Book Review
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MathSciNet review:
1567444
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Book Information:
Author:
S. C. Power
Title:
Hankel operators on Hilbert space
Additional book information:
Research Notes in Mathematics, No. 64, Pitman Advanced Publishing Program, Boston-London-Melbourne, 1982, 87 pp., $13.95. ISBN 0-273-08518-2.
V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Infinite Hankel matrices and generalized problems of Carathéodory-Fejér and F. Riesz, Funkcional. Anal. i Priložen. 2 (1968), no. 1, 1–19 (Russian). MR 0234274
V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. (N.S.) 86(128) (1971), 34–75 (Russian). MR 0298453
Sheldon Axler, I. David Berg, Nicholas Jewell, and Allen Shields, Approximation by compact operators and the space $H^{\infty }+C$, Ann. of Math. (2) 109 (1979), no. 3, 601–612. MR 534765, DOI 10.2307/1971228
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
5. C. Carathéodory and L. Fejér, Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz, Rend. Circ. Math. Palermo 32 (1911), 218-239.
Douglas N. Clark, On the spectra of bounded, Hermitian, Hankel matrices, Amer. J. Math. 90 (1968), 627–656. MR 231112, DOI 10.2307/2373546
Douglas N. Clark, On interpolating sequences and the theory of Hankel and Toeplitz matrices, J. Functional Analysis 5 (1970), 247–258. MR 0254628, DOI 10.1016/0022-1236(70)90029-7
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
R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $L^{p}$, Representation theorems for Hardy spaces, Astérisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 11–66. MR 604369
I. C. Gohberg and N. Ja. Krupnik, The algebra generated by the Toeplitz matrices, Funkcional. Anal. i Priložen. 3 (1969), no. 2, 46–56 (Russian). MR 0250082
Philip Hartman, On completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9 (1958), 862–866. MR 108684, DOI 10.1090/S0002-9939-1958-0108684-8
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
James S. Howland, Trace class Hankel operators, Quart. J. Math. Oxford Ser. (2) 22 (1971), 147–159. MR 288630, DOI 10.1093/qmath/22.1.147
M. G. Kreĭn, Integral equations on the half-line with a kernel depending on the difference of the arguments, Uspehi Mat. Nauk 13 (1958), no. 5 (83), 3–120 (Russian). MR 0102721
Wilhelm Magnus, On the spectrum of Hilbert’s matrix, Amer. J. Math. 72 (1950), 699–704. MR 41358, DOI 10.2307/2372284
Zeev Nehari, On bounded bilinear forms, Ann. of Math. (2) 65 (1957), 153–162. MR 82945, DOI 10.2307/1969670
V. V. Peller, Hankel operators of class ${\mathfrak {S}}_{p}$ and their applications (rational approximation, Gaussian processes, the problem of majorization of operators), Mat. Sb. (N.S.) 113(155) (1980), no. 4(12), 538–581, 637 (Russian). MR 602274
Stephen Power, The essential spectrum of a Hankel operator with piecewise continuous symbol, Michigan Math. J. 25 (1978), no. 1, 117–121. MR 487567
S. C. Power, $C^{\ast }$-algebras generated by Hankel operators and Toeplitz operators, J. Functional Analysis 31 (1979), no. 1, 52–68. MR 523113, DOI 10.1016/0022-1236(79)90097-1
S. C. Power, Hankel operators with PQC symbols and singular integral operators, Proc. London Math. Soc. (3) 41 (1980), no. 1, 45–65. MR 579716, DOI 10.1112/plms/s3-41.1.45
C. R. Putnam, Commutators and absolutely continuous operators, Trans. Amer. Math. Soc. 87 (1958), 513–525. MR 100226, DOI 10.1090/S0002-9947-1958-0100226-0
Marvin Rosenblum, On the Hilbert matrix. I, Proc. Amer. Math. Soc. 9 (1958), 137–140. MR 94626, DOI 10.1090/S0002-9939-1958-0094626-0
Marvin Rosenblum, The absolute continuity of Toeplitz’s matrices, Pacific J. Math. 10 (1960), 987–996. MR 114125
24. T. Takagi, On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau, Japanese J. Math. 1 (1924), 83-93.
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
- 1.
- V. M. Adamjan, D. Z. Arov and M. G. Krein, Infinite Hankel matrices and generalized Carathéodory-Fejér and Riesz problems, Funkcional Anal. i Prilozen 2 (1968), 1-19 = Functional Anal. Appl. 2 (1968), 1-18. MR 0234274
- 2.
- V. M. Adamjan, D. Z. Arov and M. G. Krein, Analytic properties of Schmidt pairs for a Hankel operator and the generalized Shur-Takagi problem, Mat. Sb. 86 (128) (1971), 34-75 = Math. USSR Sbornik 15 (1971), 31-73. MR 298453
- 3.
- S. Axler, I. D. Berg, N. Jewell and A. Shields, Approximation by compact operators and the space H + C, Ann. of Math. (2) 109 (1979), 601-612. MR 534765
- 4.
- S. Axler, S.-Y. A. Chang and D. Sarason, Products of Toeplitz operators, J. Integral Equations Operator Theory 1 (1978), 285-309. MR 511973
- 5.
- C. Carathéodory and L. Fejér, Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz, Rend. Circ. Math. Palermo 32 (1911), 218-239.
- 6.
- D. N. Clark, On the spectra of bounded, Hermitian Hankel matrices, Amer. J. Math. 90 (1968), 627-656. MR 231112
- 7.
- D. N. Clark, On interpolating sequences and the theory of Hankel and Toeplitz operators, J. Functional Analysis 5 (1970), 247-258. MR 254628
- 8.
- L. A. Coburn, The C*-algebra generated by an isometry. II, Trans. Amer. Math. Soc. 137 (1969), 211-217. MR 236720
- 9.
- R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in L, Astérisque 77 (1980), 11-66. MR 604369
- 10.
- I. C. Goldberg and N. Ja. Krupnik, The algebra generated by Toeplitz matrices, Funkcional Anal. i Prilozen 3 (1969), 45-56. MR 250082
- 11.
- P. Hartman, On completely continuous Hankel matrices, Proc. Amer. Math. Soc. 9 (1958), 362-366. MR 108684
- 12.
- P. Hartman and A. Wintner, On the spectra of Toeplitz's matrices, Amer. J. Math. 72 (1950), 359-366. MR 36936
- 13.
- J. S. Howland, Trace class Hankel operators, Quart. J. Math. Oxford Ser. (2) 22 (1971), 147-159. MR 288630
- 14.
- M. G. Krein, Integral equations on a half line with difference kernel, Uspehi Mat. Nauk (N.S.) 13 (1958), No. 5 (83), 3-120. MR 102721
- 15.
- W. Magnus, On the spectrum of Hilbert's matrix, Amer. J. Math. 72 (1950), 699-704. MR 41358
- 16.
- Z. Nehari, On bounded bilinear forms, Ann. of Math. (2) 65 (1957), 153-162. MR 82945
- 17.
- V. V. Peller, Hankel operators of the class γ, Mat. Sb. (N.S.) 113 (155) (1980), 538-581. MR 602274
- 18.
- S. C. Power, The essential spectrum of a Hankel operator with piecewise continuous symbol, Michigan Math. J. 25 (1978), 117-121. MR 487567
- 19.
- S. C. Power, C*-algebra generated by Hankel operators and Toeplitz operators, J. Functional Analysis 31 (1979), 52-68. MR 523113
- 20.
- S. C. Power, Hankel operators with PQC symbol and singular integral operators, Proc. London Math. Soc. 40 (1980), 45-65. MR 579716
- 21.
- C. R. Putnam, Commutators and absolutely continuous operators, Trans. Amer. Math. Soc. 87 (1958), 513-525. MR 100226
- 22.
- M. Rosenblum, On the Hilbert matrix. I, II, Proc. Amer. Math. Soc. 9 (1958), 137-140; 9 (1958), 581-585. MR 94626
- 23.
- M. Rosenblum, The absolute continuity of Toeplitz's matrices, Pacific J. Math. 10 (1960), 987-996. MR 114125
- 24.
- T. Takagi, On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau, Japanese J. Math. 1 (1924), 83-93.
- 25.
- A. L. Volberg, Two remarks concerning the theorem of S. Axler, S.-Y. A. Chang and D. Sarason, J. Operator Theory 7 (1982), 209-218. MR 658609
Review Information:
Reviewer:
Douglas N. Clark
Journal:
Bull. Amer. Math. Soc.
9 (1983), 98-102
DOI:
https://doi.org/10.1090/S0273-0979-1983-15171-6
