Achievements and Challenges in the Field of Convolution Operators

Albrecht Böttcher is a professor emeritus at Chemnitz University of Technology. He is known for his work in operator theory, numerical analysis, and matrix analysis, especially for his contributions to Toeplitz operators, matrices, and determinants, and their applications. In 1997, he and Yuri Karlovichwon the Ferran Sunyer i Balaguer Prize for their monograph ``Carleson Curves, Muckenhoupt Weights, and Toeplitz Operators''.Oleksiy Karlovych is an associate professor at the NOVA University Lisbon, where he has been working since 2007. His research interests lie at the intersection of operator theory and the theory of function spaces, in particular, in the study of Toeplitz and convolution-type operators on Banach function spaces.Eugene Shargorodsky is a professor at King’s College London. His research is in operator theory and partial differential equations, in particular, Toeplitz operators, pseudodifferential operators, and spectral theory of Schrödinger operators.   Ilya M. Spitkovsky is a professor emeritus at New York University Abu Dhabi and the College of William and Mary (Williamsburg, Virgina), where he worked in 2013-2024 and 1990-2016, respectively. His research is in operator theory, complex analysis and matrix analysis, in particular, Riemann-Hilbert problems, Wiener-Hopf factorization, spectral theory of Toeplitz operators, and numerical ranges.

• - Part I: Yuri Karlovich and his Work.- Yuri Karlovich’s Path in Mathematics.- List of publications of Yuri Karlovich.- Yuri Karlovich and the Metamorphosis of Spectra of Singular Integral and Toeplitz Operators.- Banach Algebras Generated by N Idempotents and Applications.- Part II: Invited contributions.- M-Local Type Conditions for the C*-Crossed Product and Local Trajectories.- Factorisation of Symmetric Matrices and Applications in Gravitational Theories.- Invertibility of Toeplitz Plus Hankel Operators on 𝒍lp-Spaces.- On Pseudodifferential Operators with Slowly Oscillating Symbols on Variable Lebesgue Spaces with Khvedelidze Weights.- Eigenvalues of the Laplacian Matrices of Cycles with one Overweighted Edge.- On the Algebra of Singular Integral Operators with Almost Periodic Coefficients.- Approximate Reconstruction of a One-Dimensional Parabolic Equation From Boundary Data.- Characteristic Determinants for a Second Order Difference Equation on the Half-Line Arising in Hydrodynamics.- On a Singular Integral Operator with two Shifts and Conjugation.