A key focus of this text is to relate the manner in which fluctuations in flow or thermodynamic variables propagate and interact in combustion systems. In this chapter, we demonstrate that combustor disturbances can be decomposed into three canonical types of fluctuations – acoustic, entropy, and vorticity disturbances. This decomposition is highly illustrative in understanding the spatial/temporal dynamics of combustor disturbances [1]. For example, the velocity field can be decomposed into acoustic fluctuations, which propagate at the speed of sound with respect to the flow, and vorticity fluctuations, which are advected by the flow. This decomposition is important because, as shown in Chapters 11 and 12, two velocity disturbances of the same magnitude can lead to very different influences on the flame, depending on their phase speeds and space–time correlation. Section 2.9 further emphasizes how this decomposition provides insight into behavior measured in a harmonically oscillating flow field.

In his very influential article [72], Halmos listed ten problems for Hilbert space operators. One of these (Problem 4) asked “Is every normal operator the sum of a diagonal operator and a compact one?” Soon after the question was raised, Berg [20] and Sikonia [129] independently of each other showed that the answer is “yes”. Much of what follows reproduces their theorem generalizing the Weyl–von Neumann theorem. Moreover, we recast this theorem in the language of the ‘Ext” group.

Essentially Normal Operators

An essentially normal operator defines an extension of C(X), where X is the essential spectrum of T, by the compact operators, or equivalently, a*-monomorphism of C(X) into the Calkin algebra and conversely. One of the main goals of this chapter is to show the advantage of studying the class of essentially normal operators using the C*-extensions they define. We begin by recalling the notion of an essentially normal operator and other notions closely related to it.

Definition 2.1.1

An operator T in is essentially normal if the self-commutator [T, T*] of T is compact. We say that T is essentially unitary if T is essentially normal and T*T − I is compact. Two operators T1 and T2 in are said to be essentially equivalent if there exists a unitary operator and a compact operator such that. In this case we write, T1 ~ T2.

Remark 2.1.2 Let T1, T2 in be essentially equivalent. Then,

• (1) T1 is essentially normal if and only if so is T2.

• (2) T1 is Fredholm if and only if so is T2. In this case, ind(T1) = ind(T2).

In particular, T1 and T2 have the same essential spectra and index data.

Note that any operator is essentially equivalent to its compact perturbation. Here is one concrete illustration of this general fact.

Example 2.1.3 Let be a sequence of positive numbers such that

The weighted Hardy space of the open unit disc, denoted by is given by

is endowed with the inner-product

The shift operator is given by

If for all integers; then is nothing but the Hardy space introduced in Example 1.5.2. In this case, Mz agrees with the shift operator.

So far, we have exclusively focussed on the original proof [26, Theorem 11.1] of Brown, Douglas, and Fillmore classifying essentially normal operators. In what follows, we will refer to this proof as the “BDF proof”. We have therefore left out other proofs that simplify parts of the BDF proof. Indeed, there is a proof of the BDF theorem proposed by O’Donovan [104] that separates the techniques obtained from homological algebra and algebraic topology used in the proof of BDF from that of techniques obtained from operator theory. Although, it might appear surprising at first, it turns out that the BDF theorem is actually equivalent to what may appear to be a much simpler statement: If the index of an essentially normal operator T is trivial, then T must be of the form N + K for some normal operator N and a compact operator K. This is Exercise 4.6.4.

As we have seen, it is not difficult to show that Ext(X) is an abelian semi-group. With a little more effort, the existence of a unique element that serves as the identity in Ext(X) is established. However, the proof in [23] of the existence of the inverse in Ext(X) is intimidating. A simpler proof due to Arveson [6] (see also [40]) appeared soon afterwards. Secondly, the BDF theorem established that Ext is a covariant functor from the category of compact metric spaces to abelian groups (Corollary 2.7.1) naturally leading to the question of its connection with other known functors from topology. This question was investigated vigorously and its connections with K-theory was eventually established on a firm footing. We discuss some of these developments in the following sections.

Finally, we conclude this short chapter with the discussion of several open problems, which include the Arveson–Douglas conjecture for semi-invariant modules of Hilbert modules over function algebras and the problem of classifying commuting “homogeneous” essentially normal operators.

Other Proofs

Here we briefly summarize the simplification due to Arveson of the proof that Ext(X) is a group. There were two other papers, one by Davie and the other by O’Donovan, that provided simplifications to parts of the BDF proof. We describe them in this section.

The purpose of this chapter is two-fold. First, it serves as a rapid introduction to some of the basics of modern operator theory. Second, it has all the prerequisites needed to prove the Brown–Douglas–Fillmore theorem.

Unless otherwise stated, all Hilbert spaces considered in this text are assumed to be complex and separable. Whenever separability is not needed or it does not simplify the situation, the same is mentioned. Throughout this text, denotes a complex Hilbert space and stands for the algebra of bounded linear operators from into. Note that is a unital C*-algebra, where the identity operator I is the unit, composition of operators is the multiplication and the uniquely defined adjoint T* of a bounded linear operator T on Hilbert space is the involution (the reader is referred to Appendix B for the definition of a C*-algebra). To avoid ambiguity, whenever necessary, we let denote the identity operator on. Given, the symbols ker T and ran T stand for the kernel and range of the operator T, respectively. As usual, we let and denote the norm and the inner product in the Hilbert space.

Partial Isometries and Polar Decomposition

If λ is a nonzero complex number, then λ = |λ|eiθ for some real number θ; this is the polar decomposition of λ. Theorem 1.1.3 provides, for operators in L(H), a similar decomposition. The challenge is to find the two factors analogous to |λ| and eiθ in. A natural choice for |λ| quickly presents itself, namely, the operator (T*T)1/2. The choice for eiθ would seem to be either a unitary or an isometry; however, none of these choices is quite correct for an operator on a Hilbert space of dimension greater than one as we will see here.

Defnition 1.1.1

An operator is a partial isometry if for all that are orthogonal to ker. If, in addition, the kernel of V is ﹛0﹜, then V is said to be an isometry. The initial space of a partial isometry V is defined as the orthogonal complement (ker V)⊥ of ker V, whereas the final space of V is the range ran V of V.

This appendix is devoted to the various forms of spectral theorems for normal operators used in this book. We also present several applications of the spectral theorem.

Spectral Theorem

Let Σ be a σ-algebra over a set Ω. A spectral measure is an orthogonal projection-valued mapping with E(Ω) = I such that

• (1) and for every

• (2) is a complex measure for every.

Example C.1.1 (Multiplication by Characteristic Function as Spectral Measure)

Consider the measure space. Note that for each is essentially bounded with. Let denote the operator of multiplication by on. We check that the mapping from into defines a spectral measure:

• (1) Notice that is real-valued, it follows that is self-adjoint. Moreover, as.

• (2) Since, we have. Further if, and hence, in this case.

• (3) Finally, note that for any,

As is integrable, defines a complex measure.

Example C.1.2 (Projection as Spectral Measures) Let be any set, be its power set, and be any separable Hilbert space. Fix a sequence in, and an orthonormal basis. By Parseval's identity, one may rewrite as. For, let Define by

Clearly,. As for every, we have is easily seen to be a spectral measure.

Here is the first version of the spectral theorem for normal operators.

Theorem C.1.3 (Spectral Theorem)

If is a normal operator, then there exists a unique spectral measure E on the Borel σ-algebra B(σ(N)) which satisfies for all,

Moreover, whenever.

Let K be a compact subset of the complex plane. Let B∞(K) denote the normed algebra of complex-valued bounded Borel-measurable functions on K endowed with the sup norm.

Suppose is normal with the spectral measure E as guaranteed by Theorem C.1.3. Then, for every, there exists a unique normal operator such that for all.

Further, the map defines a contractive algebra *-homomorphism, which is isometric on the algebra C(σ(N)) of continuous functions on σ(N).

An outline of the proof of the last two theorems will be presented later in this appendix. In the remaining part of this section, we discuss several applications of spectral theorem (see also Exercises 1.23–1.26). The following says that a *-cyclic normal operator can be realized as a multiplication operator Mz on an L2 space.

Corollary C.1.1

Let be a normal operator.

In this chapter, we discuss several applications of the BDF theorem. These include a model theorem for a family of essentially normal operators, classification of essentially homogeneous operators, description of essential spectra of essentially normal circular operators. We also discuss applications to the theories of hyponormal and m-isometric operators. We conclude this chapter with a fairly long discussion on essentially reductive quotient sub-modules.

Bergman Operators and Surjectivity of

Let G be a bounded open subset of the complex plane C and let dA denote the normalized area measure on G. Let (or stand for the Hilbert space of dA-square integrable Lebesgue measurable functions on G (with two functions being identified if they are dA-almost everywhere equal to each other). Let

The normed linear space, with the norm inherited from, is a reproducing kernel Hilbert space as shown here.

Lemma 5.1.1

For any compact subset K of G; there exists such that

In particular, for every, the point evaluation is bounded on and is a Hilbert space.

Proof Let and let. By the mean value property for analytic functions,

It follows now from the Cauchy–Schwarz inequality that

As r can be taken to be arbitrarily close to dist; we get

Let and note that by applying the estimate above, we obtain

A routine compactness argument now yields (5.1.1). To see that is complete, it suffices to check that is closed in. By (5.1.1), the convergence in yields the uniform convergence on all compact subsets of G. As uniform limit of analytic functions is analytic, the desired conclusion may be derived from the fact that every convergent sequence in has a subsequence converging pointwise almost everywhere.

By Lemma 5.1.1 and the Riesz representation theorem (see Lemma B.1.8), for any w ∈ G, there exists such that

The two-variable function, is uniquely determined by the reproducing property (5.1.2).

We now define the Bergman operator BG as the operator of multiplication by z on. As G is a bounded subset of, defines a bounded linear operator on. Moreover, for every is an eigenvalue of. Indeed, for any,

that is,. This also shows that, and more importantly, the spectrum of BG has nonempty interior.

Let us compute the essential spectra and index data of Bergman operators.

The first splitting lemma allowed us to split every extension [τ] in Ext(X) with respect to the closed cover ﹛A, B﹜ of X, where X = A ∪ B and A ∩ B = ﹛x0﹜. However, we will actually need a stronger form of splitting, one that allows any extension [τ] in Ext(X) to split with respect to the closed cover ﹛A, B﹜ of X such that A ∩ B is homeomorphic to a closed interval rather than a point. The precise statement is given at the end of first section (see Corollary 4.1.1).

Second Splitting Lemma

We begin with generalizations of Lemmas 2.2.1 and 2.2.2. The first one is attributed to von Neumann.

Lemma 4.1.1

Let be a finite dimensional subspace of and. If H is a self-adjoint operator, then there exists a finite dimensional subspace and a compact self-adjoint operator such that is reduced by and.

Proof The proof is a slight modification of that of Lemma 2.2.1. Assume that H is a self-adjoint operator. Let be a decomposition of σ(H) into a finite number of disjoint intervals of length less than and let

where E is the spectral resolution of H. Clearly, contains. For i = 1, …, N, let Ei denote the orthogonal projection of onto and. Note that H + K commutes with E. In fact,

Moreover, since E is of finite rank, K is compact. To complete the proof, we must check that. To see that, let be the mid-point of: Then,

and hence, we obtain

This completes the proof of the lemma.

Before we extend the last lemma to several compact self-adjoint operators, we formally define n-diagonal operator matrices.

Defnition 4.1.1

Let n be a non-negative integer and let for every integer. An operator matrix is said to be n-diagonal if for integers such that.

Remark 4.1.2 Note that 0-diagonal operator matrix is diagonal. Let be an integer. be a self-adjoint operator with operator matrix decomposition with respect to the decomposition. Then, is n-diagonal if and only if

for all integers

(see Exercise 4.6.1).

The following lemma ensures a decomposition of compact self-adjoint operators into an orthogonal sum of n-diagonal operators modulo compacts.

Lemma 4.1.2

For any compact self-adjoint operators in, there exist compact self adjoint operators in and an orthogonal decomposition into finite dimensional subspaces,… relative to which the operator matrix for is diagonal and that for is (n + 1)-diagonal,.

In this chapter, we address the question of decomposing the semigroup Ext(X) provided a decomposition of X is given. In particular, we introduce the notion of splitting of an extension, and note that a closed disjoint cover ﹛A, B﹜ of X yields direct sum decomposition of Ext(X) into Ext(A) and Ext(B). As the first major step in the ultimate splitting lemma (to be proved in the next chapter), we establish the first splitting lemma, which states that such a decomposition holds for a closed cover ﹛A, B﹜ if A ∩ B is a singleton.

Splitting

Let X be a compact Hausdorff space. Given a*-monomorphism and, write Tf for any operator in such that. It will always be understood that Tf is determined only up to simultaneous unitary equivalence modulo the compacts. Recall that Imτ stands for the image of. If T is in and E is an orthogonal projection in, then write TE for the operator in.

Lemma 3.1

Let e be an orthogonal projection in the Calkln algebra and τe : be a *-monomorphism such that τe(1) = e. If E is an orthogonal projection in such that π(E) =e (see Corollary 2.2.1), then we have the following statements:

• (1) (Existence) There exists a unital *-monomorphism such that

• where and.

• (2) (Uniqueness) If F is another orthogonal projection such that π(F) = e, then τe,E is equivalent to τe,F.

Proof (1) Note that implies that, that is,. Thus, the map is well defined. Similarly, one can see that the projection e commutes with, and hence,

If we decompose the operator Tf with respect to E and I − E, then by (3.1.1), the off diagonal entries are compact. Thus, the map is *-homomorphism.

(2) Let F be an orthogonal projection in such that and note that E − F is a compact operator. Let U and V be isometries in such that and are unitaries (see Corollary 1.5.1). Define and by

Since EU and FV are unitaries,

As weakly equivalent extensions are equivalent (see Proposition 2.5.1), it suffices to show that is weakly equivalent to Observe that

In the last but one equality, we have used the fact that E and F differ by a compact operator.

A normal operator on a finite dimensional inner product space can be diagonalised and the eigenvalues together with their multiplicities are a complete set of unitary invariants for the operator, while on an infinite dimensional Hilbert space the spectral theorem provides a model and a complete set of unitary invariants for such operators (refer to Appendix C). Thus, we view the theory of normal operators to be well understood. It is natural to study operators that may be thought of to be nearly normal in some sense. One hope is that it would be possible to provide canonical models and a complete set of invariants for such operators. Since an operator is normal if [T, T*] := TT* − T*T is 0, one may say an operator is nearly normal if [T, T*] is small in some appropriate sense, for example, finite rank, trace class, or compact. In these notes, we will take the last of these three measures of smallness for [T, T*] and make the following definition.

An operator T in is essentially normal if the self-commutator [T, T*] of T is compact. We say that T is essentially unitary if T is essentially normal and T*T − I is compact. Let be the set of compact operators on a complex separable Hilbert space and be the natural quotient map. Set. An operator T in is essentially normal if and only if is normal in the C*-algebra. Further, an operator U in is essentially unitary if and only if is unitary.

One of the main goals of these notes is to describe a complete set of invariants for the essentially normal operators with respect to a suitable notion of equivalence. As we are considering compact operators to be small, the correct notion of equivalence would seem to be the following.

Two operators T1 and T2 in are said to be essentially equivalent if there exist an essentially unitary operator U and a compact operator K such that UT1U* = T2 + K. In this case, we write, T1 ~ T2. However, it turns out that one may replace the essentially unitary operator in this definition with a unitary operator without any loss of generality. The BDF theorem describes, among other things, the equivalence classes ﹛essentially normal operators﹜/*.

A well-known theorem of Weyl says that every Hermitian operator on a separable complex Hilbert space is the sum of a diagonal and a compact operator. Halmos raised the question: Is every normal operator the sum of a diagonal and compact operator? Berg and Sikonia, independent of each other, proved that one may replace the Hermitain operators in Weyl’s theorem by normal operators without changing the conclusion. However, this is not true if the operator is essentially normal, that is, its self-commutator is compact. The unilateral shift provides an example of such an operator. The obstruction to the decomposition of an essentially normal operator as a diagonal plus compact operator is a certain index data. This was established in a remarkable theorem by Brown, Douglas, and Fillmore that says that the index is the only obstruction to expressing, up to unitary equivalence, an essentially normal operator as the sum of a diagonal plus compact operator. They showed that the classification of essentially normal operators modulo compact is the same as an equivalence problem of *-monomorphisms of C*-algebras. What has come to be known as the “BDF” theorem provides a solution to this problem. Their proofs are imaginative, novel and use a vast array of techniques and intuition from algebraic topology and homological algebra. This theory, cutting across several different areas, is quite sophisticated. Not surprisingly, therefore, many alternative proofs for different building blocks in the original proof were found soon afterward. So much so that the ingenious original proof in [26], which is a bit terse at some places, was all but forgotten.

These notes follow very closely the initial exposition of Brown, Douglas, and Fillmore [26]. In particular, we have resisted the temptation of giving simpler proofs to many of the arguments, which are now available. Our main goal was to occasionally fill in some details and explain few of the obscure points in the original proof of Brown, Douglas, and Fillmore, which appeared in [26], taking advantage of the exposition in the monograph [47]. In the process, we hope that the reader would develop an interest in modern operator theory and be exposed to some non-trivial techniques.

The first chapter provides standard preparatory material in basic operator theory; we largely follow the exposition of [44].

In this appendix, we present some topological ingredients used in the main body of this book. In particular, the results stated are confined mostly to the metric space set up and most of the time without the proofs.

Urysohn's Lemma and Tietze Extension Theorem

Let (X, d) be a metric space with metric d. Let A be a non-empty subset of X, and for x ∈ X, let d(x, A) = inf﹛d(x, a) : a ∈ A﹜. Then, d(x, A) is a continuous function of x. Let A and B be disjoint non-empty closed subsets of X. For x ∈ X, define

Clearly, f : X → [0, 1] is a continuous function. Note that f (a) = 0 and f (b) = 1 for every a ∈ A and b ∈ B. In particular, the disjoint non-empty closed subsets A and B of X are separated by the continuous function f. Thus, we obtain the following special case of Urysohn's lemma (refer to [96]).

Theorem A.1.1 (Urysohn's Lemma)

Let X be a metric space. Given closed non-empty disjoint subsets A and B of X, there exists a continuous function f : X → [0, 1] such that and.

The following particular consequence of Urysohn's lemma is invoked in Chapter 3.

Corollary A.1.1

Let X be a compact Hausdorff space. Let be such that g vanishes on a non-empty closed subset K of X. For any, there exists a continuous function G on X vanishing in a neighborhood of K such that

Proof Note that there exists a neighborhood V of K such that. Let U be an open subset of X such that. Apply Urysohn's lemma to U and to get such that on U and f = 1 on. Let G = gf.

We also need the following extension of Urysohn's lemma in the main body.

Theorem A.1.2 (Tietze Extension Theorem)

Any continuous real-valued function on a closed subspace of a metric space may be extended to a continuous real-valued function on the entire space.

Product Topology and and Tychonoff's Theorem

Consider an arbitrary family of topological spaces indexed by a set I. The product topology on is the topology generated by the basis

For, one may define the projection map, where. It is worth mentioning that the product topology is the smallest topology that makes all projections continuous.

In this appendix, we collect some miscellaneous topics from linear analysis referred throughout the main text.

Stone–Weierstrass Theorem

In this section, we present a proof of the Stone–Weiersrass theorem, which does not rely on the Weierstrass theorem. We closely follow the treatment of [119].

Throughout this section, K denotes a compact Hausdorff space.

Theorem B.1.1

Let be an algebra of continuous functions with the following properties:

• (1) If, then there exists such that.

• (2) For every, there exists such that.

Then, is dense in the algebra of continuous real-valued functions on K endowed with the uniform norm.

We start the proof with a lemma, which shows that under some modest assumption, pointwise convergence yields uniform convergence.

Lemma B.1.1

Let ﹛fn﹜ be a sequence in C[a, b] converging pointwise to a continuous function f. If is decreasing for all, then converges uniformly to f.

Proof Let. For, consider the closed subset

of [a, b]. As,. In particular, finite intersection of sets from ﹛Kn﹜ is non-empty if every Kn is non-empty. If each Kn is non-empty, then by Cantor's intersection theorem,. However, if, then as, for sufficiently large n. Hence, KN is empty for some N, that is, for every x ∈ [a, b] and for every.

Here is an important special case of Weierstrass’ theorem.

Lemma B.1.2

Define a sequence of polynomials by p0(x) = 0, and

If qn(x) = pn(x2), then converges uniformly to f (x) = |x| on [−1, 1].

Proof A routine calculation shows that

One may now verify inductively that

In particular, converges pointwise to |x|. Now apply Lemma B.1.1 to.

The last lemma yields some basic properties of closed subalgebras of.

Lemma B.1.3

Let be a subalgebra of Cℝ(K) and let denote the uniform closure of in Cℝ(K)., then so are and.

Proof Recall that. It may be concluded from Lemma B.1.2 that for ever. The first part now is immediate from

and finite induction, whereas the remaining part follows from.