This article deals with some ergodic properties for general sequences in the closed convex hull of the orbit of some (not necessarily power-bounded) operators in Banach spaces. A regularity condition more general than that of ergodicity is used to obtain some versions of the Esterle–Katznelson–Tzafriri theorem. Also, the ergodicity of the backward iterates of a sequence is proved under appropriate assumptions as, for example, its peripheral boundedness on the unit circle. The applications concern uniformly Kreiss-bounded operators, and other ergodic results are obtained for the binomial means and some operator means related to the Cesàro means.

We say that a complex number $\lambda$ is an extended eigenvalue of a bounded linear operator $T$ on a Hilbert space $\mathcal{H}$ if there exists a nonzero bounded linear operator $X$ acting on $\mathcal{H}$, called the extended eigenvector associated to $\lambda$, and satisfying the equation $TX=\lambda XT$. In this article, we describe the sets of extended eigenvalues and extended eigenvectors for the quasinormal operators.

A cohomology for product systems of Hilbert bimodules is defined via the $Ext$ functor. For the class of product systems corresponding to irreversible algebraic dynamics, relevant resolutions are found explicitly and it is shown how the underlying product system can be twisted by 2-cocycles. In particular, this process gives rise to cohomological deformations of the $C^{\ast }$-algebras associated with the product system. Concrete examples of deformations of the Cuntz’s algebra ${\mathcal{Q}}_{\mathbb{N}}$ arising this way are investigated, and we show that they are simple and purely infinite.

We show that the approximate hyperplane series property consequence, we obtain that the class of spaces $Y$ such that the pair $({\ell }_1,Y)$ has the Bishop–Phelps–Bollobás property for operators is stable under finite ${\ell }_p$-sums for $1\le p<\infty$. We also deduce that every Banach space of dimension at least $2$ can be equivalently renormed to have the AHSp but to fail Lindenstrauss’ property $\beta$. We also show that every infinite-dimensional Banach space admitting an equivalent strictly convex norm also admits such an equivalent norm failing the AHSp.

Every analytic self-map of the unit ball of a Hilbert space induces a bounded composition operator on the space of Bloch functions. Necessary and sufficient conditions for compactness of such composition operators are provided, as well as some examples that clarify the connections among such conditions.

We prove an upper bound for the supremum norm of homogeneous Bernoulli polynomials on the unit ball of finite-dimensional complex Banach spaces. This result is inspired by the famous Kahane–Salem–Zygmund inequality and its recent extensions; in contrast to the known results, our estimates are on the scale of Orlicz spaces instead of ${\ell }_p$-spaces. Applications are given to multidimensional Bohr radii for holomorphic functions in several complex variables, and to the study of unconditionality of spaces of homogenous polynomials in Banach spaces.

We present new results on Kottman’s constant of a Banach space, showing (i) that every Banach space is isometric to a hyperplane of a Banach space having Kottman’s constant 2 and (ii) that Kottman’s constant of a Banach space and of its bidual can be different. We say that a Banach space is a Diestel space if the infimum of Kottman’s constants of its subspaces is greater that 1. We show that every Banach space contains a Diestel subspace and that minimal Banach spaces are Diestel spaces.

In this article we determine that an operator-valued measure (OVM) for Banach spaces is actually a weak∗ measure, and then we show that an OVM can be represented as an operator-valued function if and only if it has $\sigma$-finite variation. By the means of direct integrals of Hilbert spaces, we introduce and investigate continuous generalized frames (continuous operator-valued frames, or simply CG frames) for general Hilbert spaces. It is shown that there exists an intrinsic connection between CG frames and positive OVMs. As a byproduct, we show that a Riesz-type CG frame does not exist unless the associated measure space is purely atomic. Also, a dilation theorem for dual pairs of CG frames is given.

We prove that, for every complex Hilbert space $H$, every weak 2-local derivation on $B\left(H\right)$ or on $K\left(H\right)$ is a linear derivation. We also establish that every weak 2-local derivation on an atomic von Neumann algebra or on a compact C${}^{\ast }$-algebra is a linear derivation.

We show that in an arbitrary Hilbert space, the set of group-invertible operators with respect to the core-partial order has the complete lower semilattice structure, meaning that an arbitrary family of operators possesses the core-infimum. We also give a necessary and sufficient condition for the existence of the core-supremum of an arbitrary family, and we study the properties of these lattice operations on pairs of operators.

Given an integer $n\ge 2$, in this article we provide a complete description of all additive surjective maps on the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space, preserving in both directions the set of Drazin invertible operators of index $n$.

In this article we present applications of Hardy-type and refined Hardy-type inequalities for a generalized fractional integral operator involving the Mittag-Leffler function in its kernel and for the Hilfer fractional derivative using convex and monotone convex functions.