Bulletin of the American Mathematical Society

MathSciNet review: 1567838
Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: D. F. McGhee and R. H. Picard
Title: Cordes' two-parameter spectral representation theory
Additional book information: Pitman Research Notes in Mathematics Series, Volume 177, Longman Scientific and Technical, Harlow, United Kingdom and New York, 1988, 114 pp., $41.95. ISBN 0-470-21084-2.

F. M. Arscott, A new treatment of the ellipsoidal wave equation, Proc. London Math. Soc. (3) 9 (1959), 21–50. MR 104837, DOI 10.1112/plms/s3-9.1.21

F. M. Arscott, Two-parameter eigenvalue problems in differential equations, Proc. London Math. Soc. (3) 14 (1964), 459–470. MR 165164, DOI 10.1112/plms/s3-14.3.459

F. V. Atkinson, Multiparameter spectral theory, Bull. Amer. Math. Soc. 74 (1968), 1–27. MR 220078, DOI 10.1090/S0002-9904-1968-11866-X

F. V. Atkinson, Multiparameter eigenvalue problems, Mathematics in Science and Engineering, Vol. 82, Academic Press, New York-London, 1972. Volume I: Matrices and compact operators. MR 0451001

Patrick J. Browne and B. D. Sleeman, Inverse multiparameter eigenvalue problems for matrices, Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), no. 1-2, 29–38. MR 801842, DOI 10.1017/S0308210500013615

6.

R. D. Carmichael, Boundary value and expansion problems, Amer. J. Math. 43 (1921), 69-101; ibid. 43 (1921), 232-270; 44 (1922), 129-152.

Heinz Otto Cordes, Über die Spektralzerlegung von hypermaximalen Operatoren, die durch Separation der Variablen zerfallen. I, II, Math. Ann. 128 (1954), 257–289; 373–411 (1955) (German). MR 66562, DOI 10.1007/BF01360138

8.

A. Erdélyi, Higher transcendental functions, Bateman manuscript project vol. 3, McGraw-Hill, New York, Toronto, London, 1955.

9.

D. Hilbert, Grundzuge einer allgemeinsen Theorie der Linearen Integralgleichungen, Berlin, 1912.

10.

H. Isaev, Lectures on multiparameter spectral theory, Calgary, 1985.

A. Källström and B. D. Sleeman, Solvability of a linear operator system, J. Math. Anal. Appl. 55 (1976), no. 3, 785–793. MR 417833, DOI 10.1016/0022-247X(76)90081-0

A. Källström and B. D. Sleeman, Joint spectra for commuting operators, Proc. Edinburgh Math. Soc. (2) 28 (1985), no. 2, 233–248. MR 806754, DOI 10.1017/S0013091500022677

Felix Klein, Ueber Körper, welche von confocalen Flächen zweiten Grades begränzt sind, Math. Ann. 18 (1881), no. 2, 410–427 (German). MR 1510109, DOI 10.1007/BF01445858

F. J. Murray and J. Von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), no. 1, 116–229. MR 1503275, DOI 10.2307/1968693

Anna J. Pell, Linear equations with two parameters, Trans. Amer. Math. Soc. 23 (1922), no. 2, 198–211. MR 1501198, DOI 10.1090/S0002-9947-1922-1501198-X

B. P. Rynne, Multiparameter spectral theory and Taylor’s joint spectrum in Hilbert space, Proc. Edinburgh Math. Soc. (2) 31 (1988), no. 1, 127–144. MR 930021, DOI 10.1017/S0013091500006635

Bryan P. Rynne, Uniform convergence of multiparameter eigenfunction expansions, J. Math. Anal. Appl. 147 (1990), no. 2, 340–350. MR 1050209, DOI 10.1016/0022-247X(90)90352-G

B. D. Sleeman, Multiparameter spectral theory in Hilbert space, J. Math. Anal. Appl. 65 (1978), no. 3, 511–530. MR 510467, DOI 10.1016/0022-247X(78)90160-9

Hans Volkmer, Multiparameter eigenvalue problems and expansion theorems, Lecture Notes in Mathematics, vol. 1356, Springer-Verlag, Berlin, 1988. MR 973644, DOI 10.1007/BFb0089295

1.

F. M. Arscott, A treatment of the ellipsoidal wave equation, Proc. London. Math. Soc. (3) 9 (1959), 21-50. MR 0104837

2.

F. M. Arscott, Two-parameter eigenvalue problems in differential equations, Proc. London. Math. Soc. (3) 14 (1964), 459-470. MR 165164

3.

F. V. Atkinson, Multiparameter spectral theory, Bull. Amer. Math. Soc. 74 (1968), 1-27. MR 220078

4.

F. V. Atkinson, Multiparameter eigenvalue problems: matrices and compact operators, Academic Press, New York and London, 1972. MR 451001

5.

P. J. Browne and B. D. Sleeman, Inverse multiparameter eigenvalue problems for matrices, Proc. Roy. Soc. Edinburgh Sect. (A) 100 (1985), 29-38. MR 801842

6.

R. D. Carmichael, Boundary value and expansion problems, Amer. J. Math. 43 (1921), 69-101; ibid. 43 (1921), 232-270; 44 (1922), 129-152.

7.

H. O. Cordes, Über die Spektralzerlegung von hypermaximalen Operatoren die durch Separation der Variablen zer fallen. I, Math. Ann. 128 (1954), 257-289; II, Math. Ann. (1955), 373-411. MR 66562

8.

A. Erdélyi, Higher transcendental functions, Bateman manuscript project vol. 3, McGraw-Hill, New York, Toronto, London, 1955.

9.

D. Hilbert, Grundzuge einer allgemeinsen Theorie der Linearen Integralgleichungen, Berlin, 1912.

10.

H. Isaev, Lectures on multiparameter spectral theory, Calgary, 1985.

11.

A. Källström and B. D. Sleeman, Solvability of a linear operator system, J. Math. Anal. Appl. 55 (1976), 785-793. MR 417833

12.

A. Källström and B. D. Sleeman, Joint spectra for commuting operators, Proc. Ed. Math. Soc. 28 (1985), 233-248. MR 806754

13.

F. Klein, Math. Ann. 18 (1881), p. 410, but see E. L. Ince, Ordinary differential equations, Dover, New York, 1956. MR 1510109

14.

F. J. Murray and J. von Neumann, On rings of operators, Ann. of Math. (2) 37 (1936), 116-229. MR 1503275

15.

A. J. Pell, Linear equations with two parameters, Trans. Amer. Math. Soc. 23 (1922), 198-211. MR 1501198

16.

B. P. Rynne, Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space, Proc. Roy. Soc. Edinburgh Sect. (A) 31 (1988), 127-144. MR 930021

17.

B. P. Rynne, Uniform convergence of multiparameter eigenfunction expansions, J. Math. Anal. Appl. (to appear). MR 1050209

18.

B. D. Sleeman, Multiparameter spectral theory in Hilbert space, Research Notes in Mathematics, vol. 22, Pitman, London, 1978. MR 510467

19.

H. Volkmer, Multiparameter eigenvalue problems and expansion theorems, Lecture Notes in Math., vol. 1356, Springer-Verlag, Berlin and New York, 1988. MR 973644