We use one-dimensional differential inequalities to estimate the squareness and type of Banach spaces with modulus of convexity of power type two. The estimates obtained are sharp and the constants involved moderate.

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of $Hv=f:\Omega \to ℂholomorphic:su{p}_{z\in \Omega }\left|f\left(z\right)\right|v\left(z\right)<\infty$ and investigate some isomorphism classes of $hv=f:\Omega \to ℂharmonic:su{p}_{z\in \Omega }\left|f\left(z\right)\right|v\left(z\right)<\infty$ where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either $Hv\sim l_{\infty }$ or $Hv\sim H_{\infty }$, and at least two possibilities for hv, again $hv\sim l_{\infty }$ and $hv\sim H_{\infty }$. We also discuss many new examples of weights.

We introduce the notions of pointwise modulus of squareness and local modulus of squareness of a normed space X. This answers a question of C. Benítez, K. Przesławski and D. Yost about the definition of a sensible localization of the modulus of squareness. Geometrical properties of the norm of X (Fréchet smoothness, Gâteaux smoothness, local uniform convexity or strict convexity) are characterized in terms of the behaviour of these moduli.

We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ${\left(e_i\right)}_{i\in \mathbb{N}}$; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, ${\left[e_i\right]}_{i\in A}\ncong {\left[e_i\right]}_{i\in B}$, or for a residual set of infinite subsets A of ℕ, ${\left[e_i\right]}_{i\in A}$ is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to $\oplus {\left[e_i\right]}_{i\in D}$ for any D ⊂ ℕ, and isomorphic to a denumerable Schauder decomposition...

It is shown that every uncountable symmetric basic set in an F-space with a symmetric basis is equivalent to a basic set generated by one vector. We apply this result to investigate the structure of uncountable symmetric basic sets in Orlicz and Lorentz spaces.

For a Banach space X with an unconditional basic sequence, one of the following regular-irregular alternatives holds: either X contains a subspace isomorphic to ℓ₂, or X contains a subspace which has an unconditional finite-dimensional decomposition, but does not admit such a decomposition with a uniform bound for the dimensions of the decomposition. This result can be viewed in the context of Gowers' dichotomy theorem.

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[{({ℳ}_k,{\theta }_k)}_{k=1}^l\right]$ with index $i\left({ℳ}_k\right)$ finite are either $c_0$ or ${\ell }_p$ saturated for some $p$ and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left({ℳ}_k\right)$ and the parameter ${\theta }_k$. Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[{({𝒜}_k,{\theta }_k)}_{k=1}^{\infty }\right]$ in terms of the asymptotic behaviour of the sequence $∥{\sum }_{i=1}^n{e}_i∥$ where $\left(e_i\right)$ is the canonical basis....

We prove a structural property of the class of unconditionally saturated separable Banach spaces. We show, in particular, that for every analytic set 𝓐, in the Effros-Borel space of subspaces of C[0,1], of unconditionally saturated separable Banach spaces, there exists an unconditionally saturated Banach space Y, with a Schauder basis, that contains isomorphic copies of every space X in the class 𝓐.

We give a new construction of uniformly convex norms with a power type modulus on super-reflexive spaces based on the notion of dentability index. Furthermore, we prove that if the Szlenk index of a Banach space is less than or equal to ω (first infinite ordinal) then there is an equivalent weak* lower semicontinuous positively homogeneous functional on X* satisfying the uniform Kadec-Klee Property for the weak*-topology (UKK*). Then we solve the UKK or UKK* renorming problems for Lp(X) spaces...

Every separable Banach space with $C^{\left(n\right)}$-smooth norm (Lipschitz bump function) admits an equivalent norm (a Lipschitz bump function) which is both uniformly Gâteaux smooth and $C^{\left(n\right)}$-smooth. If a Banach space admits a uniformly Gâteaux smooth bump function, then it admits an equivalent uniformly Gâteaux smooth norm.

We show that every infinite-dimensional Banach space with separable dual admits an equivalent norm which is weakly locally uniformly rotund but not locally uniformly rotund.

In questo lavoro, motivati dalla teoria di Fredholm in spazi di Banach e dalla cosiddetta teoria degli ideali di operatori nel senso di Pietsch, viene definito un nuovo concetto di semigruppo di operatori. Questa nuova definizione include quella di molte classi di operatori già studiate in letteratura, come la classe degli operatori di semi-Fredholm, quella degli operatori tauberiani ed altre ancora. Inoltre permette un nuovo ed unificante approccio ad una serie di problemi in teoria degli operatori...