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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.

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.

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.

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.

The subject of the paper is the best approximation in $\ell^1$ of a given infinite sequence by sequences in the range of a given banded lower-triangular Toeplitz operator. Necessary and sufficient conditions for the existence of a minimizing solution are established and a numerical algorithm for finding such a solution is designed and theoretically founded. It is also shown that an optimal error sequence is only finitely nonzero. The relevancy of the problem in systems theory is outlined and numerical examples are presented.

We establish a Fredholm criterion for the operators belonging to the C ∗ -algebra generated by singular integral operators with semi-almost periodic matrix coefficients. The result is applied to Toeplitz-like operators that are perturbed by integral operators with fixed singularities at infinity, in which case it leads to an effectively verifiable Fredholm criterion together with an index formula. Our approach is based on an isomorphism theorem for C ∗ -algebras associated with C ∗ -dynamical systems and the notion of canonical generalized AP factorization for almost periodic matrix functions.

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 influence of frequency modulation on the semi-Fredholm properties of Toeplitz operators with oscillating matrix symbols. The main results give conditions on an orientation-preserving homeomorphismαof the real line that ensure the following: ifbbelongs to a certain class of oscillating matrix functions (periodic, almost periodic, or semi-almost periodic matrix functions) and the Toeplitz operator generated by the matrix functionb(x)is semi-Fredholm, then the Toeplitz operator with the matrix symbolb(α(x))is also semi-Fredholm.

It is well known that amplitude modulation does not affect Fredholmness of Toeplitz operators. The same is true for frequency modulation provided the symbol of the operator is piecewise continuous. In this article, it is shown that frequency modulation can destroy Fredholmness for Toeplitz operators with almost periodic symbols; the corresponding example is based on the observation that certain almost periodic functions become semi-almost periodic functions after appropriate frequency modulation. Moreover, this article contains several results that can be employed in order to decide whether a Toeplitz operator with a frequency modulated semi-almost periodic symbol is Fredholm.

We establish a Fredholm criterion and an index formula for Toeplitz operators with semi-almost-periodic matrix symbols on the Hardy spaces Hp (1

We answer in the negative a question by Farenick and Lee as to whether or not the spectrum is a continuous set-valued function on the linear space of Toeplitz operators. Thus, we show that if \(\{a_n\}\) is any sequence of \(L^\infty \) functions which converge uniformly to a function \(a \in L^\infty \), then the spectra of the Toeplitz operators T(an) need not converge to the spectrum of the Toeplitz operator T(a) in the Hausdorff metric. The counterexample is constructed with semi-almost periodic functions an and a.

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

The index formula of Douglas is a formula which expresses the index of a Fredholm Toeplitz operator with a discontinuous symbol as the limit of the indices of a family of Fredholm Toeplitz operators with continuous symbols. This paper is concerned with Toeplitz operators on the Bergman space \( A^p \)of the unit ball of \( \mathcal{C}^m \). The symbols are supposed to be matrix functions with entries in \( C + H^\infty \) or to be certain discontinuous matrix functions which are locally elliptic in a sense. The main result reduces the index computation for the Toeplitz operators under consideration to the case of continuous matrix symbols.

The paper is a concise survey of some rigorous results on the Fox–Li operator. This operator may be interpreted as a large truncation of a Wiener–Hopf operator with an oscillating symbol. Employing theorems from Wiener–Hopf theory one can therefore derive remarkable properties of the Fox–Li operator in a fairly comfortable way, but it turns out that Wiener–Hopf theory is unequal to the task of answering the crucial questions on the Fox–Li operator.

The convergence of several Galerkin-Petrov methods is established, including polynomial collocation and analytic element collocation methods, for Toeplitz operators on the Bergman space of the unit disk. In particular, it is shown that such methods converge if the basis and test functions (or the collocation points) own certain circular symmetry, whereas unfortunate choice of the basis and test parameters produces nonconvergent Galerkin–Petrov methods.

We prove a Fredholm criterion for Toeplitz operators with piecewise quasicontinuous symbols on weighted Hardy spaces, thus uniting part of the Gohberg-Krupnik and Sarason theories. The criterion established solved the problem of describing all the subsets M of (1, ∞) with the following property: there exists a Toeplitz operator which is Fredholm on H p if and only if p belongs to M .

The first time I encountered the name Gohberg was in 1978 when Bernd Silbermann told me about Gohberg’s invertibility criterion for infinite Toeplitz matrices. It says that such a matrix is invertible if and only if the function whose Fourier coefficients are the entries of the first row and first column does not have zeros on the unit circle and has winding number zero about the origin, provided, of course, that this function is continuous. I was deeply impressed by such a fascinating interplay of operator theory, harmonic analysis, and topology, and now, 30 years later, I have to state that this theorem by Gohberg has actually determined my entire mathematical career. In the late 1970s the three books by Israel Gohberg with Israel Feldman, Naum Krupnik, and Mark Krein occupied the most prominent place on my desk and introduced me to a world full of excitement and profound beauty.

We establish a criterion for the applicability of the finite section method to Toeplitz operators with continuous matrix-symbols on the Bergman space of the polydisk.

This paper is devoted to Banach algebras generated by Toeplitz operators with strongly oscillating symbols, that is, with symbols of the form $b(e^{i\alpha(x)})$ where $b$ belongs to some algebra of functions on the unit circle and $\alpha$ is a fixed orientation-preserving homeomorphism of the real line onto itself. We prove the existence of certain interesting homomorphisms and establish conditions for the normal solvability, Fredholmness, and invertibility of operators in these algebras.

We describe the structure of asymptotically good pseudomodes for Toeplitz matrices and their circulant analogues as well as for Wiener-Hopf integral operators and a continuous analogue of banded circulant matrices. The pseudomodes of circulant matrices and their continuous analogues are extended, while those of Toeplitz matrices or Wiener-Hopf operators are typically strongly localized in the endpoints.

The essential spectra of Toeplitz operators with piecewise continuous and almost periodic symbols admit pretty nice geometric descriptions. In contrast to this, little is known about the essential spectra of Toeplitz operators with semi-almost periodic symbols, that is, with symbols belonging to the algebra generated by the continuous functions on the real line with a jump at infinity and by the continuous almost periodic functions. A classical result by Sarason enables us to decide whether the Toeplitz operatorT(a-λ) is Fredholm for a given point a in the plane, but the problem of characterizing the set ofallλ for whichT(a -λ) is not Fredholm is nevertheless intricate. This question is studied in the present paper.

Our next objective is Wiener-Hopf operators on LP (R + w). One of the technical differences between Wiener-Hopf and Toeplitz operators is the emergence of a multiplier problem in the Wiener-Hopf case. This problem is studied in the first section of this chapter where we in particular prove the important Stechkin inequality. Another point is that Wiener-Hopf factorization is in fact a technique for Toeplitz operators that does not work for Wiener-Hopf operators on LP spaces. This obstacle can be overcome by appropriately switching between different spaces and naturally leads to the consideration of pseudodifferential operators with piecewise continuous symbols.

It is not only phenomena, it is in fact some surprises (optimistic view) and all of the evil with Wiener-Hopf operators (pessimistic view) that begins with SAP symbols.

With this chapter we reach the first big goal on our trek. If a matrix function a ∈ APWN×N has a canonical right APW factorization a = a_a+, then W(a) is obviously invertible on (R). We here prove that the converse is also true, that is, the symbol a necessarily has a canonical right APW factorization if W (a) is invertible. The proof of this result requires some heavy tools. We first establish an isomorphism theorem for C*-dynamical systems, which allows us to show that the invertibility of W(a) on (R) is equivalent to the invertibility of a certain operator on the vector-valued Besicovitch space. After that we employ the Bochner-Phillips theorem on the inverse closedness of certain matrix algebras in order to prove that the invertibility of the latter operator implies the canonical right APW factorability of a.

In this chapter we establish a Fredholm criterion and an index formula for Toeplitz operators with symbols in SAP N×N on (R w), where w is a power weight. This is a quite deep result and the proof is laborious

Semi-almost periodic (SAP) symbols constitute one of the simplest classes of symbols with discontinuities beyond jumps, and as shown in Chapter 1, SAP symbols emerge naturally in several settings. Armed with the results on C+H∞,PC, and AP symbols established in Chapter 2, we here present Sarason’s theory of semi-Fredholmness for scalar Wiener-Hopf operators with SAP symbols. A key result in the analysis of the matter is Sarason’s lemma, which says that the product of an almost periodic function in H∞ with mean value zero and a function in C(S)is a function in C+H∞. Our approach is different from Sarason’s, which can be characterized as an elegant combination of localization techniques and C* -algebra arguments. We rather follow Duduchava and Saginashvili, who gave a proof of Sarason’s theorem that can be carried over to the LP case without undue difficulty.

In this chapter we show that if, Kland Krare the left and right total indices of some matrix function \(L_{N \times N}^\infty \) in with at most a finite number d of discontinuities on \(R \cup \left\{ \infty \right\}\) then \(\left| {{k_l} + {k_r}} \right|d\left( {N - 1} \right)\). Conversely, given any integersג andǫsatisfying \(\left| {\lambda + \varrho } \right|d\left( {N - 1} \right)\) there exist matrix functions in \(L_{N \times N}^\infty \) with at most d discontinuities whose left and right total indices are just A and p. We remark that such matrix functions cannot be found in \({\left[ {C + H_\pm ^\infty } \right]_{N \times N}}\) PCN*N or APN*Nbut that one can find them in SAPWN*N

Bernd Silbermann was born on 6 April 1941 in Langhennersdorf, a village in Saxony. His parents were farmers. To this day, he is proud of his ability to drive a tractor. He went to school in Langhennersdorf from 1947 to 1955, and in the subsequent two years he apprenticed to a grocer (and really sold fish). From 1958 to 1962 he attended a school in Chemnitz, and from 1962 to 1967, in the heyday of Soviet mathematics, he was a student of mathematics at the Lomonosov University in Moscow. His lecturers included such eminent mathematicians as P.S. Alexandrov, A.G. Kurosh, and G.E. Shilov. His diploma paper was supervised by E.A. Gorin and A.Ya. Helemskii and was devoted to the structure of radicals in certain normed rings. Since 1966 he has been married to Ludmilla Pavlovna; their children Sergej and Katja were born in 1971 and 1974. In 1967, Silbermann moved back to Chemnitz and started working under the supervision of Siegfried Prossdorf.

The Cauchy singular integral operator, S, is one of the main actors in the theory of Fourier convolutions, Toeplitz operators, Riemann-Hilbert problems, Wiener-Hopf and singular integral equations. While the boundedness of S has been studied for many decades, final results on the spectrum of S were obtained only in recent times. During the last few years, it was discovered that there is a surprising and undreamt-of metamorphosis of the (local) spectra of S from circular arcs via horns and logarithmic double-spirals to so-called logarithmic leaves with a halo. This article is a survey of this fascinating development. We hope it is both a feast for the eyes and an illustration of Nick Trefethen’s motto: The spectrum gives an operator a personality!

We now study the behavior of the determinants $${D_n}(a): = \det {T_n}(a): = \det \left( {\begin{array}{*{20}{c}} {{a_0}}&{{a_{ - 1}}}& \cdots &{{a_{ - (n - 1)}}} \\ {{a_1}}&{{a_0}}& \cdots &{{a_{ - (n - 2)}}} \\ \vdots & \vdots & \ddots & \vdots \\ {{a_{n - 1}}}&{{a_{n - 2}}}& \cdots &{{a_0}} \end{array}} \right)$$ as n goes to infinity. The strong Szego limit theorem says that, after appropriate normalization, the determinants Dn(a) approach a nonzero limit provided a is sufficiently smooth and T(a) is invertible. Before stating and proving this theorem, we need a few more auxiliary facts.

Let H be a Hilbert space and let A be a bounded linear operator on H. Then sp A*A ⊂[0,∞), and the non-negative square roots of the numbers in sp A*A are called the singular values of A. The set of the singular values of A will be denoted byΣ(A), $$ \sum {\left( A \right)} : = \left\{ {s \in \left[ {\left. {0,\infty } \right):{s^2} \in sp{A^*}A} \right.} \right\}. $$ (4.1)

Let 1

Chapter 1 dealt with curves only, and in Chapter 2 we focussed our attention on weights, although weights cannot exist without a curve they live on. We now turn to some problems arising when considering a curve and a weight on this curve as a whole.

This chapter is the heart of the book. By first employing a localization theorem and subsequently constructing a Wiener-Hopf factorization for the symbols of the local representatives, we will completely identify the essential spectra of Toeplitz operators with piecewise continuous symbols. We know from the preceding chapter that the essential spectrum of a classical Toeplitz operator is the union of the essential range of the symbol and of line segments joining the endpoints of each jump. We will show that in the general case these line segments metamorphose into circular arcs, logarithmic double spirals, horns, spiralic horns, and eventually into what we call leaves. The shape of the leaves can be described in terms of the indicator functions.

It is well known that for Banach algebras generated by two idempotents and the identity all irreducible representations are of order not greater than two. These representations have been described completely and have found important applications to symbol theory. It is also well known that without additional restrictions on the idempotents these results do not admit a natural generalization to algebras generated by more than two idempotents and the identity. In this paper we describe all irreducible representations of Banach algebras generated by N idempotents which satisfy some additional relations. These representations are of order not greater than N and allow us to construct a symbol theory with applications to singular integral operators.

This paper is a systematic introduction (with a number of new results) to some aspects of the theory of Toeplitz operators with oscillating symbols on the Hardy space H 2 of the unit circle. What we are interested in is obtaining answers to the question which geometric and/or algebraic properties of the argument of the symbol imply that the operator is normally solvable, semi-Fredholm, Fredholm, or even invertible. Our discussion includes well known results on symbols in C + H ∞ , SAP, or PQC and also less known and new insights into operators whose streched symbols have arguments behaving like x λ, exp(x λ), (log x)λ, or sin(x λ) with λ > 0 at infinity. We also present some new results on the finite section method for Toeplitz operators.

This paper is concerned with the spectra of Toeplitz operators with piecewise continuous symbols and with the symbol calculus for singular integral operators with piecewise continuous coefficients on L P (Γ) where 1 < p < ∞ and Γ is a Carleson Jordan curve. It is well known that piecewise smooth curves lead to the appearance of circular arcs in the essential spectra of Toeplitz operators, and only recently the authors discovered that certain Carleson curves metamorphose these circular arcs into logarithmic double-spirals. In the present paper we dispose of the matter by determining the local spectra produced by a general Carleson curve. These spectra are of a qualitatively new type and may, in particular, be heavy sets — until now such a phenomenon has only be observed for spaces with general Muckenhoupt weights.

We extend the strong Szego-Widom limit theorem on the asymptotic behavior of large Toeplitz determinants generated by matrix-valued functions to the case of operator-valued generating functions. The approach of this paper relies heavily on the Fredholm theory of Toeplitz and Hankel operators with operator-valued symbols and is based upon new results on the finite section method for Toeplitz operators induced by operator-valued functions.

Throughout this chapter we let L P and L + p (1≤P≤∞) refer to the L P spaces of Lebesgue measure on ℝ and ℝ+, respectively. The L p spaces on the unit circle will be denoted by L P (T). The operator P defined by P:L p →L p +, φ ↦ φ | ℝ+is clearly bounded for 1 ≤ p ≤ ∞.

We have already settled the Fredholm theory of the operators in alg (Corollaries 4.7 and 4.8) and stated a localization result for Toeplitz operators on l N p (Theorem 2.95). This chapter is devoted to some more delicate questions of the l P . theory of Toeplitz operators.

We begin by applying Theorem 3.42 and Corollary 3.44 to the Fredholm theory of Toeplitz operators. Although this chapter is concerned with Toeplitz operators on H2 ≅ 12, we first state some results for Toeplitz operators on H p and 1 P , because there do not arise any substantial difficulties when passing from the case p = 2 to the case p ≠ 2.

Some questions for Toeplitz operators on H p have already been answered in the preceding chapters. In particular, we settled the Fredholm theory for the operators in algℒ (H N p ) T(C N × N +H N × N ∞ ). Also notice the localization result stated in Theorem 2.96. However, many questions still remain open and it is only a small number of them which will be answered in this chapter.

This paper is concerned with Toeplitz matrices generated by symbols of the form a(t) = Φ r=1 R | t−t r | 2α r (− t ) t r β r b(t) (| t |=1) where t 1 ,…, t R are pairwise distinct points on the unit circle, b is sufficiently smooth, b(t) ≠ 0 (¦t¦ = 1) , and ind b = 0, (− t ) t r β r is defined as exp {iβ r arg ( − t t r )} with ¦arg ( − t t r )¦ , and α r , β r are complex numbers satisfying ¦ Re α r ¦ 1 2 , ¦ Re β r ¦ 1 2 . Weighted Hardy spaces L + 2 ( ϱ 1 ) and L + 2 ( ϱ 2 ) are defined such that the Toeplitz operator T ( a ): L + 2 ( ϱ 1 ) → L + 2 ( ϱ 2 ) is bounded and invertible and that the finite section method is applicable to T ( a ) considered as acting from L + 2 ( ϱ 1 ) onto L + 2 ( ϱ 2 ). These results are then applied to prove a conjecture of Fisher and Hartwig ( Adv. in Chem. Phys. 15 (1968), 333–353) which asserts that the determinants of the n × n sections of T ( a ) are asymptotically equal to G(b) n n q E (a) (n → ∞) , where q = ∑ ( α r 2 − β r 2 ) and G ( b ), E (a) are certain nonzero constants. Under quite general conditions on the smoothness of b, these constants are completely identified.

The subject of this paper is polynomial collocation for approximately solving Toeplitz integral equations on the Bergman space of the complex unit disk. It turns out that the convergence of such methods depends on the choice of the system of collocation points. We establish a result for the case where the collocation points are evenly distributed on massive subsets of the unit disk.