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This is a survey of Harold Widom’s work in Toeplitz operators, embracing his early results on the invertibility and spectral theory of Toeplitz operators, his investigations of the eigenvalue distribution of convolution operators, and his groundbreaking research into Toeplitz and Wiener–Hopf determinants.

A Hilbert space operator A is said to be core invertible if it has an inner inverse whose range coincides with the range of A and whose null space coincides with the null space of the adjoint of A. This notion was introduced by Baksalary, Trenkler, Rakić, Dinčić, and Djordjević in the last decade, who also proved that core invertibility is equivalent to group invertibility and that the core and group inverses coincide if and only if A is a so-called EP operator. The present paper contains criteria for core invertibility and for the EP property as well as formulas for the core inverse for operators in the von Neumann algebra generated by two orthogonal projections.

The paper is devoted to results connecting the eigenvalues and singular values of operators composed by $P^\ast G P$ with those composed in the same way by $QG^{−1}Q^\ast$. Here $P +Q = I$ are skew complementary projections on a finite-dimensional Hilbert space and $G$ is a positive definite linear operator on this space. Also discussed are graph theoretic interpretations of one of the results.

Toeplitz matrices are typically non-Hermitian and hence they evade the well-elaborated machinery one can employ in the Hermitian case. In a pioneering paper of 1960, Palle Schmidt and Frank Spitzer showed that the eigenvalues of large banded Toeplitz matrices cluster along a certain limiting set which is the union of finitely many closed analytic arcs. Finding this limiting set nevertheless remains a challenge. We here present an algorithm in the spirit of Richard Beam and Robert Warming that reduces testing $$O(N^2)$$ O ( N 2 ) points in the plane for membership in the limiting set by testing only O(N) points along a one-dimensional curve. For tetradiagonal Toeplitz matrices, we describe all types of the limiting sets, we classify their exceptional points, and we establish asymptotic formulas for the analytic arcs near their endpoints.

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. This algebra contains every skew projection on that Hilbert space and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm., Comment: 6 pages

There are infinitely many finite groups of orthogonal matrices all orbits of which, including those of irrational vectors, span lattices, that is, discrete additive subgroups of the underlying Euclidean space. We show that, both up to isomorphism and up to orthogonal similarity, exactly eight of these groups are irreducible: the two trivial groups in one dimension, the cyclic groups of orders 3, 4, 6 in two dimensions, and the quaternion, binary dihedral, binary tetrahedral groups in four dimensions.

Let $$F(\varvec{x})$$ be a homogeneous polynomial in $$n \ge 1$$ variables of degree $$1 \le d \le n$$ with integer coefficients so that its degree in every variable is equal to 1. We give some sufficient conditions on F to ensure that for every integer b there exists an integer vector $$\varvec{a}$$ such that $$F(\varvec{a}) = b$$ . The conditions provided also guarantee that the vector $$\varvec{a}$$ can be found in a finite number of steps.

We employ the fact certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symmetric polynomials of even degree $2p$, the positivity of which is a classical result by D. B. Hunter. We extend Hunter's result to complete homogeneous symmetric polynomials of fractional degree, which are defined via Jacobi's bialternant formula. We show in particular that these polynomials have positive real part for real degrees $��$ with $|��-2p|< 1/2$. We also prove a positivity criterion for linear combinations of the classical complete homogeneous symmetric polynomials and a sufficient criterion for the positivity of linear combinations of products of such polynomials., 13 pages

This note supplies additions to our paper quoted in the title on integral spans of tight frames in Euclidean spaces. In that previous paper, we considered the case of an equiangular tight frame (ETF), proving that if its integral span is a lattice then the frame must be rational, but overlooking a simple argument in the reverse direction. Thus our first result here is that the integral span of an ETF is a lattice if and only if the frame is rational. Further, we discuss conditions under which such lattices are eutactic and perfect and, consequently, are local maxima of the packing density function in the dimension of their span. In particular, the unit ( 276 , 23 ) equiangular tight frame is shown to be eutactic and perfect. More general tight frames and their norm-forms are considered as well, and definitive results are obtained in dimensions two and three.

We consider lattices generated by finite Abelian groups. We prove that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame. Further, our result combined with a recent theorem of R. Bacher produces (via the classical Voronoi criterion) a new infinite family of extreme lattices. Additionally, we investigate the structure of the automorphism groups of these lattices, strengthening our previous results in this direction., 12 pages

We consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight ( k , n ) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2 k . We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect.

The paper concerns the limit of the zero set S n of the Fibonacci-type polynomials G n ( x ) = G n ( a , b , x ) given by G 0 ( x ) = a , G 1 ( x ) = x + b , and G n + 2 ( x ) = x G n + 1 ( x ) + G n ( x ) . It presents an explicit set L consisting of at most two points such that S n converges to the union of [ − 2 i , 2 i ] and L in the Hausdorff metric. The proof is based on the asymptotic analysis of the eigenvalues of special non-Hermitian perturbations of Hermitian Toeplitz matrices.

This is an essay containing personal reminiscences and describing selected topics of joint mathematical work of the author and Sergei Grudsky. The topics have their focus on Toeplitz operators and large finite Toeplitz matrices.

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of $n$-by-$n$ matrices as $n$ goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the Szeg\H{o} type. Our results transfer these convergence theorems into uniform convergence statements., Comment: 15 pages, 3 figures

The paper is devoted to the structure and the asymptotics of the eigenvector matrix of Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The results extend existing results on banded Toeplitz matrices to full Toeplitz matrices with temperate decay of the entries in the first row and column. We establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors. These formulas are in terms of the Wiener–Hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue. The paper does not aim at providing numerical algorithms. Its main purpose is rather to reveal the structure underneath the eigenvectors, which are shown to be the sum of two harmonics and certain edge corrections.

This is a review of a development in the 1970s and 1980s that was concerned with index formulas for block Toeplitz operators with discontinuous symbols. We cite the classical results by Douglas and Sarason, outline Silbermann’s approach to index formulas via algebraization, and embark on the replacement of the Abel–Poisson means by arbitrary approximate identities. One question caused by this development was whether the Abel–Poisson means are the only approximative identities that are asymptotically multiplicative on (H∞,H∞), and the review closes with Wolf and Havin’s theorem, which gives an affirmative answer to this question.

The meta theorem of this paper is that Halmos’ two projections theorem is something like Robert Sheckley’s Answerer: no question about the W*- and C*-algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supersymmetry equality introduced by Avron, Seiler, and Simon. A noteworthy insight of the paper reveals that the supersymmetric approach is nothing but Halmos in different language and hence an equivalent Answerer.

An equation with a Hardy space Toeplitz operator can be solved by Wiener–Hopf factorization. However, Wiener–Hopf factorization does not work for Bergman space Toeplitz operators. The only way we see to tackle equations with a Toeplitz operator on the Bergman space is to have recourse to approximation methods. The paper is intended as a review of and an illustrated tour through several such methods and thus through some beautiful topics in the intersection of operator theory, complex analysis, differential geometry, and numerical mathematics.

It was shown in a series of recent publications that the eigenvalues of \(n\;\times\;n\) Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of n + 1. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol g(x) = (2 sin(x/2))4, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. Here we use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic expansion. This also delivers a counter-example to a conjecture by Ekstrom, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.

The collective behavior of the singular values of large Toeplitz matrices is described by the Avram-Parter theorem. In the case of Hermitian matrices, the Avram-Parter theorem is equivalent to Szeg?'s theorem on the eigenvalues. The Avram-Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the singular values and the appropriately ordered absolute values of the symbol. The purpose of this paper is twofold. Under natural hypotheses, we first strengthen the known estimates in the mean to estimates in the maximum norm, thus turning from collective results on the singular values to results on individual singular values. Secondly, we want to emphasize that the use of the quantile function eases the proofs and statements of results significantly and provides a promising language for forthcoming research into higher order asymptotics for individual singular values.

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.

Given two orthogonal projections P and Q, we are interested in all unitary operators U such that UP=QU and UQ=PU. Such unitaries U have previously been constructed by Wang, Du, and Dou and also by one of the authors. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du described all unitaries U with the required property. Their proof is via the two projections theorem by Halmos. We here give a proof based on the supersymmetric approach by Avron, Seiler, and one of the authors., 12 pages

Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

The Duduchava–Roch formula is a formula for the inverse of a Toeplitz matrix that is generated by a pure Fisher–Hartwig singularity. We cite this formula with a full proof and give several of its applications. These are the Fredholm theory of Toeplitz operators with piecewise continuous symbols, the derivation of the pure Fisher–Hartwig determinant, problems connected with lattice determinants, and Green’s function for a boundary value problem for a higher-order ordinary differential operator.

Let $$H(N)$$ denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in $$H(N)$$ whose largest coefficients are as small as possible and also for polynomials in $$H(N)$$ with minimal sum of the coefficients. Let $$h(N)$$ and $$s(N)$$ denote these minimal values. Using Fekete’s subadditive lemma we show that the $$N$$ th square roots of $$h(N)$$ and $$s(N)$$ have a limit as $$N$$ goes to infinity and that these two limits coincide. We also derive tight bounds for the common value of the limits.

We consider the determinants of compressions of Toeplitz operators to finite-dimensional model spaces and establish analogues of the Borodin–Okounkov formula and the strong Szegő limit theorem in this setting.