This is a review of a coherent body of knowlegde, which perhaps deserves the name of the geometric spectral theory of positive linear operators (in finite dimension), developed by this author and his co-author Hans Schncider (or S.F. Wu) over the past decade. The following topics are covered, besides others: combinatorial spectral theory of nonnegative matrices, Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, distinguished eigenvalues, cone-solvability theorems, the peripheral spectrum and the core, the invariant faces, the spectral pairs, and an extension of the Rothblum Index theorem. Some new insights, alternative proofs, extensions or applications of known result are given. Several new new results are proved or announced, and some open problems are also mentioned.

By a continuous degenerate semigroup we mean a strongly continuous mapping T : R+ ! L(X) having the semigroup property. Thus, T (0) is a projection which may be different from the identity. The main theorem is a Trotter-Kato type approximation result for such degenerate semigroups. It is used to study convergence of heat semigroups with respect to variable domains.

The existence, uniqueness and continuous dependence of global weak solutions of coupled sine-Gordon equations are established in the framework of variational method due to Dautray and Lions. As an application of weak solutions, we solve the quadratic optimal control problems for the control systems described by coupled sine-Gordon equations.

It is well-known that if Y is a closed subspace of a separable Banach space X; and if Y admits a locally uniformly rotund (LUR) norm, then this LUR norm on Y can be extended to a LUR norm on X: (See [4,5] and II.8 of [1].) In [2], Fabian used a new technique to extend the result to other forms of rotundity without requiring X to be separable. However in [2], the subpace Y has to be reflexive. The second-named author later showed that the condition that Y is reflexive could be removed. (See [8] and [9].) It is worth noting that the techniques used in the above three instances are completely different and each has its own strength. Fabian’s extension preserves moduli of convexity of power type. The extension given in [8] is simple and turns out to be very useful in many situations.

In this paper we present a few facts and open problems about uniformly lipschitzian mappings and lipschitzian retractions.

We give a general method for constructing symmetric sequence spaces that for $1 \lt q \lt 2$ satisfy a lower q-estimate but fail to be q-concave and, for $2 · q \lt 1$, have the q-Orlicz property but fail to be q-concave. In particular, this gives examples of spaces with the 2-Orlicz property but without cotype 2.

Let X be a Banach space, $X_2 \subseteq X$ be a two dimensional subspace of $X$, and $S(X) = \{x \in X, ||x|| = 1\}$ be the unit sphere of $X$. The relationship between the normal structure and the arc length in X is studied. Let $R(X) = \mbox{inf} \{l(S(X_2)) - \gamma(X_22) : X_2 \subseteq X\}$, where $l(S(X_2))$ is the circumference of $S(X_2)$ and $\gamma(X_2) = \mbox{sup}\{2(||x + y|| + ||x - y||) : x; y \in S(X_2)\}$ is the least upper bound of the perimeters of the inscribed parallelogram of $S(X_2)$. The main result is that $R(X) \gt 0$ implies $X$ has the uniform normal structure.

Farthest point theory is not so rich and developed as nearest point theory, which has more applications. Farthest points are useful in studying the extremal structure of sets; see, e.g., the survey paper [14]. There are some interactions between the two theories; in particular, uniquely remotal sets in Hilbert spaces are related to the old open problem concerning the convexity of Chebyshev sets. The aim of this paper is twofold: first, we indicate characterizations of inner product spaces and of infinite-dimensional Banach spaces, in terms of remotal points and uniquely remotal sets. Second, we try to update the survey paper [15], concerning uniquely remotal sets.

We consider the best approximation by projections in Banach spaces under certain suitable conditions. Furthermore, applications are discussed for multiplier operators and convolution type operators associated with strongly continuous families of bounded linear operators as well as for homogeneous Banach spaces which include the classical function spaces, as particular cases.

In this paper, we first consider an iteration scheme given by an infinite family of nonexpansive mappings in Banach spaces and then prove a strong convergence theorem for the family of nonexpansive mappings. Using this result, we consider the feasibility problem of finding a solution of infinite convex inequalities and the problem of finding a common fixed point of infinite nonexpansive mappings.

The paper deals with the general theme of what is known about the existence of fixed points and approximate fixed points for mappings which satisfy geometric conditions in product spaces. In particular it is shown that if X and Y are metric spaces each of which has the fixed point property for nonexpansive mappings, then the product space $(X \times Y )_\infty$ has the fixed point property for nonexpansive mappings satisfying various contractive conditions. It is also shown that the product space $H = (M \times K)_\infty$ has the approximate fixed point property for nonexpansive mappings wheneverM is a metric space which has the approximate fixed point property for such mappings and K is a bounded convex subset of a Banach space.

The stability of solutions of the equation $Bu^0(t) = Au(t)$ is considered, where A and B are closed linear operators on a Banach space. Under the well-posedness condition it is proved that if the imaginary part of the spectrum of the pencil (¸B ¡ A) is countable, then a bounded uniformly continuous solution u(t) of the equation is asymptotically almost periodic if and only if the functions e¸ tu(t), (¸ 2 iR), have uniformly convergent means. A condition of exponential stability also is given when the generalized eigenvectors and associated root vectors of the linear pencil (¸B ¡ A) form a Riesz basis.

Let A be a C¤ -algebra. Let z be the maximal atomic projection and p a closed projection in A¤ ¤ . It is known that x in A¤¤ has a continuous atomic part, i.e., zx = za for some a in A, whenever x is uniformly continuous on the set of pure states of A. Under some additional conditions, we shall show that if x is uniformly continuous on the set of pure states of A supported by p, or its weak* closure, then pxp has a continuous atomic part, i.e., zpxp = zpap for some a in A.