## mastermath functional analysis 2013

### general

Information on the aim and topics of the course and the compulsory book by Rudin can be found on the mastermath page of the course.

The course is taught by Marcel de Jeu and Onno van Gaans. The teaching assistants are Björn de Rijk and Willem van Zuijlen.

The lectures are scheduled on Tuesdays from 10:15 through 13:00 in Science building of the Free University in Amsterdam, room S655 (no lecture on 22 October!).

The final grade is exclusively based on the results obtained for the biweekly (in principle) homework assignments. The deadline for each assignment is three weeks after the announcement, unless stated otherwise. Please note: although it is unlikely to occur, the lecturers reserve the right to invite individual students for an oral exam if necessary and base the final grade on both the homework and the oral exam.

PLEASE NOTE THAT THERE IS NO RETAKE FOR THIS COURSE.

### homework

The homework assignments will become available below. The work can be given to the lecturer around the lectures on Tuesday. It can also be sent by mail or email to the Teaching Assistant mentioned on the homework assignment. Please make sure that work sent by email is in pdf format and consists of one file. The post address of both Björn and Willem is: Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden.

The schedule is as follows.

Homework 1. Available from September 17. Due October 8.

Homework 2. Available from October 1. Due October 22.

Homework 3. Available from October 15. Due November 5.
Note that in problem II lambda is nonzero!

Homework 4. Available from November 5. Due November 26.
The phrase `such an x' in problem II means an x in A satisfying the previous quadratic equation.
The reason that Y is assumed to be closed in problem IV(a) is that J_Y is only defined (and only of interest) for closed sets Y; it does not play a role in the proof that J_Y is closed.

Homework 5. Available from November 19. Due December 10.

Homework 6. Available from December 3. Due January 7, 2014. (One point subtraction deadline: January 14, 2014)

Homework 7. Available from December 17. Due January 21, 2014. (One point subtraction deadline: January 28, 2014)

For those who are not able to make the deadline of an assigment we offer the following special arrangement: homework handed in not more than one week after the deadline will still be graded, but one point (out of 10) will be deducted. Any homework not handed in before the extended deadline will count as zero points.

### topics of the lectures

Lecture 1. September 10, Marcel de Jeu.

We started out on Chapter 1 and covered (most of) the material up to and including 1.32. There was a rather high definition density and many basic and not so deep properties of TVS had to be collected at this stage of the theory. Still some noteworthy facts already appeared: a basic separation theorem (1.10), the characterization of continuous linear functionals by the closedness of their kernels (1.18), the uniqueness of the topology on a finite dimensional TVS (1.21), the fact that a finite dimensional subspace is always closed (1.21 again), and the metrizability theorem (1.24).

Lecture 2. September 17, Marcel de Jeu.
We mainly concentrated on the relation between seminorms on a vector space and subsets of that vector space.

If X is a TVS, then there is a bijection between the continuous seminorms on X and the convex balanced neighbourhoods of 0, which is given in one direction by assigning to a continuous seminorm its "open unit ball", and in the other direction by assigning to such a neighbourhod its Minkowski functional. This relation hints at how one should "turn the tables" and, given a separating family of seminorms on an abstract vector space, introduce a locally convex vector space topology on that space. This is described in the important Theorem 1.37. As a consequence of remark (b) on page 29, the locally convex TVS are even _precisely_ the spaces that can be topologised via the procedure of Theorem 1.37. We covered the examples of LCS in 1.44-1.46 and mentioned the L^p-spaces in 1.47, which have only the empty set and the space itself as open convex subsets, and consequently zero as the only continuous functional). By covering the material on quotient spaces in 1.40-1.42 we finished Chapter 1.

Lecture 3. September 24, Marcel de Jeu.

We covered Chapter 2. The basis for the whole chapter is Baire's theorem 2.2, which implies the Category Theorem: a complete metric space and (somewhat lesser well known) a locallly compact Hausdorff space is of the second category in itself. That is, if the space is written as a countable union, then the closure of at least one of these constituents must contain an interior point. This makes clear why F-spaces are important: the Category Theorem is available to work with. From the Categroy Theorem, a range of fundamental tools follow (all valid and perhaps already known for Banach spaces from an introductory course), such as the Uniform Boundedness Principle, the Open Mapping and the Closed Graph Theorem. We skipped 2.16 and 2.17.

Moving to Chapter 3, the usual Hahn-Banach-type results for normed space follow rather easily from Theorem 3.3. For locally convex spaces the road to, say, continuous extension theorems is much more involved than in the normed case. The basis here is the fundamental separation result Theorem 3.4, and notably part (b). Though the proof itself is not particularly long, many earlier results and ideas are used in it, and Theorem 3.4 can be regarded as the deepest result so far. The Corollary on p.60, Theorem 3.5 and Theorem 3.6 are then easily obtained and are the analogues of well-known results in the normed case. We skipped 3.7.

Skipping 3.8 for the moment we concentrated on Theorem 3.10, which we formulated in another fashion: Starting with a separating vector space X' of linear functionals, the absolute values of these functionals form a separating family of seminorms, and hence the usual procedure yields a locally convex topology on X, such that the continuous dual of the resulting space is precisely X' -- this uses Lemma 3.9. It is then also true that this topology is the weakest topology on X such that all elements of X' are continuous (this is the approach in the book for 3.10).

Lecture 4. October 1, Marcel de Jeu.
We finished Chapter 3 and concluded this miniseries on TVS.

We started by noting that, on infinite dimensional spaces, the topologies introduced by separating vector spaces of functionals have the property that each neighbourhood of 0 contains an infinite dimensional subspace. Hence such topologies on an infinite dimensiol spaces are never normable. Two prime examples of such topologies are the weak topology on a LCS X and the weak* topology on the continuous dual X* of a TVS X. If X is a Banach space, then either of these topologies is metrizable precisely when the space is finite dimensional (see Megginson's "Introduction to Banach space theory", 2.5.14 and 2.6.12 for these results).

The equality of closures of convex sets in the original and the weak topology of a LCS has remarkable consequences, as we saw in Theorem 3.13.

One of the highlights in the theory of TVS and LCS is the Banach-Alaoglu theorem 3.15, based on Tychnov's theorem. It implies, e.g., that the closed unit ball in the dual of a Banach space is weak* comnpact. If the Banach space is separable, then the weak* topology on this unit ball is metrizable (and conversely, see Conway's book V.5.1).

The second highlight we saw is the Krein-Milman theorem 3.23 (and the related results 3.24 and 3.25). Together with the Banach-Alaoglu theorem, which "provides" compactness of obviously convex sets, this is a very powerful combination. For example, it was now almost trivial that c_0 is not a dual Banach space.

We concluded with showing that for any given continuous map of a compact Hausdorff space into itself, there always exists an invariant probability measure. The proof is a mere combination of the Riesz representation theorem, the fixed point theorem 5.28 ("Brouwer"), and the Banach-Alaoglu theorem (providing the compactness needed in 5.28). One can push this a step further and invoke the Krein-Milman theorem to conclude that the (non-empty!) set of invariant probability measures must have at least one extreme point. Since these extremal points can be shown to be equal to the so-called ergodic measures, there exists at least one ergodic measure. This elaborate example is typical and shows how these rather abstract theorems from analysis in locally convex analysis have far reaching implications in concrete situations, with only little work needed.

We skipped 3.18, most of 3.19, 3.20, 3.21, 3.26-3.32.

Lecture 5. October 8, Onno van Gaans.
We have taken up a topic from Operator Theory: the Riesz-Schauder theory on the spectrum of compact operators on complex Banach spaces. This is Chapter 4 of the book. For an n by n matrix A, injectivity and surjectivity of S=A-zI are equivalent. Surjectivity means that the columns span the entire space and injectivity is equivalent to the orthogonal complement of the span of the rows being {0}. Note that the span of the rows equals the span of the columns of the transpose of A. In a similar analysis for operators on Banach spaces, we need suitable generalizations of transposes and orthogonal complements.

We have briefly discussed the duality pairing between a Banach space X and its dual X* and the corresponding two types of annihilators. We have also discussed the adjoint T* of an operator T between to Banach spaces and the relations between the range and null space of T and T* and their annihilators. The closed range theorem says that the range of an adjoint is weak* closed if and only if it is norm closed.

The space of all compact operators between two Banach spaces X and Y is a closed subspace of B(X,Y) with the operator norm. Every finite rank operator (that is, an operator with finite dimensional range) is compact and therefore operators which are limits of finite rank operators are compact. The converse has been a famous open problem for decades in Banach space theory: does every Banach space have the approximation property? It has been resolved by a counterexample by the Swedish mathematician Per Enflo in 1972.

We have defined the notions of spectrum and eigenvalue and formulated the spectral theorem for compact operators on Banach spaces. Its proof is rather elaborate. We have done most of it. For a compact operator T on a complex Banach space X, a complex number z, and S=T-zI, we have shown that T* is compact, the dimension of the null space of S is finite whenever z is nonzero, the spectrum of T contains 0 if X is infinite dimensional, the range of S is closed if z is nonzero, and if z is a nonzero eigenvalue of T then S is not surjective.

The rest of the proof will be done in Lecture 6 and then we will also start with Chapter 10.

Lecture 6. October 15, Onno van Gaans.

The first part of the lecture dealt with the remainder of the proof of the spectral theorem for compact operators. It has been shown that the number of eigenvalues of a compact operator T outside a disk centered at zero is finite. Then the proof has been completed by showing Theorem 4.25, which says that the dimensions of the null spaces and the codimensions of the ranges of (T-zI) and (T*-zI) are all equal and finite.

The vector space of bounded linear operators on a Banach space also has a multiplication, namely, the composition of two operators. The norm of the product of two operators is less than or equal to the product of the norm. This structure can be captured in a general definition: a Banach algebra. This is the topic of Chapter 10. A Banach algebra can always be extended in such a way that it has a unit and we agree that by a Banach algebra we will always mean a Banach algebra with a unit. By means of the notion of invertibility, the spectrum and spectral radius of an element in a Banach algebra can be defined.

We have shown that the set G(A) of invertible elements of a Banach algebra A is open, that the map that maps an element of G(A) to its inverse is a homeomorphism, and we have established an explicit estimate for the distance between the inverses of x and (x+h). These proofs use a geometric series expension of inverses. By an argument involving a Cauchy theorem for integrals over closed contours of Banach algebra valued holomorphic functions, we have shown that the spectrum is always a non-empty compact set.

Next lecture: Symbolic Calculus.

Lecture 7. October 29, Onno van Gaans.
For an element x of a Banach algebra and a sufficiently nice function f we aim to define f(x). For a polynomial f the element f(x) is naturally defined by means of the algebraic operations of the algebra. If f has a power series expension and the norm of x is less than the radius of convergence, then f(x) can be defined by a norm convergent series. For more general functions we will use contour integrals. After a very brief discussion of the Bochner integral and contour integrals for functions with values in Banach algebras, we have shown a Cauchy formula for f of the form f(z)=(w-z)^n. As a consequence we have shown that the integral formula definition of f(x) for a rational functional f coincides with the algebraic definition as quotient of polynomials. For each f that is holomorphic on an open set U containing the spectrum of x we have defined f(x) by the integral formula and we have shown that this induces an algebra homomorphism from the holomorphic functions on U to the Banach algebra. We have also shown a continuity property of this map, the spectral mapping theorem and a theorem on compositions. One of the ingredients from complex function theory in the proofs is Runge's theorem on the approximation of holomorphic functions by rational functions with special properties of its poles.

Lecture 8. November 5, Onno van Gaans.
Chapter 11 treats commutative Banach algebras. We started by a theorem of Banach and Mazur, which says that a Banach algebra in which every nonzero element is invertible is isometrically isomorphic to the complex numbers. In a commutative algebra we have defined ideals and we have showed with the aid of Zorn's lemma that every proper ideal is contained in a maximal ideal. The ideal space of a commutative Banach algebra A is defined as the set of all (nonzero) complex homomorphisms on A. This set is a subset of the unit ball of the Banach space dual of A and we endow it with the restriction of the weak* topology. Each element of A induces a continuous complex function on the ideal space. This map from elements of A to functions on the ideal space is called the Gelfand transform. It is an algebra homomorphism and an isomorphism if A is semisimple, which means that the intersection of all maximal ideals in A is only the zero element. The spectrum of x coincides with the range of its Gelfand transform.

The embedding of a Banach algebra A in the Banach algebra of continuous functions on its maximal ideal space becomes very nice if we assume some more structure of A, namely that it has an involution which makes it a B* algebra. We have shown the Gelfand-Naimark theorem, which says that for a B* algebra the Gelfand transform is a surjective isometric isomorphism. Moreover, the *-operation corresponds to complex conjugation of the functions.

Lecture 9. November 12, Onno van Gaans.
The Gelfand-Naimark theorem provides a symbolic calculus. For a continuous function f on the maximal ideal space of A and an element a of A, a corresponds to a continuous function on the maximal ideals space, which can be composed with f, which yields a continuous function and therefore an element of A, denoted f(a). We have shown that if the set of all polynomials of a and its adjoint is dense in A, then the map which maps f to f(a) can be extended to all continuous functions on the spectrum of a, and this map is an isometric *-isomorphism.

The theory of commutative B*-algebras can be applied to non-commutative Banach algebras with an involution. Indeed, if a is a normal element of a Banach algebra with an involution, then there is a smallest closed normal subalgebra containing a and this subalgebra is commutative. By using the Gelfand-Naimark representation theory for such subalgebras, we have proved the spectral theorem for normal elements in B*-algebras.

Lecture 10. November 19, Onno van Gaans.
The topic of Chapter 12 is the Banach algebra of bounded linear operators on a Hilbert space. We have developed a theory similar to the unitary diagonalization of normal matrices. That diagonalization can be seen as a decomposition of the normal matrix into a linear combination of orthognal projections with one-dimensional ranges. The coefficients are the eigenvalues of the matrix. In the general situation the decomposition will be an integral over the spectrum with respect to an orthogonal projection valued measure. We have started with the latter: resolutions of the identity. After some elementary properties we have constructed an integral with respect to a resolution of the identity. The integrands are arbitrary bounded measurable functions and the value of the integral is then a bounded linear operator. We have proved the spectral theory for normal operators, which states that for every normal operator there exists a resolution of the identity on its spectrum such that the operator equals the integral of the identity function with respect to that resolution. By integrating arbitrary bounded Borel measurable function instead of the identity function we have constructed a symbolic calculus. We have considered

Lecture 11. November 26, Onno van Gaans.
We have first considered several consequences of the spectral theorem for normal operators, for instance, that a normal operator is selfadjoint if and only if its spectrum is real. The general spectral theorem for normal operators does not contain any information about eigenvectors and we have investigated them in this lecture. For a complex number z, the eigen space of a normal operator T corresponding to z equals the range of the orthogonal projection obtained by evaluating the resolution of the identity at the one point set {z}. We have shown that every isolated point of the spectrum of T must be an eigenvalue, which explains the name point spectrum for the set of eigenvalues. We have proved a characterization of compactness of a normal operator in terms of its spectrum. A normal operator turns out to be positive definite if and only if its spectrum is positive. Further, every positive operator has a unique positive square root. After proving these statements we have applied them to prove the polar decomposition of invertible and of normal operators. Finally, we have shown that every B*-algebra is isometric *-isomorphic to a closed subalgebra of the algebra of bounded operators on some Hilbert space. The construction of the Hilbert space relies on existence of sufficiently many positive functionals, which we derived by Hahn-Banach. With these positive functionals we could define inner products on quotient spaces and then we constructed the Hilbert space as a Cartesian product of the completions of these quotient spaces.

Lecture 12. December 3, Marcel de Jeu.
We started the final miniseries of three lectures on unbounded operators and semigroups, i.e., Chapter 13 in the book.

After arguing that is not feasible to find an infinite dimensional Banach space of functions, together with a globally defined operator that could reasonably be called "d/dx", we set out to study the basic properties of (generally unbounded) operators defined on a linear subspace of a Hilbert space.

If the domain is dense, then it is possible to define the adjoint operators and it makes sense to speak about symmetric and self-adjoint operators. Describing an adjoint explicitly (including its domain!) can be non-trivial, as is illustrated by Example 13.4, for which we gave some background (not in the book) on absolutely continuous functions. We skipped 13.5 and 13.6.

Studying the adjoint is greatly facilitated by the description of its graph in Theorem 13.8; see 13.10 and 13.11 (the latter we did not cover) for examples. Theorem 13.12 gives "the usual" result for the double adjoint, precisely for the only class for which it could possibly hold: the densely defined closed operators.

We skipped 13.13, covered 13.15 and proved parts of 13.16 for a self-adjoint operator on our way to the important Theorem 13.19. This we proved only in the version as we will need it, namely that the Cayley transform gives a bijection between the self-adjoint operators and the unitary operators U such that 1 is not in the point spectrum of U. Since we have a spectral theorem for unitary operators (even for normal ones), this Cayley transform will enable us to establish a spectral theorem for self-adjoint operators.

The full Theorem 13.19 is still important, since it underlies 13.20, where the existence of certain extensions of a closed symmetric densely defined operator is described in terms of the so-called deficiency indices of that operator. We mentioned these results without proof.

Lecture 13. December 10, Marcel de Jeu.
We covered 13.22 through 13.31, and noted that there is also a spectral theorem for unbounded normal operators (13.33). We developed a general functional calculus for unbounded functions. Here one has to be careful about domains, see 13.23. The proofs are longer than for the bounded case, and combine truncations of functions, knowledge of the bounded case, and integration theorems (increasing sequences of positive functions, dominated convergence). Once this calculus has been developed, it is relatively easy to combine this with the Cayley-transform and the change of measure principle to obtain the spectral theorem for self-adjoint operators, 13.30. This is used in the construction of a unique positive root of a given positive operator, and this again is used in the uniqueness part of the spectral theorem for normal operators.

Lecture 14. December 17, Marcel de Jeu.
We started with an informal motivation how several partial differential equations with initial values have a related (but not equivalent) so-called abstract Cauchy problem, which is a first order (!) ordinary (!) differential equation with initial value for a Banach space valued function. These abstract Cauchy problems have everything to do with strongly continuous (one parameter) operator semigroups. This is the topic of 13.34 and further in Rudin's book, and we covered this material in this lecture.

We proved 13.35 in detail, establishing the basis facts on these semigroups. Part (d) shows that the orbit of an element of the domain of the closed and densely defined infinitesimal generator solves the abstract Cauchy problem for this generator. Part (e) shows that, for sufficiently large real part, the resolvent function is the Laplace transform of the semigroup. Though innocent, in the proofs of 13.35 and elsewhere it sometimes seems to be necessary to replace \gamma with | \gamma |, e.g., in the upper bound for the norm of M_t on page 377 between (4) and (5).

The prototypical semigroup is a "truly" exponential semigroup, and 13.36 shows that these are precisely the uniformly continuous ones. It often helps to understand results about semigroups to pretend that such an exponential formula is valid, even when it need not be.

The Hille-Yosida theorem 13.37 characterises the infinitesimal generators of semigroups. The estimates in (1), where there is a C and not C^n, is the non-trivial and key ingredient. Actually we formulated a more precise version of the theorem, II.3.8 in [1]. The necessity of the Hille-Yosida estimates follows from (4), which we derived as on page 57 of [1} from from IV.1.3 in [1], rather than from convolutions of semigroups.

Theorem 13.38, for the Hilbert space context, makes contact with spectral theory and functional calculus for normal operators. The main point is (2), showing that the functional calculus assigns an element of the semigroup to an exponential function on the spectrum.

There is an extensive literature on operator semigroups. Some relevant references:

[1] Engel and Nagel, One-parameter semigroups for linear evolution equations, Springer, 2000. 586 pages and a sometimes encyclopedic standard text with many examples.

[2] Engel and Nagel, A short course on operator semigroups, Springer, 2006. A 247 page excerpt from [1].

[3] Sell and You, Dynamics of evolutionary equations, Springer, 2002. 670 pages with many examples and also paying attention to the various types of solutions of related, but not equivalent, formulations of an initial value problem.