A Hille-Yosida theorem for Bi-continuous semigroups

Sufficient conditions for the local null controllability of non-linear functional differential systems with unbounded linear operators in Banach space are established. The results are obtained using the semigroup of linear operators, fractional powers of operators, and the Schauder fixed-point theorem. Applications to parabolic differential systems are given.

Chapter 6 gives a concise introduction into the theory of one-parameter groups or semigroups of operators with an emphasis on the interplay between groups and semigroups and their generators. In the first section, one-parameter unitary groups are investigated, and two fundamental theorems, Stone’s theorem and Trotter’s formula, are proved. Semigroups of operators are applied to Cauchy problems for abstract differential equations on Hilbert space. Then generators of semigroups of contractions on Banach spaces are studied, and the Hille–Yosida theorem is proved. Finally, generators of contraction semigroups on Hilbert space are characterized as m-dissipative operators.

initial value problems (Cauchy problems) are usually studied via operator semigroups. In many cases, the well-developed theory of C0-semigroups, i.e., one-parameter operator semigroups which are strongly continuous for the norm on a Banach space X , suffices and provides a powerful machinery to study such problems. The applications range from partial differential equations, Volterra integro-differential equations and dynamic boundary problems to delay equations. It seems that a linear (autonomous) evolution equation like { u′(t) = Au(t) u(0) = x, (EE) that is an equation that describes a system evolving from an initial state ”in time”, can be handled via the theory of C0-semigroups. However, as the most trivial example shows this is not the case. Consider the left shift semigroup S on the space of bounded, continuous functions Cb(R) S(t)f(s) := f(t+ s). It is clear that the orbit t 7→ S(t)f is continuous for the supremum norm, if and only if f is uniformly continuous. This shows that the shift semigroup does not fit into the framework of C0-semigroups, nevertheless S clearly describes an evolution system on R. The situation is not so bad either. If we replace the norm topology by the topology τc of uniform convergence on compact sets, then the orbits become τc-continuous. This leads us to weakening the notion of C0-semigroups, thus allowing to consider such ”pathological” cases, these therefore turn out to be less unpleasant than thought previously. There are numerous generalisations of the theory of C0-semigroups. Among these we find the approach of introducing new continuity notions, this method is we want to follow. Investigations on semigroups on locally convex spaces were started fairly long ago, see e.g., [14], [51], [52] and [64]. A nice exposition on the historical aspects can be found in [56]. To exploit the Banach space structure and the same time to introduce coarser topologies, the notion of bi-continuous semigroups was introduced recently by Kuhnemund in [56]. The theory developed therein covers a large part of previously known results, and puts these concrete examples in an abstract framework. The general theory is then applicable in concrete cases.

The main result is that if $\{ T(t):t \geqslant 0\}$ is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space $X$ and if the resolutions of the identity for $T(t)$ are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure $K( \cdot )$ such that $T(t) = \smallint \exp (\lambda t)dK(\lambda )$, for $t \geqslant 0$. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that $R$ is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.

The starting point for this investigation was an attempt to generalize the well known theorem of Silverman and Toeplitz [25](1) on regularity of a sequence-to-sequence transformation. This theorem may be stated as follows: If a transformation of sequences { t, } of real numbers to sequences { Sm } is defined from a matrix { amn }, m, n = 1, 2, * , by the equations Sm =ZEn amntn, the transformation is regular-that is, is defined everywhere and takes every convergent sequence { t, } into another convergent sequence with the same limit-if and only if the matrix { amn } satisfies the conditions (a) limm dn 'tamn = 1, (b) limm amn = 0 for each n, and (c) there is a K such that En I amn, < K for every m. In the special case under consideration, the fact that regularity implies condition (c) (the non-trivial part of the proof) can be derived from a theorem of Banach [3, p. 80, Theorem 5 ]: If A and B are Banach spaces, and if U, n = 1, 2, * * , are. linear operators on A to B, such that lim supn || Un(a)ll < o? for each a in A, then lim supn 11 U711 < oo If the sequence of integers is replaced by a directed set X, it is known that A, B, X, and U. can be chosen for which the similar statement relating lim supx || Ux(a)ll and lim supx || Ux,l is false; sections 1-3 of this paper consider these cases in an attempt to solve the problem of boundedness: Characterize those Banach spaces A and B, and directed sets X such that choosing the linear operators U, on A to B so that lim supx || U,(a)ll < oo for each a in A implies that lim supx || Ux|| < oo. Section 1 is a review of pertinent facts about directed sets and convergence (mostly due to Moore and Smith [19], G. Birkhoff [5], and Tukey [27]). Section 2 studies the relations among three topologies in the space of operators on A to B. In ?3 the problem of boundedness is studied but not completely solved. The second part of the paper is concerned with certain special operators on some function spaces. In ?4 the space is that of the totally measurable functions on a set Y to a Banach space B; a class of operators on this space is defined in terms of additive, real-valued set-functions and the relations among various topologies in this set of operators is given; this is used in ?6 to give a general form to a theorem of Vulich [28]. In ?5 the functions studied are the measurable functions on Y to B; the operations on this space are defined in terms of completely additive, limited, set-functions whose values are transfor-

Exponential Stability of Operators and Operator Semigroups

In this dissertation we study time and space discretization methods for approximating solutions of abstract Cauchy problems and evolution equations in a Banach space setting. Two extensions of the Hille-Phillips functional calculus are developed. The first result is the Hille-Phillips functional calculus for generators of bi-continuous semigroups, and the second is a C-regularized version of the Hille-Phillips functional calculus for generators of C-regularized semigroups. These results are used in order to study time discretization schemes for abstract Cauchy problems associated with generators of bi-continuous semigroups as well as C-regularized semigoups. Stability, convergence results, and error estimates for rational approximation schemes for bi-continuous and C-regularized semigroups are presented. We also extend the Trotter-Kato Theorem to the framework of C-regularized semigroups and combine it with the time discretization methods previously mentioned in order to obtain fully discretized schemes, provided by A-stable rational functions. Among the applications, we outline how to use rational approximation schemes to approximate solutions of nonlinear ODE's, and we show the significance of the results for bi-continuous semigroups for obtaining new numerical inversion formulas for the Laplace transform (with sharp error estimates). Furthermore, rational approximation schemes for integrated semigroups are presented with applications to the second order abstract Cauchy problem.

Stability Properties Characterizing the Spectra of Operators on Banach Spaces

Several problems of the spectral theory of linear relations in Banach spaces are considered. Linear differential inclusions in a Banach space are studied. The construction of the phase space and solutions is carried out with the help of the spectral theory of linear relations, ergodic theorems, and degenerate operator semigroups.

: The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.

The characterizations of the stable perturbation of a closed operator by a linear operator in Banach spaces

Spaces of σ(p)-nuclear linear and multilinear operators on Banach spaces and their duals

Some new perturbation theorems for generalized inverses of linear operators in Banach spaces

By ℕ, ℤ, ℝ and ℂ we denote the natural, integer, real and complex numbers, respectively. We let ℕo ≔ ℕ ∪ {0}. Banach spaces over the field K ∈ {ℝ,ℂ} of real or complex numbers will be abbreviated by the letters X,Y,Z, operators between such spaces by R,S,T. By operators we always mean continuous linear operators between Banach spaces. The (Banach) space of continuous linear operators T from X to Y under the operator norm ‖T‖:= sup {‖Tx‖Y | ‖x‖X = 1} is denoted by L(X,Y). We let L(X) = L(X,X), BX = {x ∈ X | ‖x‖X ≤ 1} and write Id = IdX for the identity map on X. If A is a subset of X, [A] or span A denotes the closed linear hull of A in X. The topological dual of X is denoted X*(= L(X, K)) and the duality pairing written or x*(x) where x ∈ X, x* ∈ X*. The dual of an operator T is denoted by T*.

This paper focuses on boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and stability of closedness of linear relations under bounded and relatively bounded perturbations is studied. The results obtained in the present paper generalize the corresponding results for single-valued linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results.