122167029
Admissible observation operators for linear semigroups
Consider a semigroupT on a Banach spaceX and a (possibly unbounded) operatorC densely defined inX, with values in another Banach space. We give some necessary as well as some sufficient conditions forC to be an admissible observation operator forT, i.e., any finite segment of the output functiony(t)=CTtx,t≧0, should be inLp and should depend continuously on the initial statex. Our approach is to start from a description of the map which takes initial states into output functions in terms of a functional equation. We also introduce an extension ofC which permits a pointwise interpretation ofy(t)=CTtx, even if the trajectory ofx is not in the domain ofC.
8267029
Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity
We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form x(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) , like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Foures and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.
115711723
The representation of regular linear systems on Hilbert spaces
119810829
The resolvent growth assumption for semigroups on Hilbert spaces
Let X be a Hilbert space and let T = (T1),,, be a C, semigroup of linear operators on X, with generator A. The aim of this paper is to show how information about the behavior of T, for large t can be obtained from information about the behavior of the resolvent (slA)-’ for s with large imaginary part, and vice versa. In particular, the type of U can easily be determined if we know on which vertical lines in the complex plane the resolvent is bounded. We start by defining a sequence of numbers (oU,), which give information about the behavior of the semigroup for large t.
19600533
H CLASSES OF HOLOMORPHIC FUNCTIONS IN TUBE DOMAINS.
124018984
Complex Methods in Harmonic Analysis
121398700
Torsion of a Shaft with a Toroidal Cavity
This paper gives the solution to the torsion problem for a near‐cylindrical shaft of circular section having a toroidal cavity. Since the interpretation of the axially symmetric problem as a flow problem in 5 dimensions is well known, the first part of the paper is devoted to developing a method for handling the flow problem about an n dimensional ring‐shaped body of general cross section. In the second part of the paper this method is applied to the case of an n dimensional torus ring. The case n=3 then yields the solution to the classical flow problem, and n=5 gives the solution to the torsion problem under consideration.
118896808
A kernel associated with certain multiply connected domains and its applications to factorization theorems
54083834
Interpolation of operators with change of measures
Introduction. This paper has two purposes. First, it is to give general interpolation theorems which allow one to change measures simultaneously with changing exponents of the Lp classes concerned. These theorems are generalizations of theorems of M. Riesz [7] and Marcinkiewicz [4]. The idea of interpolating between measure seems to have been implicit for some time. The general situation has, however, only recently been investigated. (See e.g. [2] and [8].) Our second purpose is to apply these techniques. In §§3 and 4, we shall prove theorems for unbounded orthonormal systems which generalize known results for bounded orthonormal systems. These theorems are extensions of theorems of Paley, Marcinkiewicz and Zygmund [5], and Pitt [6]. §1 will be devoted to necessary definitions and statements of needed known results. §2 will contain the general interpolation theorems. 1. Definitions and statement of general results. Let (M, 93J, p) and (N, yi, v) be two measure spaces. We will consider only measurable, real or complex-valued functions on these two spaces. In this paper we will establish a generalization of the well known convexity theorem of M. Riesz (Theorem (1.1) below) and give some applications of this new theorem to certain orthonormal series. In order to do this we will need to establish some notation and give a few definitions. Suppose T is a mapping of a class of functions on M into a class of functions on N. T is called a sublinear operator if it satisfies the following properties: (i) If/=/i+/2 and Tft (i=l, 2) are defined then Tf is defined; (ii) | T(fx+f2) | ^ | Tf\\ + | Tf2\ almost everywhere; (iii) For any scalar k, we have | T(kf)\ =\k\\ Tf\ almost everywhere. Let p, g2t 1 be two real numbers. We say that T is of type (p, q) in case T is defined for all functions/ in L"(M, W, ju) and there exists a positive number, K, independent of/such that