Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
fractional powers of operators; $L$-characteristic; Lebesgue spaces; interpolation theorems
fractional powers of operators; $L$-characteristic; Lebesgue spaces; interpolation theorems
Summary:
We describe the geometric structure of the ${\Cal L}$-characteristic of fractional powers of bounded or compact linear operators over domains with arbitrary measure. The description builds essentially on the Riesz-Thorin and Marcinkiewicz-Stein-Weiss-\linebreak Ovchinnikov interpolation theorems, as well as on the Krasnosel'skij-Krejn factorization theorem.
References:
[1] Bergh J., Löfström J.: Interpolation spaces - an introduction. Springer, Berlin, 1976. MR 0482275
[2] Brudnyj Ju.A., Krejn S.G., Semenov E. M.: Interpolation of linear operators (in Russian). Itogi Nauki Tekhniki 24 (1986), 3-163 [Engl. transl.: J. Soviet Math. 42 (1988), 2009-2113]. MR 0887950
[3] Butzer P.L., Berens H.: Semigroups of operators and approximation. Springer, Berlin, 1967.
[4] Calderón A.P.: Intermediate spaces and interpolation: the complex method. Studia Math. 24 (1964), 113-190. MR 0167830
[5] Calderón A.P., Zygmund A.: A note on the interpolation of linear operators. Studia Math. 12 (1951), 194-204. MR 0046567
[6] Hunt R.A.: On $L(p,q)$ spaces. Enseignement Math. 12 (1966), 249-276. MR 0223874 | Zbl 0181.40301
[7] Krasnosel'skij M.A.: Decomposition of operators from $L_q$ into $L_p$ (in Russian). Dokl. Akad. Nauk SSSR 82,3 (1952), 333-336.
[8] Krasnosel'skij M.A.: Properties of the root of a linear integral operator (in Russian). Dokl. Akad. Nauk SSSR 88,5 (1953), 749-751. MR 0055561
[9] Krasnosel'skij M.A.: On a theorem of M. Riesz (in Russian). Dokl. Akad. Nauk SSSR 131,2 (1960), 246-248.
[10] Krasnosel'skij M.A., Krejn S.G.: Criteria for the continuity and complete continuity of a linear operator in terms of properties of its square (in Russian). Trudy Sem. Funk. Anal. Voronezh. 5 (1957), 98-101.
[11] Krasnosel'skij M.A., Pustyl'nik E.I.: Using fractional powers of operators in the investigation of Fourier series expansion with respect to eigenfunctions of differential operators (in Russian). Dokl. Akad. Nauk SSSR 122,6 (1958), 978-981. MR 0100145
[12] Krasnosel'skij M.A., Zabrejko P.P.: On the $L$-characteristic of operators (in Russian). Uspekhi Mat. Nauk 19 (1964), 187-189.
[13] Krasnosel'skij M.A., Zabrejko P.P., Pustyl'nik E.I., Sobolevskij P.E.: Integral operators in spaces of summable functions (in Russian). Nauka, Moscow 1966 [Engl. transl.: Noordhoff, Leyden, 1976].
[14] Krejn S.G., Petunin Ju.I., Semenov E.M.: Interpolation of linear operators (in Russian). Nauka, Moscow 1978 [Engl. transl.: Math. Monogr. Amer. Math. Soc., Providence, 1982]. MR 0506343
[15] Marcinkiewicz J.: Sur l'interpolation d'opérateurs. C.R. Acad. Sci. Paris 208 (1939), 1272-1273.
[16] O'Neil R.: Convolution operators and L(p,q) spaces. Duke Math. J. 30,1 (1963), 129-142. Zbl 0178.47701
[17] Ovchinnikov V.I.: An exact interpolation theorem in $L_p$ spaces (in Russian). Dokl. Akad. Nauk SSSR 272,2 (1983), 300-303. MR 0724507
[18] Pustyl'nik E.I.: On fractional powers of unbounded positive operators (in Ukrainian). Dokl. Akad. Nauk UkrSSR 10 (1961), 1266-1270.
[19] Rao M.M., Ren Z.D.: Theory of Orlicz spaces. Dekker Inc., New York, 1991. MR 1113700 | Zbl 0724.46032
[20] Riemenschneider S.D.: The L-characteristics of linear operators on $L^{1/\alpha}([0,1])$. J. Funct. Anal. 8 (1971), 405-421. MR 0290144
[21] Riemenschneider S.D.: Linear operators on $L^{1/\alpha}(0,\infty)$ and Lorentz spaces: the Krasnosel'skii-Zabreiko characteristic sets. Studia Math. 49 (1974), 225-233. MR 0333703
[22] Riesz M.: Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires. Acta Math. 49 (1926), 465-497. MR 1555250
[23] Rutitskij Ja.B.: Scales of Orlicz spaces and interpolation theorems (in Russian). Dokl. Akad. Nauk SSSR 149,1 (1963), 32-35. MR 0160101
[24] Stein E.M., Weiss G.: An extension of a theorem of Marcinkiewicz and some of its applications. J. Math. Mech. 8,2 (1959), 263-284. MR 0107163 | Zbl 0084.10801
[25] Thorin G.O.: An extension of a convexity theorem due to M. Riesz. Comm. Sem. Math. Lund 4 (1939), 1-5. Zbl 0021.14404
[26] Zabrejko P.P., Krasnosel'skij M.A., Pustyl'nik E.I.: A problem for fractional powers of operators (in Russian). Uspekhi Mat. Nauk 20,6 (1965), 87-89. MR 0199724
[27] Zygmund A.: On a theorem of Marcinkiewicz concerning interpolation. J. Pure Appl. Math. 9 (1956), 223-248. MR 0080887 | Zbl 0070.33701
Search
Partner of
