RT Journal Article T1 Interpolation of the measure of non-compactness of bilinear operators among quasi-Banach spaces A1 Besoy, Blanca F. A1 Cobos, Fernando AB Working in the setting of quasi-Banach couples, we establish a formula for the measure of non-compactness of bilinear operators interpolated by the general real method. The result applies to the real method and to the real method with a function parameter. PB Elsevier SN 0021-9045 YR 2019 FD 2019-02-06 LK https://hdl.handle.net/20.500.14352/13173 UL https://hdl.handle.net/20.500.14352/13173 LA spa NO [1] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980) 1-72.[2] C. Bennett and R. Sharpley, Interpolation of Operators, AcademicPress, New York, 1988.[3] A. Benyi and T. Oh, Smoothing of commutators for a H�ormander class of bilinear pseudodifferential operators, J. Fourier Anal. Appl. 20 (2014) 282{300.[4] A. Benyi and R.H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141 (2013) 3609{3621.[5] J. Bergh and J. L�ofstr�om, Interpolation Spaces. An introduction, Springer, Berlin, 1976.[6] Y. Brudny and N. Krugljak, Interpolation Functors and Interpolation Spaces, Vol. 1, North-Holland, Amsterdam, 1991.[7] A.P. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964) 113-190.[8] B. Carl and I. Stephani, Entropy, Compactness and the Approximationof Operators, Cambridge Univ. Press, Cambridge, 1990.[9] F. Cobos, L.M. Fern�andez-Cabrera, A. Manzano and A. Martinez, Real interpolation and closed operator ideals, J. Math. Pures Appl. 83 (2004) 417-432.[10] F. Cobos, L.M. Fern�andez-Cabrera and A. Martinez, Abstract K and J spaces and measure of non-compactness, Math. Nachr. 280 (2007) 1698-1708.[11] F. Cobos, L.M. Fern�andez-Cabrera and A. Mart��nez, Measure of noncompactness of operators interpolated by limiting real methods, in Operator Theory: Advances and Applications 219 (2012) pp. 37-54.[12] F. Cobos, L.M. Fern�andez-Cabrera and A. Mart��nez, Interpolation of compact bilinear operators among quasi-Banach spaces and applications, Math. Nachr. 291 (2018) 2168{2187.[13] F. Cobos, P. Fern�andez-Martinez and A. Martinez, Interpolation of the measure of non-compactness by the real method, Studia Math. 135 (1999) 25-38.[14] F. Cobos, T. K�uhn and T. Schonbek, One-sided compactness results for Aronszajn-Gagliardo functors, J. Funct. Anal. 106 (1992) 274-313.[15] F. Cobos and J. Peetre, Interpolation of compactness using Aronszajn- Gagliardo functors, Israel J. Math. 68 (1989) 220-240.[16] J.M. Cordeiro, Interpolaci�on de Ciertas Clases de Operadores por Metodos Multidimensionales, Ph. D. thesis, Publicaciones del Depto. de Matematica Aplicada , Universidad de Vigo, 1999.[17] M. Cwikel and J. Peetre, Abstract K and J spaces, J. Math. Pures Appl. 60 (1981) 1-50.[18] D.E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.[19] D. E. Edmunds and Yu. Netrusov, Entropy numbers and interpolation, Math. Ann. 351 (2011) 963{977.[20] D. E. Edmunds and Yu. Netrusov, Entropy numbers of operators acting between vector-valued sequence spaces, Math. Nachr. 286 (2013) 614- 630.[21] D. E. Edmunds and B. Opic, Limiting variants of Krasnosel'skii's compact interpolation theorem, J. Funct. Anal. 266 (2014) 3265-3285.[22] D.E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers and Differential Operators, Cambridge Univ. Press, Cambridge, 1996.[23] W.D. Evans and B. Opic, Real interpolation with logarithmic functors and reiteration, Canad. J. Math. 52 (2000) 920-960.[24] W.D. Evans, B. Opic and L. Pick, Real interpolation with logarithmic functors, J. Inequal. Appl. 7 (2002) 187-269.[25] D.L. Fernandez and E.B. da Silva, Interpolation of bilinear operators and compactness, Nonlinear Anal. 73 (2010) 526-537.[26] L.M. Fernandez-Cabrera and A. Mart��nez, Interpolation of Ideal Measures by Abstract K and J Spaces, Acta Math. Sinica 23 (2007) 1357-1374.[27] L.M. Fern�andez-Cabrera and A. Martinez, On interpolation properties of compact bilinear operators, Math. Nachr. 290 (2017) 1663-1777.[28] L.M. Fern�andez-Cabrera and A. Martinez, Real interpolation of compact bilinear operators, J. Fourier Anal. Appl. 24 (2018) 1180-1203.[29] P. Fernandez-Martinez, Interpolation of the measure of noncompactness between quasi-Banach spaces, Rev. Mat. Complut. 19 (2006) 477-498.[30] J. Gustavsson, A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (1978) 289-305.[31] T. Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26(1970) 177-199.[32] G. Hu, Compactness of the commutator of bilinear Fourier multiplier operator, Taiwanese J. Math. 18 (2014) 661-675[33] S. Janson, Minimal and maximal methods of interpolation, J. Funct.Anal. 44 (1981) 50–73.[34] H. K¨onig, Eigenvalue Distribution of Compact Operators, Birkhaa¨user, Basel, 1986.[35] G. K¨oththe,e, Topological Vector Spaces I, Springer, Berlin, 1969.[36] J.-L. Lions and J. Peetre, Sur une classe d’espaces d’interpolation, Inst. Hautes EE´ tudes Sci. Publ. Math. 19 (1964) 5–68.[37] M. Mastyll-o and E.B. Silva, Interpolation of the measure of non- compactness of bilinear operators, Trans. Amer. Math. Soc. 370 (2018) 8979–8997.[38] P. Nilsson, Reiteration theorems for real interpolation and approxima- tion spaces, Ann. Mat. Pura Appl. 132 (1982) 291–330.[39] P. Nilsson, Interpolation of Calderoo´n and Ov˘˘cnnikov pairs, Ann. Mat. Pura Appl. 134 (1983) 201–232.[40] J. Peetre, A theory of interpolation of normed spaces, Lecture Notes, Brasilia, 1978.[41] L.-E. Persson, Interpolation with a parameter function, Math. Scand. 59 (1986) 199–222.[42] A. Pietsch, Operator Ideals, North-Holland, Amsterdam, 1980.[43] R. Szwedek, Measure of non-compactness of operators interpolated by the real method, Studia Math. 175 (2006) 157–174.[44] R. Szwedek, On interpolation of the measure of non-compactness by the complex method, Quart. J. Math. Oxford 66 (2015) 323–332.[45] R. Szwedek, Geometric interpolation of entropy numbers, Quart. J. Math. Oxford 69 (2018) 377–389.[46] M.F. Teixeira and D.E. Edmunds, Interpolation theory and measures of non-compactness, Math. Nachr. 104 (1981) 129–135.[47] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978. NO Ministerio de Economía y Competitividad (MINECO)/FEDER NO Ministerio de Educación y Ciencia (MEC) DS Docta Complutense RD 30 abr 2024