The basic role of the single-valued extension property (SVEP) arises in the spectral decomposition theory, since every decomposable operator T enjoys this property, as does its dual T′. Indeed, in part IV, Chapter 21 it has been shown that the decomposability of an operator may be viewed as the union of two properties, the so-called Bishop's property (β) and the property (δ). Property (β) for T implies the SVEP for T (see part IV, Chapter 21) and, as observed in part IV, Chapter 23, properties (β) and (δ) have a complete duality, so that, if T has (δ), then the dual T′ has (β) and therefore SVEP.

The main goal of this chapter is to investigate in detail a localized version of SVEP. First we shall show that local spectral theory provides a suitable frame for some characterizations of the analytical core and of the quasi-nilpotent part. Then we shall use these characterizations to describe the localized SVEP by means of a variety of conditions that involve the analytical core and the quasinilpotent part of an operator, as well as the generalized range and the generalized kernel.

The SVEP at a point

To explain the role of SVEP in local spectral theory we begin with some preliminary and well-known facts from operator theory.

In Chapter 27 we established many conditions which imply SVEP at a point. The main goal of this chapter is to show that all these implications become equivalences for an important class of operators, the class of all semi-Fredholm operators. It will be also shown that, for these operators, SVEP at a point λ0 ∈ ℂ is equivalent to the finiteness of two important quantities, the ascent and the descent of the operator λ0I – T. These equivalences also provide useful information on the fine structure of the spectrum. In particular, we shall show that many spectra originating from Fredholm theory coincide whenever T or T′ have SVEP.

Ascent, descent, and semi-Fredholm operators

Let us recall the definition of some classical quantities associated with an operator. Given a linear operator T on a vector space X, it is easy to see that ker for every n ∈ ℕ.

Definition 28.1.1The ascent of T is the smallest positive integer p = p(T), whenever it exists, such that ker Tp = ker Tp+1. If such p does not exist, set p = ∞. Analogously, the descent of T is defined to be the smallest integer q = q(T), whenever it exists, such that. If such q does not exist, set q = ∞. If both p(T) and q(T) are finite, then T has finite chains.

Multipliers

In this chapter I want to give an impression of what sort of information becomes available via local spectral theory when it is applied to a particular class of operators, namely the multipliers on a commutative Banach algebra. Throughout this chapter, the letter A will denote a commutative, complex Banach algebra.

Definition 25.1.1A linear map T: A → A is a multiplier if aT (b) = T (a)b for all a, b ∈ A. The set of multipliers on A is denoted by M(A).

The most obvious example, given A, is the multiplication operator La induced by a fixed element a ∈ A, that is, the operator La(b):= ab for all b ∈ A. If A has a unit e (A is unital) then every multiplier T is a multiplication operator. In this case (Exercise 25.3.2).

If the map a → La: A → B(A) is injective (faithful), then A is said to be a faithful algebra. Every unital algebra is faithful, as is every semisimple, and also every semiprime algebra – the latter term means, in our commutative case, that the algebra contains no non-zero nilpotent elements.

Example 25.1.2 Let A:= C0(Ω) be the Banach algebra of all continuous complex-valued functions vanishing at ∞ on the locally compact Hausdorff space Ω, and let f ∈ Cb(Ω) be a bounded continuous function on Ω. Then T := Lf (notation self-explanatory by now) is a multiplier. Conversely, it can be shown that a multiplier gives rise to a bounded continuous function on Ω with respect to which the multiplier acts by pointwise multiplication (Exercise 25.3.4).

In this chapter we shall explore the surprising relations between decomposability and properties (α) and (δ).

Duality between (β) and (δ)

We have already seen, in Theorem 21.2.8, that T is decomposable if and only if it has both (β) and (δ). But the main reason for emphasizing these two properties is not just that they together describe decomposability – surely (β), in particular, is too technical, and too non-intuitive, to gain fame just for that! Their main conceptual raison d··etre is that they possess a remarkable dual relationship: an operator will have one of them (either one!) precisely when its adjoint operator has the other one. This is the significant conclusion that is provided by the duality theory. (You should note that the adjoint operator is called the dual operator in Part I.)

The duality theory for operators that we are talking about here goes back to Errett Bishop's PhD thesis, published in Bishop (1959), where he developed a spectral theory for an arbitrary bounded linear operator on a reflexive Banach space. Bishop called his development a duality theory, because the operator and its adjoint are involved. In Bishop (1959) we can see, in more or less fully developed form, much of what are now basic tools and concepts of the field, such as the glocal spectral subspaces, and conditions (β) and (δ). Therewas even a precursor of decomposability (which Bishop called duality theory of type 3).

Knowledge of the structure of L1(G) can be used to show that derivations from L1(G) are automatically continuous when G is abelian, compact, or is in one of several other classes. However it is not known whether such derivations are continuous for every locally compact group G. Work on this problem leads to further questions in abstract harmonic analysis.

Automatic continuity of derivations

It was shown by Ringrose (1972) that every derivation from a C*-algebra A is automatically continuous. Powerful general techniques for proving automatic continuity results have been developed from this and earlier work; see Part I and the substantial account in Dales (2000). The following general method for proving automatic continuity of derivations was given by Jewell (1977).

Theorem 12.1.1Let A be a Banach algebra such that:

1. every closed two-sided ideal with finite codimension in A has a bounded approximate identity; and

2. if I is a closed two-sided ideal with infinite codimension in A, then there are sequences (anand (bn) in A such that bna1…an−1 ∉ I, but bna1…an ∈ I for each n ≥ 2.

Then every derivation D : A → X, where X is a Banach A-bimodule, is continuous.

Condition 2 of the theorem is used to show that the continuity ideal (see Part I and Exercise 12.4.1) of any derivation D : A → X cannot have infinite codimension. A variant of the Main Boundedness Theorem of Bade and Curtis (see Theorem 5.2.2) is used to show this.