A general one-sided compactness result for interpolation of bilinear operators
DOI:
https://doi.org/10.7146/math.scand.a-111424Abstract
The behavior of bilinear operators acting on the interpolation of Banach spaces in relation to compactness is analyzed, and an one-sided compactness theorem is obtained for bilinear operators interpolated by the ρ interpolation method.
References
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Cobos, F., Fernández-Martínez, P., and Martínez, A., Interpolation of the measure of non-compactness by the real method, Studia Math. 135 (1999), no. 1, 25–38.
Cobos, F., Kühn, T., and Peetre, J., Multilinear forms of Hilbert type and some other distinguished forms, Integral Equations Operator Theory 56 (2006), no. 1, 57–70. https://doi.org/10.1007/s00020-005-1412-2
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Cwikel, M., Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992), no. 2, 333–343. https://doi.org/10.1215/S0012-7094-92-06514-8
Edmunds, D. E. and Netrusov, Y., Entropy numbers and interpolation, Math. Ann. 351 (2011), no. 4, 963–977. https://doi.org/10.1007/s00208-010-0624-1
Edmunds, D. E. and Netrusov, Y., Entropy numbers of operators acting between vector-valued sequence spaces, Math. Nachr. 286 (2013), no. 5-6, 614–630. https://doi.org/10.1002/mana.201100195
Fernandez, D. L. and da Silva, E. B., Interpolation of bilinear operators and compactness, Nonlinear Anal. 73 (2010), no. 2, 526–537. https://doi.org/10.1016/j.na.2010.03.049
Fernandez, D. L. and da Silva, E. B., One-sided compactness for interpolation of bilinear operators, 8th International Conference on Function Spaces, Differential Operators, Nonlinear Analysis (FSDONA-2011), September 18–24, Tabarz, Thüringen, 2011.
Fernández-Cabrera, L. M. and Martínez, A., On interpolation properties of compact bilinear operators, Math. Nachr. 290 (2017), no. 11-12, 1663–1677. https://doi.org/10.1002/mana.201600203
Fernández-Martínez, P., Interpolation of the measure of non-compactness between quasi-Banach spaces, Rev. Mat. Complut. 19 (2006), no. 2, 477–498. https://doi.org/10.5209/rev_REMA.2006.v19.n2.16614
Grafakos, L. and Kalton, N. J., The Marcinkiewicz multiplier condition for bilinear operators, Studia Math. 146 (2001), no. 2, 115–156. https://doi.org/10.4064/sm146-2-2
Grafakos, L. and Mastyło, M., Interpolation of bilinear operators between quasi-Banach spaces, Positivity 10 (2006), no. 3, 409–429. https://doi.org/10.1007/s11117-005-0034-x
Gustavsson, J., A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (1978), no. 2, 289–305. https://doi.org/10.7146/math.scand.a-11754
Hernando, B., Approximation numbers of nuclear and Hilbert-Schmidt multilinear forms defined on Hilbert spaces, J. Math. Anal. Appl. 341 (2008), no. 2, 922–930. https://doi.org/10.1016/j.jmaa.2007.10.067
Krikorian, N., Compact multilinear transformations, Proc. Amer. Math. Soc. 33 (1972), 373–376. https://doi.org/10.2307/2038063
Lacey, M. and Thiele, C., On Calderón's conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475–496. https://doi.org/10.2307/120971
Lions, J.-L. and Peetre, J., Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math. (1964), no. 19, 5–68.
Maligranda, L., Interpolation of some spaces of Orlicz type. II. Bilinear interpolation, Bull. Polish Acad. Sci. Math. 37 (1989), no. 7-12, 453–457.
Mastyło, M., On interpolation of bilinear operators, J. Funct. Anal. 214 (2004), no. 2, 260–283. https://doi.org/10.1016/j.jfa.2003.10.004
Michels, C., Review of "Fernandez and da Silva, Nonlinear Anal. 73", zbMath (2010), no. Zbl 1200.46023.
Myśliński, A., Shape optimization of a nonlinear elliptic system, Kybernetika (Prague) 29 (1993), no. 3, 270–283.
Pisier, G., A remark on $\pi _2(\ell _p,\ell _p)$, Math. Nachr. 148 (1990), 243–245.
Szwedek, R., Measure of non-compactness of operators interpolated by the real method, Studia Math. 175 (2006), no. 2, 157–174. https://doi.org/10.4064/sm175-2-4
Szwedek, R., Interpolation of approximation numbers between Hilbert spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 343–360. https://doi.org/10.5186/aasfm.2015.4014
Szwedek, R., On interpolation of the measure of non-compactness by the complex method, Q. J. Math. 66 (2015), no. 1, 323–332. https://doi.org/10.1093/qmath/hau009
Teixeira, M. F. and Edmunds, D. E., Interpolation theory and measures of noncompactness, Math. Nachr. 104 (1981), 129–135. https://doi.org/10.1002/mana.19811040110
Cabello Sánchez, F., García, R., and Villanueva, I., On a question of Pełczyński about multilinear operators, Bull. Polish Acad. Sci. Math. 48 (2000), no. 4, 341–345.
Calderón, A.-P., Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. https://doi.org/10.4064/sm-24-2-113-190
Cobos, F., Fernández-Cabrera, L. M., and Martínez, A., Compact operators between $K$- and $J$-spaces, Studia Math. 166 (2005), no. 3, 199–220. https://doi.org/10.4064/sm166-3-1
Cobos, F., Fernández-Cabrera, L. M., and Martínez, A., Abstract $K$ and $J$ spaces and measure of non-compactness, Math. Nachr. 280 (2007), no. 15, 1698–1708. https://doi.org/10.1002/mana.200510572
Cobos, F., Fernández-Cabrera, L. M., and Martínez, A., Estimates for the spectrum on logarithmic interpolation spaces, J. Math. Anal. Appl. 437 (2016), no. 1, 292–309. https://doi.org/10.1016/j.jmaa.2016.01.003
Cobos, F., Fernández-Martínez, P., and Martínez, A., Interpolation of the measure of non-compactness by the real method, Studia Math. 135 (1999), no. 1, 25–38.
Cobos, F., Kühn, T., and Peetre, J., Multilinear forms of Hilbert type and some other distinguished forms, Integral Equations Operator Theory 56 (2006), no. 1, 57–70. https://doi.org/10.1007/s00020-005-1412-2
Cobos, F., Kühn, T., and Schonbek, T., One-sided compactness results for Aronszajn-Gagliardo functors, J. Funct. Anal. 106 (1992), no. 2, 274–313. https://doi.org/10.1016/0022-1236(92)90049-O
Cobos, F. and Peetre, J., Interpolation of compactness using Aronszajn-Gagliardo functors, Israel J. Math. 68 (1989), no. 2, 220–240. https://doi.org/10.1007/BF02772662
Cwikel, M., Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992), no. 2, 333–343. https://doi.org/10.1215/S0012-7094-92-06514-8
Edmunds, D. E. and Netrusov, Y., Entropy numbers and interpolation, Math. Ann. 351 (2011), no. 4, 963–977. https://doi.org/10.1007/s00208-010-0624-1
Edmunds, D. E. and Netrusov, Y., Entropy numbers of operators acting between vector-valued sequence spaces, Math. Nachr. 286 (2013), no. 5-6, 614–630. https://doi.org/10.1002/mana.201100195
Fernandez, D. L. and da Silva, E. B., Interpolation of bilinear operators and compactness, Nonlinear Anal. 73 (2010), no. 2, 526–537. https://doi.org/10.1016/j.na.2010.03.049
Fernandez, D. L. and da Silva, E. B., One-sided compactness for interpolation of bilinear operators, 8th International Conference on Function Spaces, Differential Operators, Nonlinear Analysis (FSDONA-2011), September 18–24, Tabarz, Thüringen, 2011.
Fernández-Cabrera, L. M. and Martínez, A., On interpolation properties of compact bilinear operators, Math. Nachr. 290 (2017), no. 11-12, 1663–1677. https://doi.org/10.1002/mana.201600203
Fernández-Martínez, P., Interpolation of the measure of non-compactness between quasi-Banach spaces, Rev. Mat. Complut. 19 (2006), no. 2, 477–498. https://doi.org/10.5209/rev_REMA.2006.v19.n2.16614
Grafakos, L. and Kalton, N. J., The Marcinkiewicz multiplier condition for bilinear operators, Studia Math. 146 (2001), no. 2, 115–156. https://doi.org/10.4064/sm146-2-2
Grafakos, L. and Mastyło, M., Interpolation of bilinear operators between quasi-Banach spaces, Positivity 10 (2006), no. 3, 409–429. https://doi.org/10.1007/s11117-005-0034-x
Gustavsson, J., A function parameter in connection with interpolation of Banach spaces, Math. Scand. 42 (1978), no. 2, 289–305. https://doi.org/10.7146/math.scand.a-11754
Hernando, B., Approximation numbers of nuclear and Hilbert-Schmidt multilinear forms defined on Hilbert spaces, J. Math. Anal. Appl. 341 (2008), no. 2, 922–930. https://doi.org/10.1016/j.jmaa.2007.10.067
Krikorian, N., Compact multilinear transformations, Proc. Amer. Math. Soc. 33 (1972), 373–376. https://doi.org/10.2307/2038063
Lacey, M. and Thiele, C., On Calderón's conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475–496. https://doi.org/10.2307/120971
Lions, J.-L. and Peetre, J., Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math. (1964), no. 19, 5–68.
Maligranda, L., Interpolation of some spaces of Orlicz type. II. Bilinear interpolation, Bull. Polish Acad. Sci. Math. 37 (1989), no. 7-12, 453–457.
Mastyło, M., On interpolation of bilinear operators, J. Funct. Anal. 214 (2004), no. 2, 260–283. https://doi.org/10.1016/j.jfa.2003.10.004
Michels, C., Review of "Fernandez and da Silva, Nonlinear Anal. 73", zbMath (2010), no. Zbl 1200.46023.
Myśliński, A., Shape optimization of a nonlinear elliptic system, Kybernetika (Prague) 29 (1993), no. 3, 270–283.
Pisier, G., A remark on $\pi _2(\ell _p,\ell _p)$, Math. Nachr. 148 (1990), 243–245.
Szwedek, R., Measure of non-compactness of operators interpolated by the real method, Studia Math. 175 (2006), no. 2, 157–174. https://doi.org/10.4064/sm175-2-4
Szwedek, R., Interpolation of approximation numbers between Hilbert spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 343–360. https://doi.org/10.5186/aasfm.2015.4014
Szwedek, R., On interpolation of the measure of non-compactness by the complex method, Q. J. Math. 66 (2015), no. 1, 323–332. https://doi.org/10.1093/qmath/hau009
Teixeira, M. F. and Edmunds, D. E., Interpolation theory and measures of noncompactness, Math. Nachr. 104 (1981), 129–135. https://doi.org/10.1002/mana.19811040110
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Published
2019-06-17
How to Cite
da Silva, E. B., & Fernandez, D. L. (2019). A general one-sided compactness result for interpolation of bilinear operators. MATHEMATICA SCANDINAVICA, 124(2), 247–262. https://doi.org/10.7146/math.scand.a-111424
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