Albrecht Böttcher

Technische Universität Chemnitz, Chemnitz, Saxony, Germany

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Publications (12)6.78 Total impact

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    Albrecht Böttcher, Hartmut Wolf
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    ABSTRACT: 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.
    Journal of Computational and Applied Mathematics 01/1996; 66(s 1–2):89–96. DOI:10.1016/0377-0427(95)00172-7 · 1.08 Impact Factor
  • Albrecht Böttcher, Harald Heidler
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    ABSTRACT: Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X X a composition operatorC a onC(X) by the rule(C a f)(x) = f(a(x)). We describe all self-mapsa for whichC a is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp a (z) and the essentially characteristic polynomialq a (z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp a (z)/q a (z).
    Aequationes Mathematicae 01/1995; 49(1):276-294. DOI:10.1007/BF01827945 · 0.55 Impact Factor
  • Albrecht Böttcher, Hartmut Wolf
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    ABSTRACT: We establish a criterion for the asymptotic invertibility of Toeplitz operators with continuous symbols on weighted Bergman spaces of the polydisk and their formal limit, the multidimensional Bargmann space.
    Asymptotic Analysis 01/1994; 8(1). · 0.42 Impact Factor
  • Albrecht Böttcher, Hartmut Wolf
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    ABSTRACT: Authors’ summary: 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.
    SIAM Journal on Numerical Analysis 06/1993; 30(3). DOI:10.1137/0730043 · 1.69 Impact Factor
  • Albrecht Böttcher, Harald Heidler
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    ABSTRACT: 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.
    Integral Equations and Operator Theory 01/1992; 15(3):389-411. DOI:10.1007/BF01200326 · 0.58 Impact Factor
  • Albrecht Böttcher, Hartmut Wolf
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    ABSTRACT: The authors consider a question of asymptotic invertibility for Bergman space Toeplitz operators with piecewise continuous symbol. They give a criterion for asymptotic invertibility of such operators. This problem is connected with applicability of the finite section method to Toeplitz operators. The case of continuous symbols was earlier considered in the previous works [A. Böttcher, Monatsh. Math. 110, No. 1, 23-32 (1990; Zbl 0727.47012); A. Böttcher, H. Wolf, Bull. Am. Math. Soc., New Ser. 25, No. 2, 365-372 (1991; Zbl 0751.47010)].
    Mathematische Nachrichten 01/1992; 156(1):129 - 155. DOI:10.1002/mana.19921560110 · 0.66 Impact Factor
  • Albrecht Böttcher, Hartmut Wolf
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    ABSTRACT: We show that circles centered at the origin play an exceptional role in connection with convergence of interpolation projections in the Bergman space of the unit disk, the mean value property of harmonic functions, and an inverse problem for the single-layer logarithmic potential.
    01/1992;
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    Albrecht Böttcher, Hartmut Wolf
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    ABSTRACT: An invertible Toeplitz operator is asymptotically invertible if for each n, its compression to the space of polynomials of degree not exceeding n is invertible, and the inverses of these compressions converge strongly to the original operator. Let μ be a rotationally invariant measure on C and suppose a is a continuous function on C whose radial limits as r→∞ define a continuous function on the unit circle. Under appropriate restrictions on μ, the author shows that if the Toeplitz operator with the symbol a is invertible, then it is automatically asymptotically invertible. A multivariate version of the situation is also considered. For appropriate μ and a based on C N , asymptotic invertibility of the Toeplitz operator with symbol a turns out to be equivalent to the invertibility of 2 N “mixed” Toeplitz operators.
    Bulletin of the American Mathematical Society 04/1991; 25(1991). DOI:10.1090/S0273-0979-1991-16078-7 · 1.17 Impact Factor
  • Source
    Albrecht Böttcher
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    ABSTRACT: 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.
    Monatshefte für Mathematik 01/1990; 110(1):23-32. DOI:10.1007/BF01571274 · 0.64 Impact Factor
  • Albrecht Böttcher, Harald Heidler
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    ABSTRACT: Let X be a compact Hausdorff space and Aut X the group of all homeomorphisms of X onto itself. Given two involutions α,β∈AutX (α 2 =e=β 2 ), let grp(α,β) be the smallest subgroup containing α and β. The aim of the paper is to find conditions for the unique solvability of ∑ g∈G a g ϕ(g(x))=f(x)(x∈X), where f∈C(X) is given, ϕ∈C(X) is the unknown function and G is a subset of grp(α,β). The case where grp(α,β) is finite has been dealt with by A. B. Antonevich [Linear functional equations: The operator theory approach (Russian), Univ. Press, Minsk (1988)]. Here the case is considered where grp(α,β) is infinite.
  • Albrecht Böttcher, Harald Heidler
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    ABSTRACT: The focus of the talk 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)?
  • Albrecht Böttcher, Hartmut Wolf
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    ABSTRACT: Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk 𝔻 N or on the Segal-Bargmann space over ℂ N . Even in the case N=1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large “finite section”, that is, by the compression A n of A to the linear span of the monomials {z 1 k 1 ⋯z N k N :0≤k j ≤n}. Unfortunately, in general the spectrum of A n does not mimic the spectrum of A as n goes to infinity. However, in the same way as in numerical analysis the question “Is A invertible?” is replaced by the question “What is |A -1 |?”, it turns out that the mysteries of Λ(A n ) for large n may be much better understood by considering the pseudospectrum of A n rather than the usual spectrum. For ε>0, the ε-pseudospectrum of an operator T is defined as the set Λ ε (T)={λ∈ℂ:|(T-λI) -1 |≥1/ε}. Our central result says that the limit lim n→∞ |A n -1 | exists and is equal to the maximum of |A -1 | and the norms of the inverses of 2 N -1 other operators associated with A. This result implies that for each ε>0 the ε-pseudospectrum of A n approaches the union of the ε-pseudospectra of A and the 2 N -1 operators associated with A. If in particular N=1, it follows that Λ(A)=lim ε→0 lim n→∞ Λ ε (A n ), whereas, as already said, the equality Λ(A)=lim n→∞ lim ε→0 Λ ε (A n ) (=lim n→∞ Λ(A n )) is in general not true. The paper does not aim at completeness, its purpose is rather to outline the ideas behind the theory, and especially, to illustrate the power of C * -algebra techniques for trackling the problem of spectral approximation. We therefore focus our attention on Segal-Bargmann space Toeplitz operators. Our main theorems include Fredholm criteria for such operators, results on the norms of the inverses of their large truncations, as well as the foundation of several approximation methods for solving equations with a Segal-Bargmann space Toeplitz operator.

Publication Stats

23 Citations
6.78 Total Impact Points

Institutions

  • 1990–1996
    • Technische Universität Chemnitz
      • Department of Mathematics
      Chemnitz, Saxony, Germany