ⩾ nm, then the decay rate of the solution of the problem is at least t−m for large times t.

The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m.

We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional $\ell _{p}$ spaces for all $1\leqslant p<\infty$. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X,Y)$ has the pointwise Bishop–Phelps–Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X,Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X,Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_{p}(\unicode[STIX]{x1D707})$ spaces fail to have this property when $p>2$. No universal pointwise BPB range space can be simultaneously uniformly convex and uniformly smooth unless its dimension is one. We also discuss a version of the pointwise BPB property for compact operators.

In this paper we present a simple (fixed point) method that yields various results concerning approximate solutions of some difference equations. The results are motivated by the notion of Ulam stability.

We say that a Banach space $X$ is ‘nice’ if every extreme operator from any Banach space into $X$ is a nice operator (that is, its adjoint preserves extreme points). We prove that if $X$ is a nice almost $CL$-space, then $X$ is isometrically isomorphic to $c_{0}(I)$ for some set $I$. We also show that if $X$ is a nice Banach space whose closed unit ball has the Krein–Milman property, then $X$ is $l_{\infty }^{n}$ for some $n\in \mathbb{N}$. The proof of our results relies on the structure topology.

Let D(A) be the domain of an m-accretive operator A on a Banach space E. We provide sufficient conditions for the closure of D(A) to be convex and for D(A) to coincide with E itself. Several related results and pertinent examples are also included.

Let $X,Y$ be two Banach spaces and $B_{X}$ the closed unit ball of $X$. We prove that if there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists an isometry $F:X\rightarrow Y^{\ast \ast }$. If, in addition, $Y$ is weakly nearly strictly convex, then there is an isometry $F:X\rightarrow Y$. Making use of these results, we show that if $Y$ is weakly nearly strictly convex and there is an isometry $f:B_{X}\rightarrow Y$ with $f(0)=0$, then there exists a linear isometry $S:X\rightarrow Y$.

We investigate ideals of the form {A ⊆ ω: Σn∈Axn is unconditionally convergent} where (xn)n∈ω is a sequence in a Polish group or in a Banach space. If an ideal on ω can be seen in this form for some sequence in X, then we say that it is representable in X.

After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a nonpathological analytic P-ideal.

We focus on the family of ideals representable in c0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable in c0, and that a tall Fσ P-ideal is representable in c0 iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in ℓ1 but not in ℝ.

Finally, we summarize some open problems of this topic.

Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon–Nikodým property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold and that $M\,=\,{{\ell }_{2}}$ is a counterexample.