American Mathematical Society

Book Review

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MathSciNet review: 1568032
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Book Information:

Author: A. B\"ottcher and B. Silbermann
Title: Analysis of Toeplitz operators
Additional book information: Springer-Verlag, New York, 1990, 512 pp., US$79.00. ISBN 3-540-52147-X.

M. B. Abrahamse, Subnormal Toeplitz operators and functions of bounded type, Duke Math. J. 43 (1976), no. 3, 597–604. MR 428097

M. B. Abrahamse, The spectrum of a Toeplitz operator with a multiplicatively periodic symbol, J. Functional Analysis 31 (1979), no. 2, 224–233. MR 525953, DOI 10.1016/0022-1236(79)90063-6

Sheldon Axler, Sun-Yung A. Chang, and Donald Sarason, Products of Toeplitz operators, Integral Equations Operator Theory 1 (1978), no. 3, 285–309. MR 511973, DOI 10.1007/BF01682841

C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, J. Funct. Anal. 68 (1986), no. 3, 273–299. MR 859136, DOI 10.1016/0022-1236(86)90099-6

Albrecht Böttcher and Bernd Silbermann, Invertibility and asymptotics of Toeplitz matrices, Mathematical Research, vol. 17, Akademie-Verlag, Berlin, 1983. MR 734173

Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9

Douglas N. Clark, On Toeplitz operators with loops. II, J. Operator Theory 7 (1982), no. 1, 109–123. MR 650196

Douglas N. Clark and Judith H. Morrel, On Toeplitz operators and similarity, Amer. J. Math. 100 (1978), no. 5, 973–986. MR 517140, DOI 10.2307/2373957

L. A. Coburn, The $C^{\ast }$-algebra generated by an isometry. II, Trans. Amer. Math. Soc. 137 (1969), 211–217. MR 236720, DOI 10.1090/S0002-9947-1969-0236720-9

L. A. Coburn and R. G. Douglas, Translation operators on the half-line, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1010–1013. MR 275228, DOI 10.1073/pnas.62.4.1010

John B. Conway, The theory of subnormal operators, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR 1112128, DOI 10.1090/surv/036

Carl C. Cowen and John J. Long, Some subnormal Toeplitz operators, J. Reine Angew. Math. 351 (1984), 216–220. MR 749683

Allen Devinatz, Toeplitz operators on $H^{2}$ spaces, Trans. Amer. Math. Soc. 112 (1964), 304–317. MR 163174, DOI 10.1090/S0002-9947-1964-0163174-9

Ronald G. Douglas, Another look at real-valued index theory, Surveys of some recent results in operator theory, Vol. II, Pitman Res. Notes Math. Ser., vol. 192, Longman Sci. Tech., Harlow, 1988, pp. 91–120. MR 976844

R. G. Douglas, Local Toeplitz operators, Proc. London Math. Soc. (3) 36 (1978), no. 2, 243–272. MR 482348, DOI 10.1112/plms/s3-36.2.243

Ronald G. Douglas, Banach algebra techniques in operator theory, Pure and Applied Mathematics, Vol. 49, Academic Press, New York-London, 1972. MR 0361893

I. C. Gohberg, Toeplitz matrices composed of the Fourier coefficients of piecewise continuous functions, Funkcional. Anal. i Priložen. 1 (1967), no. 2, 91–92 (Russian). MR 0213909

I. C. Gohberg and M. G. Kreĭn, Systems of integral equations on the half-line with kernels depending on the difference of the arguments, Uspehi Mat. Nauk (N.S.) 13 (1958), no. 2 (80), 3–72 (Russian). MR 0102720

P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933. MR 270173, DOI 10.1090/S0002-9904-1970-12502-2

Philip Hartman and Aurel Wintner, On the spectra of Toeplitz’s matrices, Amer. J. Math. 72 (1950), 359–366. MR 36936, DOI 10.2307/2372039

Philip Hartman and Aurel Wintner, The spectra of Toeplitz’s matrices, Amer. J. Math. 76 (1954), 867–882. MR 73859, DOI 10.2307/2372661

R. S. Ismagilov, On the spectrum of Toeplitz matrices, Dokl. Akad. Nauk SSSR 149 (1963), 769–772 (Russian). MR 0146670

Jerome Kaminker, Operator algebraic invariants for elliptic operators, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 307–314. MR 1077392, DOI 10.1090/conm/105

N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223, DOI 10.1007/978-3-642-70151-1

Vladimir V. Peller, Hankel operators and multivariate stationary processes, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 357–371. MR 1077396

S. C. Power, Essential spectra of piecewise continuous Fourier integral operators, Proc. Roy. Irish Acad. Sect. A 81 (1981), no. 1, 1–7. MR 635571

S. C. Power, Hankel operators on Hilbert space, Research Notes in Mathematics, vol. 64, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 666699

Marvin Rosenblum, A concrete spectral theory for self-adjoint Toeplitz operators, Amer. J. Math. 87 (1965), 709–718. MR 181901, DOI 10.2307/2373070

Marvin Rosenblum, The absolute continuity of Toeplitz’s matrices, Pacific J. Math. 10 (1960), 987–996. MR 114125

Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. Oxford Science Publications. MR 822228

Donald Sarason, Toeplitz operators with piecewise quasicontinuous symbols, Indiana Univ. Math. J. 26 (1977), no. 5, 817–838. MR 463968, DOI 10.1512/iumj.1977.26.26066

Bernd Silbermann, The $C^*$-algebra generated by Toeplitz and Hankel operators with piecewise quasicontinuous symbols, Integral Equations Operator Theory 10 (1987), no. 5, 730–738. Toeplitz lectures 1987 (Tel-Aviv, 1987). MR 904487, DOI 10.1007/BF01195799

I. B. Simonenko, Riemann’s boundary problem with a measurable coefficient, Soviet Math. Dokl. 1 (1960), 1295–1298. MR 0140698

Shun Hua Sun, Bergman shift is not unitarily equivalent to a Toeplitz operator, Kexue Tongbao (English Ed.) 28 (1983), no. 8, 1027–1030. MR 763833

Harald Upmeier, Toeplitz operators and index theory in several complex variables, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988) Proc. Sympos. Pure Math., vol. 51, Amer. Math. Soc., Providence, RI, 1990, pp. 585–598. MR 1077410, DOI 10.1017/s0022112089002193

A. L. Vol′berg, Two remarks concerning the theorem of S. Axler, S.-Y. A. Chang and D. Sarason, J. Operator Theory 7 (1982), no. 2, 209–218. MR 658609

Harold Widom, Inversion of Toeplitz matrices. II, Illinois J. Math. 4 (1960), 88–99. MR 130572

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-, Inversion of Toeplitz matrices. III, Notices Amer. Math. Soc. 7 (1960), 63.

Harold Widom, On the spectrum of a Toeplitz operator, Pacific J. Math. 14 (1964), 365–375. MR 163173

Ke He Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York, 1990. MR 1074007

Review Information:

Reviewer: Thomas Kriete

Journal: Bull. Amer. Math. Soc. 28 (1993), 387-396
DOI: https://doi.org/10.1090/S0273-0979-1993-00370-7

Bulletin of the American Mathematical Society

Bibliography