References
- Beals , R. 1979 . J. Funct. Anal. , 34 : 1
- van der Mee , C. V. M. 1981 . “Semigroup and Factorization Methods in Transport Theory” Amsterdam Math. Centre Tract No. 146
- Greenberg , W. , van der Mee , C. V.M. and Walus , W. 1984 . Int. Eqs. Oper. Theor. , 7 : 60
- Zweifel , P. F. 1982/83 . Transp. Theor. Stat. Phys. , 11 : 183
- Case , K. M. and Zweifel , P. F. 1967 . “Linear Transport Theory” , Reading, Mass. : Addison-Wesley .
- Halmos , P. R. 1967 . “A Hilbert Space Problem Book” , 91 New York : Van Nostrand . cf. 293
- 26 The reader is referred to Ref. [6], for a thoughtful discussion of the distinction between invertibility in the settheoretic sense, i.e. bijectivity, and invertibility as an operator having a bounded inverse
- Hangelbroek , R. J. and Lekkerkerker , C. G. 1977 . SIAM J. Math. Anal. , 8 : 458
- Lekkerkerker , C. G. 1976 . Proc. Edinburgh Math. Soc. , 75A : 283 In these works, half-space theory for one-speed neutron transport with isotropic scattering is discussed at great length
- Greenberg , W. , van der Mee , C. V.M. and Walus , W. “Strong Solutions of Stationary Equations in Abstract Kinetic Theory,” . SIAM J. Math. Anal. , submitted.
- van der Mee , C. V. M. 1984 . Transp. Theor. Stat. Phys. , 13 : 341
- Lp ((O,τ);H) is the Banach space of strongly measurable functions f:(O,τ) → H satisfying ‖f(.) ‖H ε Lp(O,τ), endowed with the Lp-norm. The integral in Eq. (6) is to be interpreted as a Bochner integral
- In fact, , where we have to assume ‖ g(x)-g(y)‖H M|x-y|y for some M <∞ and γ ε(0,1) with x, y ε|O,τ|.
- Beals , R. 1981 . J. Math. Phys. , 22 : 954
- 1983 . J. Math. Phys. , 24 : 1932
- Beals , R. and Protopopescu , V. 1983 . J. Stat. Phys. , 32 : 565
- Beals , R. 1985 . J. Diff. Eqs. , 56 : 391
- Case , K. M. and Zweifel , P. F. 1963 . J. Math. Phys. , 4 : 1376 Ref. [5], Appendix D; also
- Mullikin , T. W. and Victory , H. D. 1977 . J. Math. Anal. Appl. , 58 : 605
- Hovenier , J. W. and van der Mee , C. V. M. 1983 . Astron. Astrophys. , 128 : 1
- Germogenova , T. A. and Konovalov , N. V. 1978 . “The Spectrum of the Characteristic Equation in the Theory of Transfer with Polarization Taken into Account” , Moscow : Inst. Appl. Math., Acad. Sci. U.S.S.R. . Preprint no. 62 [Russian]
- van der Mee , C. V. M. 1984 . Math. Meth. Appl. Sci. , 6 : 393
- Beals , R. 1984 . “The Multi-dimensional Linear Transport Equation,” Unpublished manuscript
- Hovenier , J. W. and van der Mee , C. V. M. The proof of Ref. [18] uses positive cone arguments. A simple proof, requiring no such reasoninK, has been given recently by (in preparation). In both of these references the statement appears in the form of inequalities for certain expansion coefficients (related to the eigenvalues of B) and must be reDhrased in the present context