Toeplitz Matrices and Operators

This is an extremely thorough introduction to the theory of Toeplitz matrices (and their siblings the Hankel matrices; known collectively by the portmanteau Ha-plitz). It provides most of the needed background, and probably could be used for very advanced undergraduates but seems more like a second-year graduate text. This is a translation of the French-language text Matrices et Opérateurs de Toeplitz published in 2017 by Calvage & Mounet.

 

A Toeplitz matrix is an infinite matrix in which each row is the left-shift of the row below it (and therefore the diagonals are constant); in symbols, the elements \( t_{i,j} \) satisfy \( t_{i,j}=t_{i+k,j+k} \) all \( k \). A Hankel matrix is an infinite matrix in which each anti-diagonal is constant; in symbols, the elements satisfy \( t_{i,j}=t_{i+k,j-k} \)for all \( k \). A Toeplitz (Hankel) operator is one whose matrix in some orthonormal basis is Toeplitz (Hankel).

 

The theory of Toeplitz matrices originated in a 1911 paper of Otto Toeplitz, although the ideas had appeared earlier in studies of integral transforms and equations by Riemann, Hilbert, Volterra, Fredholm, and others. The theory is primarily about Hardy spaces, and in particular the space \( H^{2}(\mathbb{T} ) \) of the Hardy space on the circle, but many other function spaces can be used, and the book deals with these also.

 

The book has one (relatively short) chapter of applications; most of these are the historical applications to integral equations. There are a number of appendices that give needed background on various types of spaces, algebras, and operators that are used in the book. Among the attractive features of the book are: thirty-two several-page biographies of researchers who developed the field; extensive inline sets of exercises with complete solutions; and extensive chapter endnotes that give the history and explore some topics further.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web site is allenstenger.com. His mathematical interests are number theory and classical analysis.