## Functional Analysis Seminar

We intend to study the book "A short course in spectral theory" by Bill Arveson., which appeared with Springer in 2002. See the publisher's description for more information.
As it is with a first printing, there are a number of slight mistakes and typos in it, see Arveson's own list of errata.

Number of ECTS

Studying the book in this seminar means a combination of (mostly) reading, attending other participants' lectures and (optionally) giving a lecture.
As to the reading part, I will select four sets of exercises which you should solve, one set for each of the chapters. An average grade of 5.5 or higher then yields 6 ECTS. Solving the exercises together is certainly allowed and encouraged, but I expect you to really understand the solution once you write it down in your own words. I do not intend to select exercises which are particularly difficult (some are rather non-trivial extensions of the text), but I will try to select them in such a way that solving them needs mastering the ideas and results of the text.
People who additionally give a lecture receive 10 ECTS. The idea of such a lecture (indication: 2x45 min.) could be either to present some material which is needed for the book, but which is not in it, to explain a part of the book in which Arveson might be particularly compact in his writing, or perhaps present some other material which extends or further clarifies the contents of the book. I will suggest topics to people who are interested in giving a lecture.
It goes without saying that we will all do our very best to attend other people's lectures.

Schedule: generalities

We have ample time in the schedule, on Tuesdays and Wednesdays. At the time of writing, it seems sufficient to use the Wednesdays only. We will meet in room 401 if a lecture is scheduled for that Wednesday. If no lecture is scheduled, I will be available at my office (218) to answer questions. Of course, you can also ask me questions at other occasions if you wish, but if it is possible I would like to protect my time a little from too much fragmentation.

Schedule of lectures

Wednesday February 23, 13.45 hr: Ta Ngoc Tri will cover locally convex spaces, seminorms, weak topologies and the Banach-Alaoglu theorem.
Wednesday March 23, 13.45 hr: Jasper Lukkezen will speak on the generalized Stone-Weierstrass theorem.
Wednesday April 27, 13.45 hr: Sophie Raynor will speak on the holomorphic functional calculus (1.12 in Arveson, with extensions).

Home work assignments

First assignment (Chapter 1, deadline Friday March 18):
Page 7: the exercises 3 and 4. Don't be misled too much by the Banach space context: this helps to find the answer in 3, but 3 is actually true in any ring with a unit element, as should be obvious from your final proof. As a consequence, the result in 4 holds in any Banach algebra.
Page 18: the exercises 3, 5 and 6. In 6 you may find it helpful to use the polynomial spectral mapping theorem, which is obtained by replacing the inclusion in equation (1.11) with equality (we did this in the FA course, but then for operators).
Page 27: exercise 2 (in my opinion this could/should have been a theorem in the text).
Page 33: exercise 1.
For a two-point bonus (so that you may obtain a 12 for this assignment): page 33, exercise 3. This not-so-easy exercise, which uses a substantial part of the Gelfand theory for commutative BA's, is a special case of "the" theorem of Runge. Incidentally, the fact that this exercises is placed following 1 and 2 seems to suggest that 1 and 2 both have to be used when solving 3, but I could manage without 2 (I needed 1, though).

Second assignment (Chapter 2, deadline Friday April 29):
Page 45: 3 and 4. Hint for 4: can you write down an operator with one-dimensional range?
Page 50: 8. You can use the result of exercise 4 on page 45 here.
Page 51: 3. Hint: if f is a polynomial, then this is trivial. For general continuous f, think of the Stone-Weierstrass theorem on a sufficiently large disk.
Page 56: 1.
Page 67: 2.
Page 77/78: 1 and 2. In 2, take H infinite-dimensional. Otherwise K has a unit already, and then there is no need to adjoin one (and strictly speaking the unitization is then not even defined, according to the text).
Page 80: 1. You may want to use another book here.

Third assignment (Chapter 3, deadline Tuesday May 17):
Page 92: 2 and 4. State Ascoli's theorem as part of your solution to 4 (which you may want to look up).
Page 95: 1 and 2. Hint for 1: choose a closed complement M for TE (why is this possible?). Then there is a canonical map from the direct sum of E/Ker T and M to E. Now think of the theorem of the bounded inverse.
Page 99/100: 1, 2 and 3. Hint for 1a: Apply the result of page 92 exercise 2b to the kernel of T. Also, choose a complement of the range of T. This complement has the same dimension as the kernel of T, so how can you construct an invertible operator now, using T? Hint for 1b: there is a Fredholm operator T such that BT=1+K with K compact (why?). What is then the index of TA?

Fourth assignment (Chapter 4, deadline Friday June 17):
Page 112: 1 (this result is used in the proof of Theorem 4.3.2). You may use the result in exercise 2 on page 74.
Page 112: 3. Think of the continuous functional calculus to get a result about C(T). And next?
Page 113: 4. You need only prove that the Toeplitz extension does not split, since we did the rest already in exercise 3 on page 100.
Page 113: 7. You will also need the result of exercise 3 on page 112 here.
Page 121: 1.
Page 126: 3 and 4. The result of 3b should be compared to Corollary 1 on page 128: for non-real z the corresponding linear functionals satisfy all hypotheses in this Corollary, but they are not states.
And, to conclude, an exercise which is not in the book, but which could have been included on page 129: if A is a unital C*-algebra, then the set M_n(A) of n by n matrices with coefficients from A is an involutive algebra in a natural way. How? In fact, it can even be made into a C*-algebra, but it is not so obvious how to define the norm and how to show that it is then complete. Can you do this, by interpreting the elements of M_n(A) as operators on some Hilbert space?

Each week, I will suggest parts of the book to read, so that we can hopefully cover the book in four months, finishing by the end of May. You can try your hand at the corresponding exercises if you wish, but this is not necessary to understand the book and I do not expect you to try them. It will be fun to do so, but it may also cost you a lot of time, so be your own judge in this.

Week 5 (Jan 31 - Feb 4):
Section 1.1: read at leisure, since this is mainly an informal motivation for the notion of spectrum and invertibility. The examples 1.1.2 and 1.1.3 are very sketchy (and you don't need to know the Plancherel theorem). People who followed my FA course will recognize the result in 1.1.4, otherwise you can simply accept it.
Section 1.2: this recalls what you probably already know about the spectrum of an operator. Later on, B(E) will be replaces with an arbitrary Banach algebra with identity, and then the definition of the spectrum is precisely what you would guess (but one cannot say meaningfully what e.g. point spectrum is in this general context).

Week 6 (Feb 7 - Feb 11)
Section 1.3: here the BA part really starts. Don't worry too much about Example 3.11 and the Haar measure, since the general theory of topological groups is not core material of the book. Topological groups (and their representations) are rather important in analysis (you can fill a small library with all books about the subclass of Lie groups alone), but here it suffices to know that there is such a general not necessarily abelian notion, of which Z, R and the circle are abelian examples. The corresponding L_1(G) space, which is defined in Example 1.3.11 in the general context, specializes to Example 1.3.7 for G=Z (Haar measure is then counting measure) and to Example 1.3.8 for G=R.
Section 1.4: The main point here are the last two lines in Theorem 1.4.2: if a BA has a unit element, then we can conveniently assume that it has norm one. The statement that separate continuity of the multiplication implies simultaneous continuity is less important for us.
Section 1.5: this is elementary but important material. You will recognize Corollary 1 and 2 from Banach space theory. The proofs in that context used only that B(E) is a Banach algebra, and not that the elements of this structure are operators, so it is true for BA's in general. Don't pay attention to the K-group on page 15.

Week 7 (Feb 14 - Feb 18)
Sections 1.6, 1.7 and 1.8: this is all essential material. You may recognize Theorem 1.7.3, this is the spectral radius formula which was also mentioned the previous semester, but then only for operators on a complex Banach space.
Incidentally: we won't be needing much algebra. If you know what an ideal in an associative algebra is and how to form the quotient if the ideal is two-sided, then that's basically it. The algebra does not go beyond that in Section 1.8. It's curious that this addition of elementary algebra to more serious analysis gives such strong results.

Week 8 (Feb 21 - Feb 25)
Section 1.9: Ta Ngoc Tri will be telling us about the weak*-topology, which is mentioned in Proposition 1.9.3.
The basic observation (due to Gelfand) about unital commutative Banach algebras is that they can be mapped homomorphically into the algebra of continuous functions on a suitable compact Hausdorff space (namely: the spectrum of the algebra). This map (the Gelfand transformation) is often also injective, so that one can think about the algebra as being an algebra of continuous functions on a well-behaved space. General topological notions then also come into view as an additional tool to study such an algebra. This is all very convenient.
If the algebra does not have a unit element, then there is a Gelfand transformation into the algebra of continuous functions vanishing at infinity on a locally compact Hausdorff space. The latter space can be constructed by first adjoining a unit element to the algebra and then removing one point of the (compact) spectrum of the unital algebra. The point one removes is the complex homomorphism of the larger algebra which is zero on the algebra you started with.
Section 1.10: The Gelfand transformation does not do anything special for C(X). The point evalutions are obviously non-zero complex homomorphisms, and that is all there is. With this identification, the Gelfand transformation is just the identity map and nothing is gained. But have a look at example 1.10.5! I already mentioned in my introduction that this is the one which put Banach algebras on the map: a seemingly very deep analytic fact becomes almost trivial when looked at from the right perspective.
Section 1.11: After this section we're done with the basics about the spectrum. Corollary 2 is the way to think about it: if you shrink the algebra then it is more difficult to be invertible, so the spectrum is likely to get larger. And if it gets larger, then this is by filling in holes. You can actually prescribe the holes you want to be filled in and prove that there is then a subalgebra which gives just that as spectrum, but we will not pursue this.
Section 1.12: we skip this for the moment. I will ask someone to give a lecture on this material (with some additions) since it is important (though not for this book), but there is no hurry to study it right now.

Week 9 (Feb 28 - Mar 4)
Section 2.1: the beginning of this section will be familiar. A von Neumann algebra can alternatively be characterized as a self-adjoint subalgebra of B(H) which is equal to its own double commutant. That this really is an alternative characterization is not trivial and this result is known as the Double Commutant Theorem, due to von Neumann. As long as we don't have to use the weak and strong operator topologies on page 42, this is perhaps the most convenient way to think about von Neumann algebras. A prototypical abelian von Neumann algebra is described in Theorem 2.1.3. This is a result to remember, with Theorem 2.1.4 as a nice companion.
Section 2.2: this is a key section. The importance of Theorem 2.2.4 (known as the commutative Gelfand-Naimark Theorem) can hardly be overestimated. It gives an appealing equivalent description of commutative unital C*-algebras, which is also very valuable in the non-commutative context, since one can easily find commutative subalgebras of non-commutative ones. The proof uses the Stone-Weierstrass theorem: a self-adjoint subalgebra of C(X), where X is a compact Hausdorff space, which contains the constants and separates the points (i.e., for different points x and y there is a function f in the algebra such that f(x) and f(y) are different) is dense in C(X).
The first part of Corollary 2 is quite remarkable: the spectrum does not depend on the algebra! The equality of norm and spectral radius for self-adjoint elements is actually true for normal ones: in the last line of the proof, consider the unital algebra which is generated by x and x*.

Week 10 (Mar 7 - Mar 11)
Section 2.3: harvesting time! You may recall that Rynne and Youngson also established Theorem 2.3.1, but with some ad hoc arguments. The present proof shows what is really happening here: the continuous functional calculus is an almost trivial application of the commutative Gelfand-Naimark theorem.
Section 2.4: there is als a more detailed versions of the Spectral Theorem 2.4.5, stating that any normal operator can be built up from multiplication operators corresponding to measures living on its spectrum and that these measures determine the operator up to unitary equivalence. But 2.4.5 is interesting enough as it stands already: any normal operator on a separable Hilbert space becomes multiplication in a suitable model.

Week 11 (Mar 14 - Mar 18)
Section 2.5: the question of finding and studying homomorphic images of a structure, where the image is supposed to consist of operators (in a purely algebraic context, or in an analytic one as in our case) is the core of representation theory. This section contains the basics of this field in the direction of C*-algebras. Incidentally, the representation theory of topological groups can be reduced to that of the so-called group C*-algebra which is associated with it. The fundamental theorems of decomposition of representations of topological groups all stem from C*-theory via this connection, so this section is not only about algebras.
Section 2.6: the continuous functions on the spectrum of a normal operator are a C*-subalgebra of the bouded Borel measurable functions on this spectrum. This section show that the continuous functional calculus can be extended to a representation of this larger algebra. There are a number of things to be verified in the construction, causing the length of the proof to exceed its degree of difficulty. The main things to remember are the possibility of the extension and formula (2.11).

Week 12 (Mar 21 - Mar 25)
Section 2.7: not too much this week, just this clarifying link between the Borel calculus and spectral measures. The symbolic integrals in this section, where the "measure" has orthogonal projections as its values, are not only used by mathematicians but are also very popular with physicists. The definition of such a spectral integral is as in equation (2.11), and this is the way I personally think about these integrals.

Week 13 (Mar 28 - Apr 1)
Section 2.8: much of this section on compact operators on Hilbert spaces will be familiar, but perhaps not the results on Hilbert-Schmidt operators in the second part. The Hilbert-Schmidt operators form a Hilbert space under the HS inner product, this is the content of Exercise 4 (the text on page 72 gives Exercise 3, but this is a mistake). Aside, let me mention that the notation L^2 for the HS-operators is rightfully suggestive: one can meaningfully define subspaces L^pF of B(H) for all finite p greater than or equal to 1 and thus obtain the so-called Schatten p-classes. These are Banach spaces in the correponding p-norm (can you guess from the penultimate line on page 71 what this norm might be?) and they are two-sided ideals of B(H) (but not closed in the infinite dimensional situation). We won't be using these classes, however.

Week 14 (Apr 4 - Apr 8)
Section 2.9: Banach algebras with unit are more pleasant to work with than ones without. There is a general procedure of adjoining one, and in case of a C*-algebra this can be done so that the unital algebra which is thus obtained is again a C*-algebra. Note also the quite remarkable Proposition 2.9.3: a morphism between C*-algebras is always continuous (and has norm at most 1).
Section 2.10: for the general theory of C*-algebras, this section is indispensable. Theorem 2.10.2 is quite surprising. As to Theorem 2.10.4: this is as good as you could get. Note that, for a continuous map between Banach spaces, the image need not be closed - but for morphisms between C*-algebras this is the case.

Week 15 (Apr 11 - Apr 15)
Section 3.1: here the Calkin algebra is defined, which is a crucial object in Fredholm theory in Section 3.3. It is also motivated why this algebra can be said to carry all asymptotic information of operators, but it is sufficient to read this rather globally (and you can skip the part on the corona space).
Section 3.2: this is the Riesz theory of compact operators on a Banach space which should be familiar.

Week 16 (Apr 18 - Apr 25)
Section 3.3 and 3.4: speaking imprecisely, Fredholm operators are operators which are almost regular: their kernel is small (only finite dimensional) and their image is large (only of finite codimension). Compact operators are "small", and Fredholm properties are not disturbed by such small operators. Sticking to these analogies, one could hope that a Fredholm operator is almost invertible, i.e., invertible up to a small portion, and this is in fact true: this is Atkinson's theorem (3.3.2).
The Fredholm theory is partly built on Riesz' theory of compact operators, and parts of this theory reappear as more general statements in Fredholm theory. As an example, the identity is obviously Fredholm of index zero, and hence the same is true for 1-T for any compact operator: this gives Lemma 3.2.8 again.
Fredholm theory reappear in the final chapter, see, e.g., the lines just above Theorem 4.5.4.

Week 17 (Apr 25 - Apr 29)
The last chapter contains the applications and some more general theoretical material.
Section 4.1: Von Neumann algebras are C*-algebras, but of a very special type. They are realized in a Hilbert space context by definition, and then they are the sub C*-algebras of B(H) which are weakly closed (or strongly closed, this is equivalent for convex sets in B(H)). The famous Double Commutant Theorem of Von Neumann states that a self-adjoint subalgebra of B(H), which contains the identity, is a von Neumann algebra precisely when it is equal to its own double commutant. If N is a normal operator, then its spectral projections do in general NOT lie in the C*-algebra generated by N, but they do lie in the smallest von Neumann algebra containing N (and in fact they even generated this as a C*-algebra). So here is an example of a natural occurrence of von Neumann algebras, but there are many more. This section gives a first very brief impression for the commutative case. The literature on the subject is vast.

Week 18 (May 2 - May 6)
Section 4.2: in applications, matrices as on page 109 can occur as matrix-representations of bounded operators on l_2. Theorem 4.2.4 tells us how to describe such operators more conceptually, in terms of essentially bounded functions on the circle which act on H^2.
The notation H^2 suggests that there is not only a second Hardy space, but also a p-th one: H^p. This is indeed the case, and these spaces are well-studied. There are two ways of defining H^p, for 1\leq p\leq infty. The first one (as used by Arveson) is to let H^p consist of those L^p-functions on the circle with zero negative Fourier coeffients ("analytic L^p-functions", so to speak). The second one is to consider functions which are holomorphic on the unit disk, which are in L^p on each origin-centered circle in the unit disk, and such that their L_p-norm stays bounded as the circles approach the unit circle. It is then the case (but this is not at all trivial) that for such a function the radial limit to points on the unit circle exists for almost all points on the unit circle, and that in this way one obtains precisely the L^p-functions on the unit circle with vanishing negative Fourier coefficients as in the first definition. See, e.g., Conways's "Functions of one complex Variable II", Theorem 3.13 on page 282.

Week 19 (May 9 - May 13)
Section 4.3: Here C*-algebras enter the story about Toeplitz operators. Unfortunately there is some confusion possible here by the constant occurrence of the word "Toeplitz". Please note that the Toeplitz operators form a vector space but not an algebra, that not every Toeplitz operator is an element of the Toeplitz algebra (although the ones with continuous symbol are, see Theorem 4.3.2), and that the Toeplitz algebra does not consist of Toeplitz operators alone. Corollary 1 indicates that Fredholm theory is relevant for Toeplitz operators, and this will become even more manifest in the sequel (see, e.g., the remark preceding Theorem 4.5.4 and the conclusion of the story on page 121).
If you have not done so already, you should download Arveson's list of errata, since there are some particularly unfortunate mistakes on page 110/111.
Section 4.4: The main point here is Theorem 4.4.3, which shows that an index is (minus) a winding number. This winding number (around the origin) is what you think it is: if a continuous function f on the unit circle does not have 0 as a value, then, as z runs over the circle, f(z) sweeps out a curve in the complex plane which does not contain 0 and which "therefore" must wind around the origin a well-defined number of times. That all this is OK is in the preparatory first part of the paragraph.

Week 20 (May 16 - May 20)
Section 4.5: The theory of H^p-spaces is a nice mixture of complex function theory and functional analysis. The rather famous Theorem 4.5.1 is an illustration of this. Incidentally, it is very rare that one has a more or less concrete description of the closed invariant subspaces of a bounded operator, so Theorem 4.5.1 is a fortunate but untypical case.
Section 4.6: all the work has been done by now. All that is left to do is put the pieces together to conclude what the spectrum and the essential spectrum of a Toeplitz operator with continuous symbol are.

Week 21 (May 23 - May 27)
Section 4.7 and 4.8: This is the basis for representation theory of C*-algebras. The key result is that giving a cyclic representation and giving a positive functional is essentially the same thing; this is Theorem 4.7.3. Combined with the theory of commutative Banach algebras and the Hahn-Banach theorem this then leads to the conclusion that there are enough representations to separate the points (see for yourself how this works in the proof of Corollary 2 on page 128). The Gelfand-Naimark theorem is then an easy consequence. Of no less importance is Remark 4.8.5: there must also be sufficiently many irreducible representations as a consequence of the Krein-Milman theorem, combined with the Banach-Alaoglu theorem.
Incidentally, the orginal Gelfand-Naimark was proved with additional axioms in the definition of C*-algebras, which only after 15 years or so could shown to be redundant. It was Segal who realized this and who abstracted what is now known as the GNS-construction from the paper by Gelfand and Naimark. So the theory has not always been as smooth as it is now.