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Properties of Toeplitz operators on the space of all holomorphic functions

M. Jasiczak: Toeplitz operators on the space of all holomorphic functions on finitely connected domains, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas (117)(1) (2023) https://doi.org/10.1007/s13398-022-01380-9

The beginning of both functional analysis and operator theory is intimately connected with the research of Stefan Banach (1892-1945), a great polish mathematician, in particular with the book he authored "Teorja Operacyj. Tom I: Operacje liniowe", which was published in Warsaw in 1931. It is this book, where such objects as linear operators were defined. Linearity is easy to define. This can be done just as for functions in primary school. Operators must be continuous, as well. This means that close arguments must be mapped onto close values. The project concerns a special class of linear operators called Toeplitz operators. These operators are characterized by a certain infinite matrix of numbers, or a table of numbers. The terms of this matrix need to be constant on the diagonals. In general, it is not an easy task to describe the properties of a linear operator. They are rather complicated creatures. In the case of Toeplitz operators it is easier. This is because there are deep connections of the Toeplitz operators theory with such elementary objects as functions. To be precise, a special class of functions, named holomorphic functions. Holomorphic functions are among the most beautiful and regular objects of mathematics. The theory of Toeplitz operators is a striking example, how seemingly different branches of mathematics are interconnected. In this case, this is complex analysis, which studies holomorphic functions and operator theory. It is a really interesting fact that Toeplitz operators, named after a great german mathematician Otto Toeplitz (1881-1940), were not defined for the first time in his book "Zur Theorie der quadratischen and bilinearen Formen von unendlichen Verandlichen, part I: Theorie des L-Formen" [Toeplitz 1911]. In fact, they appeared later and ever since they are in the main stream of research in mathematical analysis.

In the article we investigated properties of Toeplitz operators on the space of all holomorphic functions on finitely connected domains in the complex plane. In other words, on domains with finitely many holes. Holes in domains are the object of study of yet another branch of mathematics: topology. Not surprisingly, topology played an important role in the research, as well.
Nowadays there exists a classical theory of Toeplitz operators on the so called Hardy spaces. These spaces consist of holomorphic functions, which additionally satisfy a certain growth condition. So, not of all holomorphic functions. It is natural to ask what happens if we take into account all holomorphic functions. From the mathematical point of view, the difference is significant, since we leave the world created by Banach. It turns out that the theory, which we built, shows strong similarities with the classical case. There are however also rather subtle differences. For example, finite sets, where certain holomorphic functions take the value zero, are important in the Hardy space case. In our project such finite sets are almost negligible. What is vitally important, are the infinite sets of zeros. The properties of the Toeplitz operators vary dramatically depending on whether such infinite zero sets exist or not. In these terms we managed to describe such operator theoretic properties as the Fredholm property, the semi-Fredholm property or the closedness of the image. Can this theory be applied in other branches of mathematics, do "these objects live somewhere else"? Surprisingly enough, they do. They enter the picture in a decive way in the proof of classical Lindemann's theorem that pi is a transcendental number. That is, there exists no no-trivial polynomial with rational coefficient with pi as its root.

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