On the $\Cal L$-characteristic of fractional powers of linear operators
Appell, Jürgen ; Simões, Marilda A. ; Zabrejko, Petr P.
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994), p. 21-34 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1994-01-01
Classification:  46E30,  46M35,  47A57,  47A60,  47A99,  47B37,  47B38 
@article{118637,
     author = {J\"urgen Appell and Marilda A. Sim\~oes and Petr P. Zabrejko},
     title = {On the $\Cal L$-characteristic of fractional powers of linear operators},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {35},
     year = {1994},
     pages = {21-34},
     zbl = {0806.47014},
     mrnumber = {1292579},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118637}
}

Appell, Jürgen; Simões, Marilda A.; Zabrejko, Petr P. On the $\Cal L$-characteristic of fractional powers of linear operators. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 21-34. http://gdmltest.u-ga.fr/item/118637/

Bergh J.; Löfström J. Interpolation spaces - an introduction, Springer, Berlin, 1976. | MR 0482275

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]. (1986) | MR 0887950

Butzer P.L.; Berens H. Semigroups of operators and approximation, Springer, Berlin, 1967.

Calderón A.P. Intermediate spaces and interpolation: the complex method, Studia Math. 24 (1964), 113-190. (1964) | MR 0167830

Calderón A.P.; Zygmund A. A note on the interpolation of linear operators, Studia Math. 12 (1951), 194-204. (1951) | MR 0046567

Hunt R.A. On $L(p,q)$ spaces, Enseignement Math. 12 (1966), 249-276. (1966) | MR 0223874 | Zbl 0181.40301

Krasnosel'Skij M.A. Decomposition of operators from $L_q$ into $L_p$ (in Russian), Dokl. Akad. Nauk SSSR 82,3 (1952), 333-336. (1952)

Krasnosel'Skij M.A. Properties of the root of a linear integral operator (in Russian), Dokl. Akad. Nauk SSSR 88,5 (1953), 749-751. (1953) | MR 0055561

Krasnosel'Skij M.A. On a theorem of M. Riesz (in Russian), Dokl. Akad. Nauk SSSR 131,2 (1960), 246-248. (1960)

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. (1957)

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. (1958) | MR 0100145

Krasnosel'Skij M.A.; Zabrejko P.P. On the $L$-characteristic of operators (in Russian), Uspekhi Mat. Nauk 19 (1964), 187-189. (1964)

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].

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

Marcinkiewicz J. Sur l'interpolation d'opérateurs, C.R. Acad. Sci. Paris 208 (1939), 1272-1273. (1939)

O'Neil R. Convolution operators and L(p,q) spaces, Duke Math. J. 30,1 (1963), 129-142. (1963) | Zbl 0178.47701

Ovchinnikov V.I. An exact interpolation theorem in $L_p$ spaces (in Russian), Dokl. Akad. Nauk SSSR 272,2 (1983), 300-303. (1983) | MR 0724507

Pustyl'Nik E.I. On fractional powers of unbounded positive operators (in Ukrainian), Dokl. Akad. Nauk UkrSSR 10 (1961), 1266-1270. (1961)

Rao M.M.; Ren Z.D. Theory of Orlicz spaces, Dekker Inc., New York, 1991. | MR 1113700 | Zbl 0724.46032

Riemenschneider S.D. The L-characteristics of linear operators on $L^{1/\alpha}([0,1])$, J. Funct. Anal. 8 (1971), 405-421. (1971) | MR 0290144

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. (1974) | MR 0333703

Riesz M. Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires, Acta Math. 49 (1926), 465-497. (1926) | MR 1555250

Rutitskij Ja.B. Scales of Orlicz spaces and interpolation theorems (in Russian), Dokl. Akad. Nauk SSSR 149,1 (1963), 32-35. (1963) | MR 0160101

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. (1959) | MR 0107163 | Zbl 0084.10801

Thorin G.O. An extension of a convexity theorem due to M. Riesz, Comm. Sem. Math. Lund 4 (1939), 1-5. (1939) | Zbl 0021.14404

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. (1965) | MR 0199724

Zygmund A. On a theorem of Marcinkiewicz concerning interpolation, J. Pure Appl. Math. 9 (1956), 223-248. (1956) | MR 0080887 | Zbl 0070.33701