Long time growth bounds

In most applications of semigroup theory one is given the generator Z explicitly, and has to infer properties of the solutions of the evolution equation ƒ′ (t) = Zƒ(t), i.e. of the semigroup Tt. This is not an easy task, and much of the analysis depends on obtaining bounds on the resolvent norms. We devote this section to establishing a connection between the spectrum of Z and the long time asymptotics of Tt. Before starting it may be useful to summarize some of the results already obtained. These include

1. (i) If ∥Tt∥ ≤ Meat for all t ≥ 0 then Spec(Z) ⊆ {z : Re(z) ≤ a} and ∥Rz∥ ≤ M/(Re(z) − a) for all z such that Re(z) > a. (Theorem 8.2.1)

2. (ii) If Rz is the resolvent of a densely defined operator Z and ∥Rx∥ ≤ 1/x for all x > 0, then Z is the generator of a one-parameter contraction semigroup, and conversely. (Theorem 8.3.2)

3. (iii) For every ε > 0 there exists a densely defined operator Z acting in a reflexive Banach space such that ∥Rx∥ ≤ (1+ ε)/x for all x > 0, but Z is not the generator of a one-parameter semigroup. (Theorem 8.3.10)

4. (iv) Tt is a bounded holomorphic semigroup if and only if there exist α > 0 and N < ∞ such that the associated resolvents satisfy ∥Rz∥ ≤ N|z|−1 for all z such that |Arg(z)| ≤ α + π/2. (Theorems 8.4.1 and 8.4.2)

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This volume is halfway between being a textbook and a monograph. It describes a wide variety of ideas, some classical and others at the cutting edge of current research. Because it is directed at graduate students and young researchers, it often provides the simplest version of a theorem rather than the deepest one. It contains a variety of examples and problems that might be used in lecture courses on the subject.

It is frequently said that over the last few decades there has been a decisive shift in mathematics from the linear to the non-linear. Even if this is the case it is easy to justify writing a book on the theory of linear operators. The range of applications of the subject continues to grow rapidly, and young researchers need to have an accessible account of its main lines of development, together with references to further sources for more detailed reading.

Probability theory and quantum theory are two absolutely fundamental fields of science. In terms of their technological impact they have been far more important than Einstein's relativity theory. Both are entirely linear. In the first case this is in the nature of the subject. Many sustained attempts have been made to introduce non-linearities into quantum theory, but none has yet been successful, while the linear theory has gone from triumph to triumph. Nobody can predict what the future will hold, but it seems likely that quantum theory will be used for a long time yet, even if a non-linear successor is found.

The fundamental equations of quantum mechanics involve self-adjoint and unitary operators. However, once one comes to applications, the situation changes.

Lp spaces

The serious analysis of any operators acting in infinite-dimensional spaces has to start with the precise specification of the spaces and their norms. In this chapter we present the definitions and properties of the Lp spaces that will be used for most of the applications in the book. Although these are only a tiny fraction of the function spaces that have been used in various applications, they are by far the most important ones. Indeed a large number of books confine attention to operators acting in Hilbert space, the case p: = 2, but this is not natural for many applications, such as those to probability theory.

Before we start this section we need to make a series of standing hypotheses of a measure-theoretic character. We recommend that the reader skims through these, and refers back to them as necessary. The conditions are satisfied in all normal contexts within measure theory.

1. (i) We define a measure space to be a triple (X, Σ, μ) consisting of a set X, a σ-field Σ of ‘measurable’ subsets of X, and a non-negative countably additive measure μ on Σ. We will usually denote the measure by dx.

2. (ii) We will always assume that the measure μ is σ finite in the sense that there is an increasing sequence of measurable subsets Xn with finite measures and union equal to X.

3. (iii) We assume that each Xn is provided with a finite partition εn, by which we mean a sequence of disjoint measurable subsets {E1, E2, …, Em(n)}, each of which has positive measure |Er| : = μ(Er). The union of the subsets Er must equal Xn.

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In this chapter and the next we discuss spline spaces defined on triangulations of the unit sphere S in ℝ3. The spaces are natural analogs of the bivariate spline spaces discussed earlier in this book, and are made up of pieces of trivariate homogeneous polynomials restricted to S. Thus, they are piecewise spherical harmonics. As we shall see, virtually the entire theory of bivariate polynomial splines on planar triangulations carries over, although there are several significant differences. This chapter is devoted to the basic theory of spherical splines. Approximation properties of spherical splines are treated in the following chapter.

Spherical Polynomials

In this section we introduce the key building blocks for spherical splines. Throughout the chapter we write ν for a point on the unit sphere S in ℝ3. When there is no chance of confusion, at times we will also write v for the corresponding unit vector. Before introducing spherical polynomials, we need to discuss spherical triangles and spherical barycentric coordinates.

Spherical Triangles

Suppose ν1, ν2 are two points on the sphere which are not antipodal, i.e., they do not lie on a line through the origin. Then the points ν1, ν2 divide the great circle passing through ν1, ν2 into two circular arcs. We write 〈ν1, ν2〉 for the shorter of the arcs. Its length is just the geodesic distance between ν1 and ν2.

Definition 13.1.Suppose ν1, ν2, ν3are three points on the unit sphere S which lie strictly in one hemisphere.