Victor Khatskevich

Fixed point theorems for holomorphic mappings and operator theory in indefinite metric spaces

In this paper we establish several new results on the existence and uniqueness of a fixed point for holomorphic mappings and one-parameter semigroups in Banach spaces. We also present an application to operator theory on spaces with an indefinite metric.

Some global properties of fractional-linear transformations

We consider some properties of fractional-linear transformations (such as focusedness and compactness of the image) which may be applied to the study behavior of solution of evolution problems (see, for example, [1], [2], [3]). Consider a solution x(t) = U(t)x 0, U(0) = I, of an evolution problem in a Hilbert space ℌ with the scalar product (·,·). The space ℌ is endowed with the corresponding indefinite metric V(t), the operator U(t) is a plus-operator (see below) with respect to V(t): (V(t)U(t)x, U(t)x) ≥ 0 for all x ∊ ℌ such that (V(t)x,x) ≥ 0. The study of the behavior of x(t) is naturally divided into the following three cases: 1) U(t) is invertible, i.e. (U(t))−1 exists, is defined on the whole of ℌ and is bounded (hyperbolic); 2) U(t) is bounded (parabolic); 3) U(t) is unbounded (elliptic).

Bistrict Plus-operators in Krein Spaces and Dichotomous Behavior of Irreversible Dynamical Systems

We consider the ball of angular operators determined in a Krein space, i.e. in a Hilbert space with indefinite metric, for which both positive and negative components can be infinite-dimensional in general. The weak compactness of the image and the preimage of this ball by a fractional-linear transformation is established. This transformation is generated by a continuous linear operator, which is not continuously invertible in general. We assume that this operator is a bistrict plus-operator acting from a Krein space to an another one. We apply the above result to the study of dichotomous behavior of solutions to a non-autonomous linear differential equation with an unbounded operator coefficient in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. As an example we study a diffusion process on the real axis.

One-parameter Semigroups of Fractional-linear Transformations

In 1964 F. Forelli [7] showed that to each strongly continuous one-parameter group of isometries on a Hardy space H P ,p≠2, p≥1, there corresponds a group of fractional-linear automorphisms of the unit disk in the complex plane. His study has been continued by many mathematicians (see, for example, [2], [3], [15], [8]). In particular, E. Berkson and H. Porta [3] developed a generation theory for one-parameter semigroups of holomorphic self-mappings of the unit disk and used their results to study the eigenvalue problem for composition operators in Hardy spaces.

Global implicit function and fixed point theorems for holomorphic mappings and semigroups

We establish several versions of a global implicit function theorem for holomorphic mappings in Banach spaces under invariance and flow invariance conditions. We then use these results to study the stationary points of one-parameter nonlinear semigroups with holomorphic generators.

Ergodic methods for the construction of holomorphic retractions

Let D be a bounded convex domain in a complex Banach space, and let F be a holomorphic self-mapping of D with a nonempty fixed point set. In this paper we study the flow generated by the mapping I — F on D, and use the asymptotic behavior of its Cesaro averages to construct a holomorphic retraction of D onto the fixed point set of F.

Nonlinear equations with holomorphic operators

Throughout this chapter we will consider complex Banach spaces only. In this case, many local features related to the solvability of equations with operators that are differentiable in the complex sense, turn out to be global.

Integration in normed spaces

Let us remind that by a function we mean a mapping having the domain in the scalar field \( \mathbb{K} \) of real or complex numbers (see Chapter 0, §1). In this chapter, we will consider functions with values in normed spaces — called vector-functions —, as well as functions with values in the field of scalars, which are called scalar-functions. The symbol f(t) means either the function f itself, or the value of f at t; the precise meaning will follow from the context. Prom now on the term “vector-function” will be used in a broader sense, for any mapping \( f:\mathfrak{X} \to \mathfrak{Y} \) with the domain in a finite dimensional space and with values in a normed space.

Operators on spaces with indefinite metric

Let us consider a complex Banach space \( \mathfrak{B} \)endowed with a norm ∥·∥, and decomposed in a topological direct sum