×

zbMATH — the first resource for mathematics

On the Cohen strongly \(p\)-summing multilinear operators. (English) Zbl 1121.47013
Let \(X_{1},\dots,X_{m},Y\) be Banach spaces and \(p\in[1,\infty].\) An \(m\)-linear operator \(T:X_{1}\times\cdots\times X_{m}\rightarrow Y\) is called Cohen strongly \(p\)-summing if there is a constant \(C>0\) such that for any \(x_{1}^{j},\dots,x_{n}^{j}\in X_{j}\) (\(j=1,\dots,m)\) and any \(y_{1}^{\ast},\dots,y_{n}^{\ast}\in Y^{\ast}\), we have that \[ \left\|\left(\langle T(x_{i}^{1},\dots,x_{i}^{m}),y_{i}^{\ast}\rangle\right)_{i}\right\|_{\ell^{1}(n)}\leq C\left(\sum_{i=1}^{n}\prod_{j=1}^{m}\left\| x_{i}^{j}\right\|_{X_{j}}^{p}\right)^{1/p}\sup_{\left\| y\right\|\leq1}\left\| y_{i}^{\ast}(y)\right\|_{\ell^{p^{\ast}}(n)}. \] Using Ky Fan’s lemma, a Pietsch type domination theorem is proved within this class of multilinear operators. Also, a result of Q.-Y. Bu [Contemp. Math. 328, 145–149 (2003; Zbl 1066.47019)] is extended, providing a framework under which every \(p\)-dominated \(m\)-linear operator is Cohen strongly \(p\)-summing.

MSC:
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47H60 Multilinear and polynomial operators
46G25 (Spaces of) multilinear mappings, polynomials
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alencar, R., Multilinear mappings of nuclear and integral type, Proc. amer. math. soc., 94, 1, 33-38, (1985) · Zbl 0537.47011
[2] Alencar, R.; Matos, M.C., Some classes of multilinear mappings between Banach spaces, Publ. dep. an. mat. univ. complut. Madrid, 12, (1985)
[3] Bombal, F.; Pérez-García, D.; Villanueva, I., Multilinear extension of Grothendieck’s theorem, Q. J. math., 55, 441-450, (2004) · Zbl 1078.46030
[4] Bu, Q., Some mapping properties of p-summing operators with Hilbertian domain, (), 145-149 · Zbl 1066.47019
[5] Cohen, J.S., Absolutely p-summing, p-nuclear operators and their conjugates, Math. ann., 201, 177-200, (1973) · Zbl 0233.47019
[6] Dimant, V., Strongly p-summing multilinear mappings, J. math. anal. appl., 278, 182-193, (2003) · Zbl 1043.47018
[7] Defant, A.; Floret, K., Tensor norms and operator ideals, North-holland math. stud., (1993) · Zbl 0774.46018
[8] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely summing operators, (1995), Cambridge Univ. Press · Zbl 0855.47016
[9] S. Geiss, Ideale multilinearer Abbildungen, Diplomarbeit, 1984
[10] Matos, M.C., On multilinear mappings of nuclear type, Rev. mat. comput., 6, 61-81, (1993) · Zbl 0807.46022
[11] Matos, M.C., Absolutely summing holomorphic mappings, An. acad. brasil. cienc., 68, 1-13, (1996) · Zbl 0854.46042
[12] Matos, M.C., Fully absolutely summing and hilbert – schmidt multilinear mappings, Collect. math., 54, 1-111, (2003) · Zbl 1078.46031
[13] Matos, M.C.; Tonge, A., Polynomials and the Pietsch domination theorem, Math. proc. R. ir. acad., 2, 195-212, (1999) · Zbl 0973.46037
[14] Pietsch, A., Ideals of multilinear functionals (designs of a theory), (), 185-199
[15] Pietsch, A., Absolut p-summierende abbildungen in normierten Räumen, Studia math., 28, 333-353, (1967) · Zbl 0156.37903
[16] Pérez-García, D.; Villanueva, I., Multiple summing operators on Banach spaces, J. math. anal. appl., 285, 86-96, (2003) · Zbl 1044.46037
[17] Pérez-García, D.; Villanueva, I., Multiple summing operators on \(C(K)\) spaces, Arkiv. mat., 42, 153-171, (2004) · Zbl 1063.46032
[18] Schneider, B., On absolutely p-summing and related multilinear mappings, Brandenburg. landeshochsch. wiss. Z., 35, 105-117, (1991) · Zbl 0777.47016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.