zbMATH — the first resource for mathematics

Invariant Banach limits and applications. (English) Zbl 1205.46012
Let \(\ell _{\infty }\) be the space of all bounded sequences \(x=(x_{1},x_{2},\ldots )\) with the norm \(\|x\|_{\ell_{\infty}} = \text{ sup}_n |x_n|\) and let \(L(\ell _{\infty })\) be the set of all bounded linear operators on \(\ell _{\infty }\). A linear functional \(B\) in the dual space \(\ell_\infty^\ast\) is said to be a Banach limit if \(B(1,1, \dots)=1\), \(B \geq 0\) and \(B(Tx)=B(x)\) for every \(x \in \ell_\infty\), where \(T\), is the translation operator, that is, \(T(x_1,x_2, \dots)=(x_2,x_3, \dots)\).
Following an approach similar to that of W. F. Eberlein for regular Hausdorff transformations in the classical paper [Proc. Am. Math. Soc. 1, 662–665 (1950; Zbl 0039.12102)] and motivated by their own recent contributions for the dilation operator \(\sigma_n\), \(n \in \mathbb{N}\), and the classical Cesàro operator \(C\), the authors present a set of easily verifiable sufficient conditions on an operator \(H \in L(\ell _{\infty })\), guaranteeing the existence of a Banach limit \(B\) on \(\ell _{\infty }\) such that \(B=BH\). They apply their results to the above mentioned Cesàro operator \(C\) on \(\ell _{\infty }\) and give a necessary and sufficient condition for an element \(x \in \ell _{\infty }\) to have fixed value \(Bx\) for all Cesàro invariant Banach limits \(B\). Finally, they apply the preceding description to obtain a characterization of “measurable elements” from the (Dixmier-)Macaev-Sargent ideal of compact operators with respect to an important subclass of Dixmier traces generated by all Cesàro-invariant Banach limits. It is shown that this class is strictly larger than the class of all “measurable elements” with respect to the class of all Dixmier traces.

46B99 Normed linear spaces and Banach spaces; Banach lattices
46L51 Noncommutative measure and integration
40J05 Summability in abstract structures (should also be assigned at least one other classification number from Section 40-XX)
Full Text: DOI
[1] Banach, S., Théorie des óperations linéaires, (1995), Chelsea Publishing Co. New York, (in French) · Zbl 0067.08902
[2] Carey, A.; Phillips, J.; Sukochev, F.A., Spectral flow and Dixmier traces, Adv. math., 173, 68-113, (2003) · Zbl 1015.19003
[3] Carey, A.L.; Sukochev, F.A., Dixmier traces and some applications in non-commutative geometry, Russian math. surveys, 61, 1039-1099, (2006) · Zbl 1151.46053
[4] Connes, A., Noncommutative geometry, (1994), Academic Press San Diego · Zbl 1106.58004
[5] Dodds, P.G.; de Pagter, B.; Sedaev, A.A.; Semenov, E.M.; Sukochev, F.A., Singular symmetric functionals and Banach limits with additional invariance properties, Izv. math., 67, 6, 1187-1212, (2003) · Zbl 1075.46028
[6] Dodds, P.G.; de Pagter, B.; Sedaev, A.A.; Semenov, E.M.; Sukochev, F.A., Singular symmetric functionals, J. math. sci., 2, 4867-4885, (2004) · Zbl 1090.46020
[7] Dodds, P.G.; de Pagter, B.; Semenov, E.M.; Sukochev, F.A., Symmetric functional and singular traces, Positivity, 2, 1, 47-75, (1998) · Zbl 0915.46021
[8] Eberlein, W.F., Banach-Hausdorff limits, Proc. amer. math. soc., 1, 662-665, (1950) · Zbl 0039.12102
[9] Lord, S.; Sedaev, A.; Sukochev, F.A., Dixmier traces as singular symmetric functionals and applications to measurable operators, J. funct. anal., 244, 1, 72-106, (2005) · Zbl 1081.46042
[10] Lorentz, G.G., A contribution to the theory of divergent sequences, Acta. math., 80, 167-190, (1948) · Zbl 0031.29501
[11] Macaev, V.I., A class of completely continuous operators, Dokl. akad. nauk SSSR, 139, 548-551, (1961), (in Russian)
[12] Pietsch, A., About the Banach envelope of \(l_{1, \infty}\), Rev. mat. complut., 22, 1, 209-226, (2009) · Zbl 1175.46004
[13] Raimi, R.A., Factorization of summability-preserving generalized limits, J. lond. math. soc., 22, 398-402, (1980) · Zbl 0419.40007
[14] Sargent, W.L.C., Some sequence spaces related to the \(l_p\) spaces, J. lond. math. soc., 35, 161-171, (1960) · Zbl 0090.03703
[15] Sucheston, L., Banach limits, Amer. math. monthly, 74, 308-311, (1967) · Zbl 0148.12202
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.