Introduction to the Theory of Toeplitz Operators with Infinite Index

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This work explores various aspects of Toeplitz operators, focusing on infinite index examples and their properties. It begins with an overview of the space Lp(?, ?) and the operator S?, discussing classes of functions and normally solvable operators. The text delves into factorization, including (p, ?)-factorization and its relationship with Toeplitz operators of infinite index, highlighting inner-outer factorization and examples of relevant functions. The discussion progresses to model subspaces, detailing their construction, boundary behavior, and bases, including interpolation conditions and sine-type functions. It also examines Toeplitz operators with oscillating symbols, addressing almost periodic functions and their discontinuities, as well as well-posed problems related to the Toeplitz equation. Further, the work covers generalized factorization of u-periodic and matrix functions, including block Toeplitz operators and conditions for infinite indices. The final sections focus on Toeplitz operators with symbols that have zeros, addressing normalization principles and examples of polynomial degeneracy. The book provides a comprehensive framework for understanding the intricacies of Toeplitz operators, their applications, and the mathematical theories underpinning them.