# Obituary for Bent Fuglede

With the death of Bent Fuglede, Denmark has lost one of its very important mathematicians and the Department of Mathematics at the University of Copenhagen has lost a highly valued teacher, researcher and colleague.

Bent Fuglede (8.10.1925 - 7.12.2023) is survived by his son Einar and daughter-in-law Dorthea and two grandsons. Bent's wife Ólafia Einarsdóttir of more than 60 years died in 2017.

Bent received the high school diploma from Skt. Jørgens Gymnasium in 1943, and he graduated from University of Copenhagen as mag. scient. and cand. mag. in 1948. After employment as scientific assistant at the Polytechnical Highschool, he was appointed associate professor at the Department of Mathematics at the University of Copenhagen. Having defended the dissertation “Extremal Length and Closed Extensions of Partial Differential Operators”, he was appointed professor at Denmark's Technical Highschool in 1960.

He returned to the University of Copenhagen as professor in 1965. He started by preparing excellent lecture notes for Mathematics 6, the major course in functional analysis that completed the first part of the cand. scient. study in mathematics. In the years that followed, he taught at all levels, always with a clear and well-worked presentation and with an easy-to-read blackboard script. His accompanying lecture notes often contained new results and elegant proofs. In 1992 he allowed himself to retire, but continued as an active researcher and professor emeritus at the Department of Mathematics and was active until shortly before his death.

Bent spent the years 1949-51 in the USA and stayed partly at Stanford University and partly at the Institute for Advanced Study in Princeton. It was during the visit at Stanford in November 1949 that “Fuglede's Theorem”, was communicated by John von Neumann to the Proceedings of the National Academy of Sciences. This result was the first and probably most famous of Fuglede's many theorems. lt states, that if a bounded operator on a Hilbert space commutes with a normal operator, then it also commutes with the adjoint operator. It is a surprisingly deep result, which today is included in all advanced textbooks on operators in Hilbert spaces.

During Bent's stay at the Institute for Advanced Study, he started collaboration with the American mathematician Richard V. Kadison on, among other things, determinant theory in infinite dimension. This led to Kadison coming to Copenhagen, and he gained a lifelong connection to Denmark, because he met his wife here. The strong position of operator algebra at our Institute is undoubtedly related to the influence of Kadison. The Fuglede-Kadison determinant continues to be used, e.g. by Lück in his development of $L^2$-cohomology and in the construction of the Brown measure for non-normal operators.

Subsequently, Bent studied partial differential operators, which led to the aforementioned doctoral dissertation, and also to research on potential theory. This theory deals with the solutions to Laplace's equation, i.e. harmonic functions, Newton potentials, etc. In this subject, he quickly became a leading figure worldwide, and he often gave lectures at the potential theory seminar in Paris “Séminaire Brelot-Choquet-Deny”, named after the three leading French mathematicians in the subject. In the 1960s, Brelot and his students had developed an abstract potential theory called harmonic spaces. A harmonic space is a locally compact topological space, where a vector space of continuous real-valued functions is specified for each open subset, modelling the harmonic functions in open sets of a Euclidean space. Brelot had established a few fundamental axioms that these abstract spaces had to satisfy in order that many of the key results of potential theory could be deduced. The significance of the abstract theory was that one could obtain these key results for solution spaces to PDEs related to the Laplace operator by simply checking the axioms.

In harmonic spaces superharmonic functions can be introduced, but as in the classical theory, these are not always continuous, but only lower semi-continuous. In 1940, Henri Cartan got the idea to replace the classical Euclidean topology by a topology having more open sets, so that superharmonic functions became continuous. Cartan called this new topology the fine topology. The fine topology also appears in the theory of Brownian motion and similar stochastic processes, because a finely open set can be characterized as follows: If the process is in the finely open set at a specific time $t_0$, then it remains there almost surely for a positive period of time. When Bent succeeded proving that the fine topology is connected and locally connected, he was inspired to construct what he called ”finely harmonic functions in finely open sets”, and this was the start of a new era in potential theory and complex analysis. Bent published a comprehensive presentation of the theory in a Springer Lecture Notes: “Finely Harmonic Functions”, 1972, but already at the International Congress of Mathematicians (ICM) in Nice, 1970, he gave an invited lecture on it.

The topic became ”hot”, and was further developed in the following years by Bent himself and with contributions from many others. Bent experienced the great honour that Heinz Bauer's plenary lecture on potential theory at the ICM in Vancouver, 1974, was largely about Fuglede's fine harmonic theory.

Just as harmonic functions in the plane are closely related to holomorphic functions, Bent succeeded in developing a rich theory for finely holomorphic functions in finely open sets of the complex plane. It also turned out that Fuglede's theory was a natural continuation of Borel's theory for ”monogenic functions”, which was published as a monograph as early as 1917.

Bent's work inspired a number of young mathematicians from abroad to new results, e.g. Terry Lyons (England) and Bernt Øksendal (Norway). Bent corresponded with a large number of mathematicians from home and abroad, and he was always generous with advice and improvements to manuscript drafts. He had a rare ability to penetrate the heart of a mathematical problem, and through this, he was often able to provide a surprising solution to the problem.

In 1974, Bent wrote a paper in the Journal of Functional Analysis: ”Commuting self-adjoint partial differential operators and a group theoretic problem”. It has today received 298 citations. The work, which was inspired by a question from Irving Segal, associates a geometric property of an $n$-dimensional domain $G$ with a property of the Hilbert space $L^2(G)$. The result is later known as Fuglede's Conjecture. The Danish mathematician Steen Pedersen from Ohio has published a number of papers on the subject and later Fields medal recipient Terence Tao published two papers: “Fuglede's Conjecture holds for convex planar domains”, (2001) and ”Fuglede's Conjecture is false in 5 and higher dimensions”, (2003).

Bent decided to retire in the spring of 1992. He felt his health a little failing after a year's stay at the Institute for Advanced Study in Princeton, where he was accompanied by Ólafia. Fortunately, his health turned out to be fine and Bent achieved an emeritus period of approx. 30 years, during which he was very active. Of the 114 of his works mentioned in Math. Sci. Net, approx. 40 is written after he became emeritus. During this period, he made significant contributions to many different areas, e.g. isoperimetric inequalities, Riemannian manifolds and moment problems. It should also be mentioned that he published a research monograph together with James Eells: ”Harmonic maps between Riemannian Polyhedra”, Cambridge Tracts in Mathematics vol. 142, 2001. Together with Natalia Zorii from Kyiv, he has since 2016 written 8 papers dealing with energy problems with respect to Riesz kernels, an impressive achievement for a person over 90 years of age.

Bent was a member of the Royal Danish Academy of Sciences and Letters, the Finnish Academy of Sciences and the Bavarian Academy of Sciences.

On his retirement, the department celebrated him with a symposium. I had the pleasure of telling about Bent's mathematical results, and there were lectures by guests from several countries. The Danish Mathematical Society celebrated Bent's 70th birthday and named him an honorary member. For a number of years, he was a member of the editorial board of the journal Expositiones Mathematicae.

With his great and broad knowledge in many areas of mathematics, he was an obvious member of assessment committees at home and abroad. He has thereby had a great influence on how the department of mathematics developed.

Fuglede was active in bringing foreign capacities to the Institute on a one-year basis, including Masanori Kishi (Nagoya) and Mikhail Sodin (Kharkiv), both specialists in potential theory. Their appointments have had a great impact on my own research, and Sodin has subsequently been a regular guest at the Institute.

All theories have their time and then the interest in them decreases. This also happened to the abstract theory of harmonic spaces. Fuglede's theory for finely harmonic functions was formulated in Brelot's abstract theory. It was therefore difficult for new generations to utilize and appreciate Fuglede's theory, and this made Sodin emphasize at regular intervals that Bent should write a new presentation of the theory about finely harmonic functions, but only in connection with the classical potential theory.

Sodin succeeded in convincing Bent about the project, and he started to prepare a manuscript, which should include the results of his work on classical fine potential theory done in the period between 1970 and 2010. Bent had for many years corresponded with the Moroccan mathematician Mohamed El Kadiri about potential theory and invited him to collaborate on the book project "Classical Fine Potential Theory". They agreed on the collaboration and also to enlarge the scope to include results on finely holomorphic functions in one or several complex variables as well as a chapter on fine pluripotential theory. They also decided to include their joint recent research on Martin boundary for finely open sets. At the end of October 2023, the book was sent to Springer Nature.

The mathematical community can look forward to the publication of this fine closing of Bent Fuglede's career.

Glory be to his memory.

*Christian Berg, 27.12.2023.*