Problems in Operator Theory


October 13-14, 2022, VU University, Amsterdam
in honor of the 67th birthday of André Ran

[Andre Ran]

Titles and abstracts:

  • Harm Bart Let F be a scalar analytic function, and let D be a non-empty open subset of the complex plane the closure of which is contained in the domain of F, and having an appropriate boundary along which the contour integral of F'/F is well defined. Divided by 2πi this integral gives the number of zeros of F in D; in particular this is a non-negative integer. What can be said if, instead of being a scalar function, F has its values in a unital (complex) Banach algebra B? In other words, what kind of elements are the logarithmic residues in Banach algebras? It turns out that sums of idempotents are always logarithmic residues. Although the converse is generally not true, for large classes of Banach algebras positive answers have been obtained (e.g., commutative algebras, matrix algebras, approximately finite dimensional algebras, algebras of quasitriangular operators). In this lecture we consider the case when B is a Banach (sub)algebra (of a larger unital Banach algebra A) generated by a single element b. More specifically, the connection with sums of the spectral idempotents associated with b is discussed. The analysis involves the application of certain celebrated results from classical advanced plane topology, such as the Alexander Addition Theorem, the Alexander Duality Theorem and the Schoenflies Theorem. The work reported on has been done jointly with Torsten Ehrhardt from Santa Cruz and Bernd Silbermann from Chemnitz.

  • Arjan van der Schaft Abstract: As shown in earlier work, formally skew-adjoint linear matrix differential operators acting on vectors of functions depending on a scalar spatial variable give rise to Dirac structures on any bounded part of the spatial domain, where the boundary variables are obtained through integration by parts. This can be immediately extended to pairs of linear differential operators defining a formally skew-adjoint relation. A key tool in the procedure is the use of two-variable matrix polynomial calculus in order to streamline repeated integration by parts. Applying the same approach to formally self-adjoint linear differential operators results in infinite-dimensional Lagrangian subspaces on the bounded spatial domain. The combination of such infinite-dimensional Dirac structures with infinite-dimensional Lagrangian subspace results in a large class of boundary control port-Hamiltonian systems. This will be illustrated on a number of physical examples. The obtained port-Hamiltonian representation is useful for analysis and control. (Based on joint work with Bernhard Maschke, Lyon.)

  • Felix Schwenninger Tba

  • Teun Koetsier In 2019 I published a book on the history of technology titled "The Ascent of GIM, the Global Intelligent Machine". In my lecture I will start with the book and then I will consider the history of mathematics in the context of the history of technology. It is surprising - at least to me - that the two histories before the 20th century are in fact disjoint, apart from simple calculations and simple geometric insight. I will use the mathematics of external ballistics as an example. The story illustrates the importance of pure mathematics performed as a noble challenging game completely ignoring the question of utility.

  • Jan Brandts Matrices that are the Gramian of a nonnegative matrix are called completely positive. They form a convex cone. Quite a few NP-hard optimization problems can be reformulated as a linear optimization problem over this convex cone (this reformulation is called a copositive program). The bottleneck in the algorithms that perform this linear optimization is to decide whether a given matrix actually belongs to the cone, and also to find the nonnegative matrix of which it is the Gramian. We present a new algorithm that aims to find such a nonnegative matrix, or in other words, to factorize a given completely positive matrix whenever possible. Before doing so, we will give a gentle elementary introduction to the topic and its relevance.

  • Jan Wiegerinck Some 30 years ago examples were given of holomorphic functions on the unit disc with the property that their graph is precisely the infinity set of a plurisubharmonic function on C2. This led to the conjecture that this happens for every nowhere extendable holomorphic function on the disc. While some support for this conjecture was found, it turned out that the conjecture is false. There are counterexamples with seemingly strange behavior. This behavior can be understood by considering finely holomorphic functions instead of the usual ones. In the talk I will discuss all this and explain the relevant notions.

  • Derk Pik Problem solving has been in the spotlight in high school math education for quite some time. Is that possible: can you become proficient in solving problems? The math magazine for young people Pythagoras has already 60 years of experience with problem solving and experimental mathematics. In the years 2013-2015, seventeen articles on Erdös' open problems were published. Readers were challenged to participate in computational research. Many of these articles are examples of Computational Thinking, a discipline now entering high school education. In this lecture, we will discuss one of these problems from this point of view.

  • Hans Zwart Let A be the generator of a strongly continuous, exponentially stable, semigroup on a Hilbert space. Furthermore, let the scalar function g be bounded and analytic on the left-half plane. By using the Toeplitz operator associated to g, we construct the (unbounded) operator g(A). In general it is unknown whether this operator is bounded. However, when additionally A is dissipative, we show by using system theoretic techniques, that g(A) is bounded, and its bound cannot exceed that of g.

  • Jan H. van Schuppen Positive control systems are used as models in engineering, in the life sciences, and in economics. The inputs, the outputs, and the states of such a system are vectors over the semi-ring of the positive real numbers which represent, for example, concentrations of chemicals or amounts of capital. A control theoretic problem is to characterize reachability or controllability of a linear positive control system. The current literature about reachability of linear positive systems is not satisfactory. Giorgio Picci has argued that the state set of a finite-state stochastic system is a polyhedral cone in the positive orthant. W.M. Wonham has focused attention on controllable subspaces. Using their approaches, an equivalent condition will be stated with respect to which the reachable set will be a polyhedral cone. Reachable polyhedral subcones of the state set can then be considered. Further research issues will be mentioned. The lecture is based on joint research with Yashar Zeinaly and Bart de Schutter.

  • Freek van Schagen The continuous time Leech problems considered in this talk are based on stable rational finite dimensional operator-valued functions G and K. Here stable means that G and K do not have poles in the closed right half plane including infinity, and the Leech problem is to find a stable rational operator solution X such that G(s)X(s) = K(s) for s ∊ ℂ+ and sup{||X(s)|| : Re s ≥ 0} < 1. A solution of the Leech problem is expressed in integral operators. This solution is then transformed into a state space realization. In the latter realization the finite dimensional operators involved are expressed in the operators of state space realizations of the functions G and K. The solutions to the discrete time Leech problem on the unit circle are easier to develop and have been solved earlier. The talk is based on joint work with A.E. Frazho and M.A. Kaashoek.

  • Bob Planque Single celled organisms such as bacteria are able to tune enzyme levels that catalyze the reaction pathways by which they eventually make new copies of themselves. Depending on nutrient conditions, more or less enzyme is invested in different parts of their reaction network, so that reaction rates are constantly high, and cellular growth rate is maximised. In this talk I will present an adaptive control mechanism designed to solve this problem. It involves an ODE system with two sets of algebraic equations attached. Through a detailed analysis of the steady state and maximisation problems, we show that the adaptive control is actually globally stable for a wide variety of pathways. This suggests that real bacteria might actually be able to perform such adaptive control mechanisms using gene regulation.