# Spectral approximation for Segal-Bargmann space Toeplitz operators

Albrecht Böttcher; Hartmut Wolf

Banach Center Publications (1997)

- Volume: 38, Issue: 1, page 25-48
- ISSN: 0137-6934

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topBöttcher, Albrecht, and Wolf, Hartmut. "Spectral approximation for Segal-Bargmann space Toeplitz operators." Banach Center Publications 38.1 (1997): 25-48. <http://eudml.org/doc/208634>.

@article{Böttcher1997,

abstract = {Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk $^\{N\}$ 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(An) 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) = limε → 0 limn → ∞ Λε(An), 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 tackling 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.},

author = {Böttcher, Albrecht, Wolf, Hartmut},

journal = {Banach Center Publications},

keywords = {Toeplitz operator; continuous symbol; Bergman space; Segal-Bargmann space; compression; -pseudospectrum; -algebra techniques; spectral approximation; Fredholm criteria; norms of the inverses; large truncations; approximation methods},

language = {eng},

number = {1},

pages = {25-48},

title = {Spectral approximation for Segal-Bargmann space Toeplitz operators},

url = {http://eudml.org/doc/208634},

volume = {38},

year = {1997},

}

TY - JOUR

AU - Böttcher, Albrecht

AU - Wolf, Hartmut

TI - Spectral approximation for Segal-Bargmann space Toeplitz operators

JO - Banach Center Publications

PY - 1997

VL - 38

IS - 1

SP - 25

EP - 48

AB - Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk $^{N}$ 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(An) 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) = limε → 0 limn → ∞ Λε(An), 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 tackling 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.

LA - eng

KW - Toeplitz operator; continuous symbol; Bergman space; Segal-Bargmann space; compression; -pseudospectrum; -algebra techniques; spectral approximation; Fredholm criteria; norms of the inverses; large truncations; approximation methods

UR - http://eudml.org/doc/208634

ER -

## References

top- [1] G. R. Allan, Ideals of vector-valued functions, Proc. London Math. Soc. (3) 18 (1968), 193-216. Zbl0194.44501
- [2] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Comm. Pure Appl. Math. 14 (1961), 187-214. Zbl0107.09102
- [3] C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, J. Funct. Anal. 68 (1986), 273-299. Zbl0626.47031
- [4] C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), 813-829. Zbl0625.47019
- [5] A. Böttcher, Truncated Toeplitz operators on the polydisk, Monatsh. Math. 110 (1990), 23-32.
- [6] A. Böttcher, Pseudospectra and singular values of large convolution operators, J. Integral Equations Appl. 6 (1994), 267-301. Zbl0819.45002
- [7] A. Böttcher and B. Silbermann, The finite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols, Math. Nachr. 110 (1983), 279-291. Zbl0549.47010
- [8] A. Böttcher, B. Silbermann, Analysis of Toeplitz operators, Springer, Berlin, 1990. Zbl0732.47029
- [9] A. Böttcher and H. Wolf, Finite sections of Segal-Bargmann space Toeplitz operators with polyradially continuous symbols, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 365-372. Zbl0751.47010
- [10] A. Böttcher, H. Wolf, Large sections of Bergman space Toeplitz operators with piecewise continuous symbols, Math. Nachr. 156 (1992), 129-155. Zbl0779.47022
- [11] A. Böttcher and H. Wolf, Galerkin-Petrov methods for Bergman space Toeplitz operators, SIAM J. Numer. Anal. 30 (1993), 846-863. Zbl0779.65036
- [12] A. Böttcher, H. Wolf, Asymptotic invertibility of Bergman and Bargmann space Toeplitz operators, Asymptotic Anal. 8 (1994), 15-33. Zbl0816.47023
- [13] L. A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973), 433-439. Zbl0271.46052
- [14] R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972. Zbl0247.47001
- [15] R. V. Duduchava, Discrete convolution operators on the quarter-plane and their indices, Math. USSR-Izv. 11 (1977), 1072-1084. Zbl0426.47031
- [16] G. B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, N.J., 1989. Zbl0682.43001
- [17] I. Gohberg and I. Feldman, Convolution Equations and Projection Methods for Their Solution, Amer. Math. Soc., Providence, R.I., 1974.
- [18] I. Gohberg and N. Krupnik, On the algebra generated by Toeplitz matrices, Functional Anal. Appl. 3 (1969), 119-127. Zbl0199.19201
- [19] A. V. Kozak, On the reduction method for multidimensional discrete convolutions, Mat. Issled. 8 (1973), 157-160 (in Russian).
- [20] G. McDonald, Toeplitz operators on the ball with piecewise continuous symbol, Illinois J. Math. 23 (1979), 286-294. Zbl0438.47031
- [21] L. Reichel and L. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Linear Algebra Appl. 162 (1992), 153-185. Zbl0748.15010
- [22] P. Schmidt and F. Spitzer, The Toeplitz matrices of an arbitrary Laurent polynomial, Math. Scand. 8 (1960), 15-38. Zbl0101.09203
- [23] I. Segal, Lectures at the Summer Seminar on Applied Mathematics, Boulder, Col., 1960.
- [24] B. Silbermann, Lokale Theorie des Reduktionsverfahrens für Toeplitzoperatoren, Math. Nachr. 104 (1981), 137-146.
- [25] H. Widom, Singular integral equations in ${L}^{p}$, Trans. Amer. Math. Soc. 97 (1960), 131-160. Zbl0109.33002
- [26] H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants II, Adv. Math. 21 (1976), 1-29. Zbl0344.47016
- [27] H. Widom, On the singular values of Toeplitz matrices, Z. Anal. Anwendungen 8 (1989), 221-229. Zbl0692.47028

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