This is a survey of the author’s recent results on the Kadison and Halmos similarity problems and the closely connected notion of “length” of an operator algebra.

In this survey article, we explore various natural situations where the results about strict convexity of X¤ like Vlasov’s Theorem or Taylor- Foguel Theorem are actually seen to be locally consequences of properties of rotund points.

This is a survey of results about norm-one projections and 1- complemented subspaces in K¨othe function spaces and Banach sequence spaces. The historical development of the theory is presented from the 1930s to the newest ideas. Proofs of the main results are outlined. Open problems are also discussed. Every effort has been made to include as complete a bibliography as possible.

We show that the Brouwer fixed point theorem is equivalent to a number of results closely related to the Euclidean spaces or n-simplexes or n-balls. Among them are the Sperner lemma, the KKM theorem, some intersection theorems, various fixed point theorems, an intermediate value theorem, various non-retract theorems, the non-contractibility of spheres, and others.

In this paper, some recent advances and open problems on perturbations and extensions of the isometric operators are presented. It is shown that for the spaces $B(L^\beta(\mu) \longrightarrow L^\beta(\nu)) (0 \lt \beta \lt 1), B(L^\infty(\mu) \longrightarrow L^\infty(\nu))$ (with some special measure spaces), $B(E_{(2)} \longrightarrow L^1(\mu)), B(E_{(n)} \longrightarrow C(\Omega))$ and $B(X \longrightarrow L^\infty(\mu))$, where $X$ is a uniformly smooth Banach space with $\dim X=1$ or $\infty$ and $(\Omega, \mu)$ is a purely atomic or purely non-atomic finite measure space and so on, the answer to the problem of the isometric approximations is positive. However, for the spaces $B(l^1 \times l^\infty), B(L^1(\mu) \longrightarrow L^\infty(\nu))$ and $B(L^1(\mu) \longrightarrow C_b(\Delta))$, the answer is negative.

This paper is devoted to static bifurcation theory for a class of degenerate boundary value problems for diffusive logistic equations with indefinite weights which model population dynamics in environments with spatial heterogeneity. The purpose of this paper is to discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem.

This paper deals with some Feller semigroups acting on a particular weighted function space on [0;+1[ whose generators are degenerate elliptic second order differential operators. We show that these semigroups are the transition semigroups associated with suitable Markov processes on [0;+1]. Furthermore, by means of a sequence of discrete-type positive operators we introduced in a previous paper, we evaluate the expected value and the variance of the random variables describing the position of the processes and we give an approximation formula (in the weak topology) of the distribution of the position of the processes at every time, provided the distribution of the initial position is given and possesses finite moment of order two.

The aim of this paper is to discuss some recent developments of the perturbation method introduced first by Kato for the Kolmogoroff equation and later extended by Voigt and Arlotti to deal with a range of problems related to the solvability of the so-called Master Equation. The paper consists of two parts. In the first we recall and unify some abstract results on generation of substochastic semigroups. In the second we perform a detailed analysis of an equation from the polymer degradation theory to demonstrate a number of possible generation cases.

We present here various facts and examples concerning the evaluation of the minimal displacement $d(T) = \mbox{inf} ||T x -x||$ for some classes of Lipschitz self mappings of balls in Banach spaces