# Emmanuel Fricain

# Research axes

## Presentation of my main field of research

My field of research concerns functional analysis, a branch of mathematical analysis which developed at the beginning of the twentieth century thanks to the work of Banach, Volterra and Hilbert. The historical roots of functional analysis lie in the study of spaces of functions and their transformations. It has now branched out in many different subjects. My work is mainly devoted to topics in operator theory (study of linear transformations) and classical function theory. I have been interested a lot in the interplay between these two areas, which has been under development for well over 50 years. It is the reservoir of much beautiful mathematics, and it remains very active today. One of my favourite toy is the *de Branges--Rovnyak spaces*, a Hilbert space of analytic functions on the openn unit disc of the complex plane, and which is a cousin of the Hardy space. I will now try to explain the interest and importance of these spaces, through the *invariant subspaces problem.*

The theory of dynamical systems studies the long-term behavior of evolving systems. It has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, and medicine. Assume that the possible states of a (physical, biological, economic,...) system are described by the elements from a set X and that evolution of the system is described by a map T: X→X ; that is if xn is the state of the system at time n, then xn+1 = T(xn). In other words, if we start from point x0, then the evolution of the system is described by the orbit of x0 under the action of T, that is by the set denoted by orbit(T; x0) ={Tn(x0) : n>0}, where Tn is the composition of T with itself n-times. Since in general we are interested in measuring changes in the values xn, we require the underlying space X to be a metric space. And since, we want small changes in xn only to result in small changes in xn+1, we require continuity of T. Hence, a discrete dynamical system is a pair (X,T) consisting of a metric space X and a continuous map T: X→X. A typical question in dynamical theory is to know if there exists dense orbit? Dense here means that you can approach (as close as you want) any point of X by a point of the (dense) orbit. In other words, can your system reach all points of the space X in an approximate way? In many situations, the set X is a Hilbert space (think to Rn endowed with the Euclidean norm) and then you can ask if the orbit generates a dense subspace. A particular interesting situation is when the map T is linear (meaning it respects, in some sense, the structure of the vector space). Since the orbit of a point x0 under T is invariant with respect to T, the problem of the existence of a point x0 such that its orbit generates a dense subspace can be seen as a particular case of the invariant subspace problem (ISP), which is the greatest unsolved problem in operator theory, and one of the most fundamental open problems in mathematics. Given an infinite complex Hilbert space H and a linear bounded (continuous) operator T acting on H, does T have a non-trivial closed invariant subspace? More explicitly, does there exist a closed subspace E of H which is invariant with respect to T (TE is contained in E) and which is non-trivial in the sense that E is not reduced to {0} nor H. In the finite dimensional case, the classical linear algebra tells us that T has eigenvalues and they provide non-trivial invariant subspaces (the corresponding eigenvector spaces). In the infinite dimensional case, we know that T does not necessarily have eigenvalues and thus the problem becomes considerably more difficult.

As a matter of fact, the ISP was initially stated in the Banach space setting and, for half a century, remained a challenging open problem. Then an outstanding counter example was given by Enflo in 1976. Another construction was provided by C. Read in 1984. They constructed a Banach space and a linear bounded operator T on it such that T has no non-trivial invariant subspace. At the same time, a very active branch of research emerged to find classes of operators which have non-trivial invariant subspaces. One of the famous result in this direction is due to Lomonosov in 1973 who proved that if T is a bounded operator on X, commuting with a non-trivial compact operator, then T has a non-trivial invariant subspace. Despite numerous attempts, the ISP in the Hilbert space setting is still unsettled.

There are many results now which give sufficient conditions insuring that an operator T on a Banach space X has a non-trivial invariant subspace. But, describing all the non-trivial invariant subspaces of T is in general very difficult. Maybe the first and most spectacular example was given by Beurling [Acta Math., 1948]. He was interested in describing all the invariant subspaces of the shift operator S on the space l_{2 }of square summable sequences. His beautiful idea was to translate this problem in the Hardy space H^{2} consisting of analytic functions on the open unit disc D whose Taylor coefficients are in l_{2}. In this context, the shift operator S translates (is unitarily equivalent) to the shift operator S on H^{2} défined by (Sf)(z) = zf(z). Since both operators are unitary equivalent, describe the closed invariant subspaces of one reduces to describe the closed invariant subspaces of the other one, and *vice et versa. *Using the theory of complex and harmonic analysis, Beurling was able to describe all the closed invariant subspaces of S on H^{2}. They are all of the form θH^{2}={θf: f dans H^{2}} where θ is an inner function, that is an analytic and bounded function on the open unit disc D whose radial limits are of modulus one almost everywhere on the unit circle T. Moreover, it turns out that we know an explicit formula for inner functions. In particular, the class of inner functions contains the class of Blaschke products (which are infinite products of automorphism of the unit disc). This amazing result of Beurling found many applications and in particular in functional model theory. More precisely, in the sixties, B. Sz.-Nagy and C. Foias, inspired by the ideas of Krein's school, developed a functional model theory for Hilbert spaces contractions. In this theory, the model operator which is involved is of the basic type, that is the backward shift operator (the adjoint of S), which to an analytic function f(z) associates the function (f(z)-f(0))/z (on Taylor coefficients, it corresponds to a left shift). This operator acts of the model space K_Θ, corresponding to to the orthogonal complement of Beurling subspaces ΘH^{2} in H^{2}. Roughly speaking, the theory of Sz.-Nagy--Foias says that any contraction T on a Hilbert space H is unitary equivalent to S^{*}: K_Θ→ K_Θ for a certain inner function Θ associated to T. In parallel, L. De Branges and J. Rovnyak developed another functional model theory where the model space K_Θ s replaced by the reproducing kernel Hilbert space H(b), whose reproducing kernel is given by (1-\bar{b(λ)}b(z))/(1-\bar{λ} z), where λ and z are in D. Here b is an analytic function on D, which is bounded by one. It turns out that when b=Θ is an inner function, then both spaces H(b) and K_Θ coincide. The theory of de Branges--Rovnyak gives a a similar result to the Sz.-Nagy–Foias theory, replacing the space K_Θ by the space H(b). The principal idea of these two theories was thus to replace the study of a possibly complicated operator T by a simpler one, the backward shift operator. Also since Θ and b are analytic, and the spaces K_Θ and H(b) consist of analytic functions, we can use complex analysis. Using the approach of Sz.-Nagy and C. Foias, D. Sarason completely classified the invariant subspaces of the Voltera operator. Even though these approaches did not give a complete solution to the invariant subspace problem, they lead to interesting new territories in mathematical analysis. Apart from model theory, it is now a fact that spaces K_Θ and de Branges--Rovnyak spaces play a key role in many other questions of function theory (solution of the Bieberbach conjecture by L. de Branges, rigid functions of the unit ball of H^{1}, Schwarz--Pick inequalities), operator theory (invariant subspaces problem, Toeplitz and composition operators), approximation theory (Nevanlinna domains,..), and linear system theory. Despite a lot of efforts by many mathematicians like Baranov, Dyakonov, Nikolski, Poltoratski, Sarason, there are still many interesting questions related to these spaces and it is a very active area of research. One of the main part of my research has consisted to make progress in the understanding of these spaces.

## Some of my recent results

- In
*Bounded symbols and reproducing kernel thesis for truncated Toeplitz operators*,**Journal of Functional Analysis (2010)**, in collaboration with A. Baranov, I. Chalendar, J. Mashreghi and D. Timotin, we were interested in the class of truncated Toeplitz operators, introduced by D. Sarason in 2007, and which is a subject in great expansion. This class of operators is closely linked with the classes of Toeplitz and Hankel operators on the Hardy space H^{2 }and to the theory of interpolation. Indeed, the truncated Toeplitz operators are the compression of Toeplitz operators on the model space K_Θ. In the classical case, a Toeplitz operator T_φ has a unique symbol φ and the operator is bounded on H^{2}if and only if its symbol φ is bounded. The situation for truncated Toeplitz operators is more complicated : there is no unicity of the symbol. In his seminal paper, D. Sarason asked the following question: does every bounded truncated Toeplitz operator has a bounded symbol? This question is important because the existence of a bounded symbol enables to work easily with these operators. In our paper, we showed that the answer to the question of Sarason is negative: we proved that a bounded truncated Toeplitz operators may have no bounded symbols. This was a fundamental and unexpected observation and created a lot of new research activities on TTO. This result says in particular that contrary to its appearance, TTO behave more like Hankel operators than like Toeplitz operators. This paper changes the perspective we had on this class of operators and is a cornerstone for the whole theory. - In
*Multipliers between model spaces*,**Studia Mathematica (2018)**, with A. Hartmann and W. Ross, we studied the problem of multipliers between model spaces. It turns out that a model space is not an algebra. In particular, the problem of characterizing the functions φ such that the multiplication by φ sends K_Θ into itself is a natural and important problem. Unfortunately, even in this case, the only multipliers are the constant functions. Nevertheless, if we are given two different inner functions Θ_{1}and Θ_{2 }, then the set of multipliers from K_Θ_{1}into K_Θ_{2}, that is the set of functions φ such that for every function f in K_Θ_{1 }the function φf is in K_Θ_{2}, can be non trival. In our paper, we have established an interesting characterization in terms of Carleson measures for the model spaces and injectivity of Toeplitz operators. Let us mention that Carleson measures have been introduced by L. Carleson in a paper published at Acta Math. (1964) in connection with interpolating sequences, and he got the prestigious Abel price for this work. Since then, Carleson measurs became a crucial object in function theory. On the other hand, the injectivity of Toeplitz operators is linked, since the important work of Makarov–Poltoratski (Invent. Math 2010), to the Beurling--Malliavin theory which is so important in harmonic analysis. One of our most intriguing result of our paper is that we can construct, in many cases, unbounded multipliers between model spaces. - In
*Asymptotically orthonormal basis and Toeplitz operators*,**J. Math. Anal. Appl. (2019),**with R. Rupam, we were interested in the study of exponential systems. The classical theory of Fourier series says that the trigonometric system (e^{int})_{n\in Z} forms an orthonormal basis of L^{2}(0;2π). A natural question is to know what happens if we pertubate a little bit the trigonometric system. Do we keep properties of basis in a certain sense? Recently, M. Mitkovski has given a criterion for Riesz basis property for exponential systems (with complex frequencies) in terms of invertibility of a certain Toeplitz operators naturally accociated to the problem, in the spirit of works of Khrushchëv–Nikolski–Pavlov (Lecture Notes In Math. no. 864, 1981). In our paper, we extend some of the results of Mitkovski for sequences of reproducing kernels of the model spaces K_Θ where Θ is a meromorphic inner function. We also give an analogue for the property of asymptotically orthonormal basis (a close notion of orthonormal basis). - In
*Reducing subspaces of C*,_{0}(N) operators**New York Journal of Mathematics (2021)**, with C. Benhida and D. Timotin, we were interested in a question linked to the invariant subspace problem, those of reducing subspaces. Recall that given a linear and continuous map T on a Hilbert space H, a closed subspace E of H is said to be*reducing*if E and its orthogonal are invariant with respect to T. In this case, the matrix of T with respect to the decomposition of H into E and its orthogonal is diagonal, which explains why we are interested in reducing subspaces. The reducing subspaces enable us to reduce the study of T to small parts (in the spirit of diagonalization in the finite dimensional case). Using the model theory of Sz-Nagy–Foias, in our paper, we completely characterize the contractions which admit a reducing subspace of index one (these index measure in a certain sense the distance between contractions and isometries). We then apply this general characterization to certain truncated Toeplitz operators and complete some study iniated recently by Li–Yang–Lu [New York Math. J., 2018 et CAOT, 2020] et Gu [Journal of Operator Theory, 2020]. In particular, we refute a conjecture of Gu on the existence of reducing subspaces for a certain class of truncated Toeplitz operators.