We continue the study undertaken in [1] of left democracy function $$h_l(N)=\inf_{\#\Lambda=N}||\sum_{n\in \Lambda_N}x_n||$$ of an unconditional basis in a Banach space $X$. We provide an example of a basis with $h_l$ non-doubling. Then we show that for bases with non-doubling $h_l$ the greedy projection is not optimal. Together with results from [1] improved by C. Cabrelli, G. Garrigós, E. Hernandez and U. Molter we get that the basis is greedy if and only if the greedy projection is optimal.

We study rearrangement invariant spaces on which the classes of strictly singular and compact operators coincide. The relation between this property and the fact that every normalized disjoint sequence in the space has a subsequence equivalent to the unit vector basis of $\ell_2$ is analyzed.

We construct a rearrangement invariant space $X$ on $[0,1]$ with the property that all bounded linear operators from $\ell^p$, $1<p<\infty$, to $X$ are compact, but there exists a non-compact operator from $\ell^\infty$ to $X$. The techniques used allow to give a new proof of the characterization given by Hern\'andez, Raynaud and Semenov of the rearrangement invariant spaces on $[0,1]$ for which the canonical embedding into $L^1([0,1])$ is finitely strictly singular.

We introduce and analyse the notion of slice continuity between operators on Banach spaces in the setting of the Daugavet property. It is shown that under the slice continuity assumption the Daugavet equation holds for weakly compact operators. As an application we define and characterise the Daugavet property for bilinear maps, and we prove that this allows us to describe some $p$-convexifications of the Daugavet equation for operators on Banach function spaces that have recently been introduced.

We present a characterization of Banach function lattices with the Fatou property generated by the interpolation sum applied to infinite families of Banach lattices with the Fatou property. We also discuss the Köthe duality between the sum and the intersection constructions for Banach spaces.

Let $X_a$ denote the order continuous part of a Banach lattice $X$, and let $\Gamma$ be an uncountable set. We extend Drewnowski's theorem on the comparison of linear dimensions between Banach spaces having uncountable symmetric bases to a~class of discrete Banach lattices, the so-called $D$-spaces. We show that if $X$ and $Y$ are two $D$-spaces and there are continuous linear injections (not necessarily embeddings) from $X$ into $Y$ and vice versa, then $X$ and $Y$ are order-topologically isomorphic. In the proof we apply a theorem on the extension of an order isomorphism from $X_a$ onto $Y_a$ to an order isomorphism from $X$ onto $Y$, the classical Drewnowski's theorem, and a supplement of Troyanski's theorem on embeddings of $\ell_1(\Gamma)$ spaces into a Banach space with an uncountable symmetric basis. Our result applies to the class of Orlicz spaces $\ell_\varphi(\Gamma)$, where $\varphi$ is an Orlicz function.

In this note we characterize Banach envelopes of $p$-Banach lattices, $0<p<1$, such that their positive cones are $1$-concave. In particular we show that the Banach envelope of Cesàro sequence space $\widehat{ces_p(v)}$, $0<p<1$, coincides isometrically with the weighted $\ell_1(w)$ space where $w(n) = ||e_n||_{ces_p(v)}= (\sum_{i=n}^\infty i^{-p} v(i))^{1/p}$ and $e_n$ are the unit vectors. For Cesàro function space $Ces_p(v)$, $0<p<1$, its Banach envelope $\widehat{Ces_p(v)}$ is isometrically equal to $L_1(w)$ with $w(t) = (\int_t^\infty s^{-p} v(s) ds)^{1/p}$, $t\in (0,\infty)$.

Let $(T(t))_{t\geq 0}$ be a strongly continuous $C_0$-semigroup of bounded linear operators on a~Banach space $X$ such that $\lim_{t\to\infty}||T(t)/t||=0$. Characterizations of when $(T(t))_{t\geq 0}$ is uniformly mean ergodic, i.e., of when its Cesàro means $r^{-1}\int_0^r T(s)ds$ converge in operator norm as $r\to\infty$, are known. For instance, this is so if and only if the infinitesimal generator $A$ has closed range in $X$ if and only if $\lim_{\lambda\downarrow 0^+}\lambda R(\lambda, A)$ exists in the operator norm topology (where $R(\lambda,A)$ is the resolvent operator of $A$ at $\lambda$). These characterizations, and others, are shown to remain valid in the class of quojection Fréchet spaces, which includes all Banach spaces, countable products of Banach spaces, and many more. It is shown that the extension fails to hold for all Fréchet spaces. Applications of the results to concrete examples of $C_0$-semigroups in particular Fréchet function and sequence spaces are presented.

The criteria for non-containment of $l^1$ for the classes of Banach and Fréchet spaces are extended to the class of locally complete locally convex spaces the bounded sets of which are metrizable.

In a joint paper with Grahame Bennett and Toivo Leiger (cf. [3]) the second author introduced for real sequence spaces the Hahn property and the notion of Hahn spaces. The aim of this paper is to extend these notions and a series of results in [3] to subspaces of Fréchet spaces by replacing the space $\omega$ of all real sequences and the set $\chi$ of all sequences of 0's and 1's by any Fréchet space $H$ and a suitable subset $\chi$ of $H$, respectively. Applications of the general considerations to the original case of Hahn spaces of real sequences are also a main subject matter of the paper.

If $X$ is an infinite Tichonov space, we show that the weak dual $L_{p}(X)$ of the continuous function space $C_{p}(X)$ cannot be barrelled, bornological, or even quasibarrelled. Indeed, of the fourteen standard weak barrelledness properties between Baire-like and primitive, $L_{p}(X)$ enjoys precisely the four between property (C) and primitive if $X$ is a P-space, and none otherwise. Since $L_{p}(X)$ is $S_{\sigma}$, it must admit an infinite-dimensional separable quotient. Under its Mackey topology, $L_{p}(X)$ enjoys eleven of the properties if $X$ is discrete, nine if $X$ is a nondiscrete P-space, and none otherwise.

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.