Böttcher, Albrecht; Wolf, Hartmut Finite sections of Segal-Bargmann space Toeplitz operators with polyradially continuous symbols. (English) Zbl 0751.47010 Bull. Am. Math. Soc., New Ser. 25, No. 2, 365-372 (1991). 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 \(\mu\) be a rotationally invariant measure on \(C\) and suppose \(a\) is a continuous function on \(C\) whose radial limits as \(r\to\infty\) define a continuous function on the unit circle. Under appropriate restrictions on \(\mu\), 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 \(\mu\) 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. Reviewer: E.Azoff (Athens / Georgia) Cited in 1 Review MSC: 47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A53 (Semi-) Fredholm operators; index theories 45L05 Theoretical approximation of solutions to integral equations Keywords:sections of Segal-Bargmann space; polyradially continuous symbols; invertible Toeplitz operator; asymptotically invertible; compression; rotationally invariant measure PDFBibTeX XMLCite \textit{A. Böttcher} and \textit{H. Wolf}, Bull. Am. Math. Soc., New Ser. 25, No. 2, 365--372 (1991; Zbl 0751.47010) Full Text: DOI References: [1] Sheldon Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1 – 50. · Zbl 0681.47006 [2] C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, J. Funct. Anal. 68 (1986), no. 3, 273 – 299. · Zbl 0626.47031 [3] C. A. Berger and L. A. Coburn, Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), no. 2, 813 – 829. · Zbl 0625.47019 [4] Albrecht Böttcher, Truncated Toeplitz operators on the polydisk, Monatsh. Math. 110 (1990), no. 1, 23 – 32. · Zbl 0727.47012 [5] Albrecht Böttcher and Bernd Silbermann, The finite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols, Math. Nachr. 110 (1983), 279 – 291. · Zbl 0549.47010 [6] Albrecht Böttcher and Bernd Silbermann, Analysis of Toeplitz operators, Springer-Verlag, Berlin, 1990. · Zbl 0732.47029 [7] R. G. Douglas and Roger Howe, On the \?*-algebra of Toeplitz operators on the quarterplane, Trans. Amer. Math. Soc. 158 (1971), 203 – 217. · Zbl 0224.47015 [8] I. C. Gohberg and I. A. Fel\(^{\prime}\)dman, Convolution equations and projection methods for their solution, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by F. M. Goldware; Translations of Mathematical Monographs, Vol. 41. [9] A. V. Kozak, The reduction method for multidimensional discrete convolutions, Mat. Issled. 8 (1973), no. 3(29), 157 – 160, 184 (Russian). [10] V. S. Pilidi, Multidimensional bisingular operators, Dokl. Akad. Nauk SSSR 201 (1971), 787 – 789 (Russian). · Zbl 0239.47024 [11] I. B. Simonenko, Multidimensional discrete convolutions, Mat. Issled. 3 (1968), no. vyp. 1 (7), 108 – 122 (Russian). [12] Harold Widom, Asymptotic behavior of block Toeplitz matrices and determinants, Advances in Math. 13 (1974), 284 – 322. , https://doi.org/10.1016/0001-8708(74)90072-3 Harold Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II, Advances in Math. 21 (1976), no. 1, 1 – 29. · Zbl 0344.47016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.