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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.

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 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.

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.

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

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.

Let P and Q be idempotents in a real or complex associative algebra and consider the list of products P , Q , PQ , QP , PQP , QPQ , PQPQ , QPQP , … . The number of factors is called the order of the product. We say that P and Q are tightly coupled if the list contains two products which take the same value and whose orders differ by at most 1 . The main result of the paper is the classification of all algebras which are generated by two tightly coupled idempotents. In other words, we provide a list of algebras such that every algebra generated by two tightly coupled idempotents is isomorphic to exactly one algebra of the list. For example, it follows that up to isomorphisms there are exactly 16 copies of such algebras in which the equality PQP = PQPQ holds.

Using results on block Toeplitz matrices, in particular the Szegő–Widom limit theorem and Gohberg and Feldman’s convergence theorem for the finite section method, we compute the spontaneous magnetization of an Ising model. In this way, the transition from the ferromagnetic phase to the paramagnetic phase may be understood as the transition from invertibility to non-invertibility of a certain block Toeplitz operator associated with the model.

We present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle, respectively. The approach is based on solving a nonlinear system by Newton’s method. The Jacobi matrix is a polynomial of a companion matrix and thus a Bezoutian times the inverse of a triangular Toeplitz matrix. After getting deeper understanding of the displacement structure of the Bezoutian, each Newton step can be reduced to a few applications of discrete Fourier or cosine transforms and the LU-decomposition of a conceived linear system with a Cauchy-like matrix. These structural features are employed to design a fast algorithm, which shows excellent numerical behavior. For example, in the case of the extremely ill-conditioned test polynomial ( 2 z n + ∑ k = 0 n - 1 z k ) ( 2 + ∑ k = 1 n z k ) of the degree 2 n = 20000 , we obtain the factorization after 22 Newton steps with an error of 3.4 · 10 - 8 in the Euclidean norm, the execution time on a laptop being a couple of minutes. The local quadratic convergence of our Newton method is proved and explicit convergence balls are given on the basis of estimates for the Lipschitz constant which occurs in Kantorovich’s theorem. We also design and test two superfast versions of our method, which, for polynomials of degree 20000, typically need about half a minute to produce an initial coefficient vector and then perform the Newton iteration within 1 s.

In the theory of traces on operator ideals, it is desirable to treat not only the complex case. Several proofs become much easier when the underlying operators are represented by real matrices. Motivated by this observation, we prove two theorems which, to the best of our knowledge, are not available in the real setting: (1) every operator is a finite linear combination of orthogonal operators, and (2) every skew-symmetric compact operator S is a commutator [A, T], where certain properties of S are inherited to T. In our opinion, theses results are interesting for their own sake. They will also be used in future studies of trace theory by the second-named author.

The paper is devoted to the eigenvectors of Hessenberg Toeplitz matrices whose symbol has a power singularity. We describe the structure of the eigenvectors and prove an asymptotic formula which can be used to compute individual eigenvectors effectively. The symbols of our matrices are special Fisher–Hartwig symbols, and the theorem of this paper confirms and makes more precise a conjecture by Dai, Geary, and Kadanoff of 2009 in a particular case.

This paper is concerned with algebras generated by two idempotents P and Q satisfying ( PQ ) m = ( QP ) m and ( PQ ) m - 1 ≠ ( QP ) m - 1 . The main result is the classification of all these algebras, implying that for each m ⩾ 2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given.

The main topic of the paper is best constants in Markov-type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves. The norm is the L 2 norm with Laguerre weight. The leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term, which is the operator norm of a Volterra operator, are given. For best constants in inequalities of the Wirtinger type, the limit is computed and an asymptotic formula for the error term is presented.

The paper is concerned with Hermitian Toeplitz matrices generated by a class of unbounded symbols that emerge in several applications. The main result gives the third order asymptotics of the extreme eigenvalues and the first order asymptotics of the extreme eigenvectors of the matrices as their dimension increases to infinity.

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.

Uniform boundedness of sequences of variable-coefficient Toeplitz matrices is a delicate problem. Recently we showed that if the generating function of the sequence belongs to a smoothness scale of the Holder type and if α is the smoothness parameter, then the sequence may be unbounded for α 1. In this paper we prove boundedness for all α > 1/2.

Questions in probability and statistical physics lead to the problem of finding the eigenvectors associated with the extreme eigenvalues of Toeplitz matrices generated by Fisher-Hartwig symbols. We here simplify the problem and consider pseudomodes instead of eigenvectors. This replacement allows us to treat fairly general symbols, which are far beyond Fisher-Hartwig symbols. Our main result delivers a variety of concrete unit vectors xn such that if Tn(a) is the n × n truncation of the infinite Toeplitz matrix generated by a function a ∈ L1 satisfying mild additional conditions and λ is in the range of this function, then ‖Tn(a)xn − λxn‖ → 0 . Mathematics subject classification (2000): 47B35, 15A18, 41A80, 46N30.

If the matrix of a square linear system is nonsingular but has very small singular values, then tiny perturbations of the right-hand side may cause drastic changes in the solution. We show that the probability for this to happen is very close to zero if sufficiently many singular values of the matrix are bounded away from zero. Mathematics subject classification (2000): 65F35, 15A12, 60H25, 65F22.

We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of Gerhard Hiss on the factorization of functions with particular divisor support into lines and their inverses., Comment: 15 pages

At present there exist numerous different approaches to results on Toeplitz determinants of the type of Szego’s strong limit theorem. The intention of this paper is to show that Jacobi’s theorem on the minors of the inverse matrix remains one of the most comfortable tools for tackling the matter. We repeat a known proof of the Borodin–Okounkov formula and thus of the strong Szego limit theorem that is based on Jacobi’s theorem. We then use Jacobi’s theorem to derive exact and asymptotic formulas for Toeplitz determinants generated by functions with nonzero winding number. This derivation is new and completely elementary.

We present two elementary derivations of the formula for the Toeplitz determinant generated by a pure Fisher-Hartwig singularity.

Numerical experiments show that the Frobenius norm of the commutator of two large matrices typically clusters sharply around a certain value, which, moreover, is much smaller than one would predict. The purpose of this paper is to give a rigorous foundation of this phenomenon. We also discuss the question how big the Frobenius norm of commutator of the two matrices can actually be.

When working with structured matrices, structured condition numbers are more meaningful than usual condition numbers. The usual condition numbers of Toeplitz matrices may grow exponentially with the dimension of the matrix. The purpose of this note is to show that in this case the structured condition numbers do with high probability also increase exponentially, which supports the claim that one is not likely to win much on the average Toeplitz input by passing from the usual condition numbers of Toeplitz matrices to their structured counterparts. Copyright © 2004 John Wiley & Sons, Ltd.

Rambour and Seghier established two deep results on the first order asymptotics of the entries of the inverses of certain Toeplitz matrices. The determination of the constants in the leading terms of the asymptotics turns out to be nontrivial. We here show how these constants can be obtained from a well known identity by Duduchava and Roch.

This paper investigates the possible spectra of large, finite dimensional Toeplitz band matrices with perturbations (impurities, uncertainties) in the upper-left m × m block. The main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations, provided they have sufficiently small norm. This follows from analysis of structured pseudospectra (structured spectral value sets). In contrast, for typical non-Hermitian Toeplitz matrices there exist certain rank-one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large-dimensional limit.

The problem of polynomial factorization is translated into the problem of constructing a Wiener–Hopf factorization, and three algorithms are designed for the solution of the latter problem. These algorithms are based on solving linear systems with large (but finite) circulant and Toeplitz matrices. The algorithms are of low complexity and, perhaps most importantly, they are extremely lucid. An upper bound for the condition number of the problem of polynomial factorization is given in terms of the condition number of a certain Toeplitz matrix.

This report is concerned with the union $sp_{\Omega}^{(j,k)}T_{n}(a)$ of all possible spectra that may emerge when perturbing a large $n \times n$ Toeplitz band matrix $T_{n}(a)$ in the $(j,k)$ site by a number randomly chosen from some set $\Omega$. The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$. Also discussed are the cases of small and large sets $\Omega$ as well as the "discontinuity of the infinite volume case", which means that in general $sp_{\Omega}^{(j,k)}T_{n}(a)$ does not converge to something close to $sp_{\Omega}^{(j,k)}T_{n}(a)$ as $n \to \infty$, where $T(a)$ is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case. The second author was supported by UK Enginering and Physical Sciences Research Council Grant GR/M12414

This paper is concerned with the change of the spectra of infinite Toeplitz and Laurent matrices under perturbations in a prescribed finite set of sites. The main result says that the spectrum of a Toeplitz matrix with a non-constant rational symbol is not affected by small localized impurities, while such impurities can nevertheless enlarge the spectrum of the corresponding Laurent matrix. We also study the spectra that may emerge when randomly perturbing Toeplitz or Laurent matrices in a randomly chosen single site.

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk≥1, the approximation numberss k(An) converge to the corresponding approximation numbers k(A) asn→∞. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA n asn goes to infinity.

We prove that ifn≥2 and ϱ, χ are two given vectors inZ n, then there exists a matrix function inL ∞ n×n (T) which has a right Wiener-Hopf factorization inL 2 with the partial indices ϱ and a left Wiener-Hopf factorization inL 2 with the partial indices χ.

LetAPΣ+(Rn) denote the Banach algebra of all continuous almost periodic functions onRn whose Bohr-Fourier spectrum is contained in an additive semi-group Σ⊂[0, ∞)n. We show that the maximal ideal space ofAPΣ+(Rn) may have a nonempty corona and we characterize all Σ for which the corona is empty. Analogous results are established for algebras of almost periodic functions with absolutely convergent Fourier series.

Given a continuous function a on the complex unit circle, let T(a) denote the infinite Toeplitz matrix generated by a and let Tn(a) stand for the (n+1)×(n+1) principal section of T(a). We think of T(a) and Tn(a) as operators on l2 spaces. A classical result by Gohberg and Feldman says that if T(a) is invertible, then so is Tn(a) for all sufficiently large n≥n0 and \(\). Only in 1994 did we realize that in fact \(\). In this paper, we provide estimates for the speed with which \(\) converges to \(\). We prove that in the “generic case” the convergence speed can be estimated by the smoothness of a, whereas in some “exceptional cases” (e.g., if T(a) is Hermitian or triangular) it is not the smoothness of $a$ but the orders of certain zeros which determine the convergence speed. Some of the results are extended to operators on lp spaces.

This paper is devoted to asymptotic estimates for the (spectral or Euclidean) condition numbers κ ( T n ( a )) = ∥ T n ( a )∥ ∥ T −1 n ( a )∥ of large n × n Toeplitz matrices T n ( a ) in the case where the symbol a is an L ∞ function and Re a ≥ 0 almost everywhere. We describe several classes of symbols a for which κ ( T n ( a )) increases like (log n ) x , n x , or even e xn .

We consider Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients onL p (Γ,w) where 1

We establish Fredholm criteria and index formulas for one-dimensional zero-order pseudodifferential operators with piecewise continuous generating functions onLp spaces with Muckenhoupt weights. The Fredholm symbol of such operators is shown to be a matrix function defined on a set which, roughly speaking, is a cylinder with a certain collection of horn shaped handles. The presence of these horns implies that, unlike the case ofLp spaces without weight or with so-called power weights, the spectrum may contain heavy parts, i. e. the set of the interior points of the spectrum need not be empty. Our proof makes essential use of recent results by Finck, Roch, Silbermann, Gohberg, and Krupnik on the inverse closedness of certain Banach algebras.

We establish Fredholm criteria and index formulas for Toeplitz operators with semialmost periodic matrix-valued symbols on the Hardy spaces corresponding toL p spaces on the real line with Muckenhoupt weight. In the scalar case, these results imply descriptions of the spectrum and, in particular, invertibility criteria.

Recently, Borodin and Okounkov [2] established a remarkable identity for Toeplitz determinants. Two other proofs of this identity were subsequently found by Basor and Widom [1], who also extended the formula to the block case. We here give one more proof, also for the block case. This proof is based on a formula for the inverse of a finite block Toeplitz matrix obtained in the late seventies by Silbermann and the author.

The focus of this paper is on the following problem. Given a linear space F of complex-valued functions on a set X and a polynomial p(z), is there an algebraic composition operator on F whose characteristic polynomial equals p(z)? We show that the supply of all the polynomials p(z) for which the answer to this question is affirmative depends heavily on the structure of the space F.

Local principles associate with every element of a Banach algebra a family of local objects in terms of which the properties of the original element can be studied. In this paper some general relations between three such principles, usually affiliated with the names of Simonenko, Allan/Douglas, and Gohberg/Krupnik, are discussed. Special attention is paid to the question on how the norm of an element can be expressed in terms of the norms of the local objects associated with it. The general theory is illustrated by some concrete results on singular integral operators and Toeplitz operators.

The spectral value set spεB,CA of a bounded linear operator A on l2 is the union of the spectra of all operators of the form A+BKC, where ∥K∥