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We convey the fundamental behind the concept “functional calculus” (Section 1.1). Then we present a of certain ideas common to many functional calculus constructions. In particular, we introduce a method of extending an elementary functional calculus to a larger algebra (Section 1.2). In Section 1.3 we introduce the notion of a for a closed operator on a Banach space and specialise the abstract results from Section 1.2. As an important example we treat (Section 1.4). In Section 1.3.2 we prove an abstract composition rule for a pair of meromorphic functional calculi.

In Section 2.1 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation. In Section 2.2 we introduce some notation for certain spaces of holomorphic functions on sectors. A functional calculus for sectorial operators is constructed in Section 2.3 along the lines of the abstract framework of Chapter 1. Fundamental properties like the composition rule are proved. In Section 2.5 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible. In this way a panorama of functional calculi is developed. In Section 2.6 some mixed topics are discussed, e.g., adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results. Section 2.7 contains a spectral mapping theorem.

In this chapter we present the basic theory of fractional powers of a sectorial operator A, making efficient use of the functional calculus developed in Chapter 2. In Section 3.1 we introduce fractional powers with positive real part and give proofs for the scaling property, the laws of exponents, the spectral mapping theorem, and the Balakrishnan representation. Furthermore, we examine for variable > 0 the behaviour of ( + ), and the behaviour of for variable . In Section 3.2 we generalise the results from Section 3.1 to fractional powers with arbitrary real part. (Here, the operator has to be injective.) In Section 3.3 we introduce the Phillips calculus for generators of bounded semigroups. Then the definition and the fundamental properties of holomorphic semigroups are presented in Section 3.4. The usual generator/semigroup correspondence is extended to the case of multi-valued operators. In Section 3.5 the logarithm of an injective sectorial operator is defined and Nollau’s theorem is proved. Finally, the connection of log with the family of imaginary powers () of .

As a straightforward abstraction of the logarithm of an injective sectorial operator we introduce the notion of a strip-type operator (Section 4.1). Since the resolvent of a strip-type operator by definition is bounded outside a horizontal strip, a functional calculus based on Cauchy integrals can be set up (Section 4.2). Section 4.3 is devoted to prove the main result, which states equality between the spectral angle of an injective sectorial operator and the spectral height of the strip-type operator log . As a corollary one obtains an important theorem of and , saying that in the case where ∈ BIP, the group type of () is always larger than the spectral angle of . In Section 4.4 the problem of ‘inversion’ is discussed, namely the question, which strip-type operators are actually logarithms of sectorial operators. Here we present a theorem of Monniaux, slightly generalised. In Section 4.5 we construct the example of an injective sectorial operator ∈ BIP on a UMD space with the property that the group type of () is larger than π.

This chapter mainly provides technical background information. We start with the so-called (Section 5.1) and some fundamental boundedness and approximation results (Section 5.2). Then we prove equivalence of boundedness of -functional calculi for different subalgebras of (Section 5.3). We also introduce and study the -angle (Section 5.4) and present permanence results with respect to additive perturbations (Section 5.5). Finally, we prove a technical lemma which indicates the connections with Harmonic Analysis (Section 5.6).

In the present chapter we examine the connections between functional calculus and interpolation spaces. As an ‘appetiser’, in Section 6.1 we present two central ideas: a model describing the real interpolation spaces (, ()) using the functional calculus and a theorem of . In Section 6.2 we examine the first of these ideas, proving several representation results for the spaces (, ()). Then we introduce extrapolation spaces for injective operators (Section 6.3). With the help of these spaces in Section 6.4 we derive two fundamental results (Theorem 6.4.2 and Theorem 6.4.5) that lead to more characterisations of interpolation spaces by functional calculus (Section 6.5.1) and a generalisation of Dore’s theorem (Section 6.5.3). In Section 6.6 we establish all the common properties of fractional domain spaces as intermediate spaces: density, the moment inequality, reiteration. Moreover, we prove the intriguing fact that for an operator ∈ BIP() the fractional domain spaces equal the interpolation spaces (Theorem 6.6.9). Finally we characterise growth conditions like sup ‖( + )‖ < ∞ in terms of interpolation spaces (Section 6.7).

We start with providing necessary background information on the functional calculus on Hilbert spaces. In Section 7.1 we show how numerical range conditions account for the boundedness of the -calculus. In the sector case this is essentially von Neumann’s inequality (Section 7.1.3), in the strip case it is a result by and (Section 7.1.5). Prom von Neumann’s inequality we obtain certain ‘mapping theorems for the numerical range’ (Section 7.1.4). Section 7.2 is devoted to -groups on Hilbert spaces. We discuss Liapunov’s direct method (Section 7.2.1) from Linear Systems Theory and apply it to obtain a remarkable decomposition and similarity result for group generators (Section 7.2.2). This allows us to prove a theorem of and on the boundedness of the -calculus on strips for such operators. This result can be approached also in a different way, yielding in addition a characterisation of group generators (Section 7.2.3). Section 7.3 is devoted mainly to the connection of the functional calculus with similarity theorems. The main tool is fundamental result on the boundedness of the -calculus (Section 7.3.1). The similarity questions we are interested in deal with operators defined by sesquilinear forms. We introduce these operators in Section 7.3.2 and obtain in Section 7.3.3 a characterisation of such operators up to similarity. Afterwards, several theorems on similarity are proved, related also to the so-called . We give an example of a -semigroup which is not similar even to a quasi-contractive semigroup (Section 7.3.4). Finally, we present applications to generators of cosine functions, showing in particular that after a similarity transformation those operators always have numerical range in a horizontal parabola (Section 7.4).

We treat constant coefficient elliptic operators on the euclidean space ℝ. The main focus lies on the connection of functional calculus with Fourier multiplier theory. The -theory is the subject of Section 8.1 while the -case is presented in Section 8.2. Then we apply the obtained results to the negative Laplace operator (Section 8.3). The universal extrapolation space for the Laplace operator is identified with a space of certain (equivalence classes) of tempered distributions. In Sections 8.4 and 8.5 we treat the derivative operator on the line, the half-line and finite intervals. The UMD property of a Banach space is characterised by functional calculus properties of the derivative operator on -valued functions.

In this chapter we present three different topics related to the functional calculus. The first (Section 9.1) is a review of some counterexamples; the common feature here is the use of spaces with bases that are not unconditional. The second (Section 9.2) is an application of functional calculus methods in theoretical numerical analysis, namely stability and convergence results for rational approximation schemes. The final Section 9.3 is devoted to regularity questions of solutions of inhomogeneous Cauchy problems, in particular to the so-called .