İnvaryant Ortalamalı Vektör Değerli Çarpanlar ve Kompakt Toplam Operatörleri

Yıl 2025, Cilt: 15 Sayı: 1, 298 - 307, 01.03.2025

Mahmut Karakuş

https://doi.org/10.21597/jist.1534364

Öz

Çarpan yakınsaklık gösterimiyle, bir dizi uzayının genelleştirilmiş Köthe-Toeplitz duali kavramı yeniden tanımlanabilir. Bir dizi uzayı N nin (e^n) ile verilen bazı (v_n)∈N^β dizisini domine ettiğinden, N nin β-(genelleştirilmiş Köthe-Toeplitz) duali N^β={(v_n )│(e^n ) > ̃(v_n ) } şeklinde temsil edilebilir. Alışılmış terminoloji ve kavramları kullanarak, bu makalede, sınırlı (sürekli) lineer operatörler dizisinin yanı sıra σ -toplanabilirlik yöntemi aracılığıyla yeni vektör değerli çarpan uzaylarını tanıtıyoruz. Bu alt uzaylar sup norm topolojisi ile donatılmışlardır. Normlu uzayların tamlığı esasına dayanarak, çarpan uzayları ve genel normlu uzaylar arasında verilen S toplam operatörünün bazı özelliklerini ayrıntılı bir şekilde inceliyoruz. Bu araştırma, operatörün çeşitli özelliklerinin detaylı bir karakterizasyonunu gerektirir. Bu özellikleri bazı tip çarpan serileri çerçevesinde inceleyerek, operatörün davranışının kapsamlı ve rafine bir analizini sunarak, işlevsel özelliklerine ilişkin daha geniş ve zenginleştirilmiş bir bakış açısı sağlıyoruz.

Anahtar Kelimeler

σ -yakınsaklık, Sınırlı çarpan yakınsak seriler, c_0 (X)-çarpan yakınsak seriler, Toplam operatörler, Toplanabilme

Vector Valued Multipliers of Invariant Means and Compact Summing Operators

Öz

In notation of multiplier convergence, one can redefine the notion generalized Köthe-Toeplitz dual of a sequence space. Since the basis (e^n) of a sequence space N dominates the sequence (v_n)∈N^β, the β-(generalized Köthe-Toeplitz) dual of N can be represented as N^β={(v_n)|(e^n)> ̃(v_n)}. Employing usual terminology and concepts, in this paper, we introduce novel vector-valued multiplier spaces through the σ-summability method alongside a sequence of bounded linear operators. These spaces are equipped with the sup norm topology. Building on the foundational comprehension of completeness of normed spaces, we examine some properties of the summing operator S in detail, which acts between multiplier spaces and general normed spaces. This investigation entails a meticulous characterization of the operator's various properties. By examining these properties through the frameworks of some types of multiplier series, we deliver a thorough and refined analysis of the operator’s behavior, providing a more expansive and enriched perspective on its functional characteristics.

Anahtar Kelimeler

σ – convergence, Bounded multiplier convergent series, c_0 (X)-multiplier convergent series, Summing operators, Summability

Kaynakça

• Aizpuru, A. & Pérez-Fernández, F. J. (1999). Characterizations of series in Banach spaces. Acta Math. Univ. Comenian. (N.S.), 58(2), 337-344.
• Aizpuru, A., Gutiérrez-Dávila, A. & Sala. A. (2006). Unconditionally Cauchy series and Cesàro summability. J. Math. Anal. Appl., 324, 39-48.
• Aizpuru, A., Armario, R. & Pérez-Fernández, F. J. (2008). Almost summability and unconditionally Cauchy series. Bull. Belg. Math. Soc. Simon Stevin, 15, 635-644.
• Aizpuru, A., Pérez-Eslava, C. & Seoane-Sepúlveda, J. B. (2009). Matrix summability methods and weakly unconditionally Cauchy series. Rocky Mountain J. Math., 39 (2), 367-380.
• Aizpuru, A., Armario, R., García-Pacheco, F. J. & Pérez-Fernández, F. J. (2014). Vector-valued almost convergence and classical properties in normed spaces. Proc. Indian Acad. Sci. Math., 124(1), 93-108.
• Albiac, F. & Kalton, N. J. (2006). Topics in Banach Space Theory. New York: Springer.
• Altay, B. & Kama, R. (2018). On Cesàro summability of vector valued multiplier spaces and operator valued series. Positivity, 22 (2), 575-586.
• Akın, N. P. (2020). Invariant summability and unconditionally Cauchy series. IJAA, 18(4),663-671.
• Başar, F. (2012). Summability Theory and Its Applications. İstanbul: Bentham Science Publishers.
• Boos, J. (2000). Classical and Modern Methods in Summability. New York: Oxford University Press.
• Diestel, J. (1984). Sequences and Series in Banach spaces. New York: Springer-Verlag.
• Eberlein, W.F. (1950). Banach-Hausdorff limits. Proc. Amer. Math. Soc., 1(5), 662-665.
• Kama, R. & Altay, B. (2017). Weakly unconditionally Cauchy series and Fibonacci sequence spaces. J. Inequal. Appl., 133, 1-9.
• Kama, R., Altay, B. & Başar, F. (2018). On the domains of backward difference matrix and the spaces of convergence of a series. Bull. Allahabad Math. Soc., 33 (1), 139-153.
• Karakuş, M. (2019). On certain vector valued multiplier spaces and series of operators. J. Math. Anal., 10(2), 1-11.
• Karakuş, M. & Başar, F. (2019). A generalization of almost convergence, completeness of some normed spaces with wuC series and a version of Orlicz-Pettis theorem. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 113(4), 3461-3475.
• Karakuş, M. & Başar, F. (2020a). Vector valued multiplier spaces of f_λ-summability, completeness through c_0 (X)-multiplier convergence and continuity and compactness of summing operators. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114:169, 1-17.
• Karakuş, M. & Başar, F. (2020b). Operator valued series, almost summability of vector valued multipliers and (weak) compactness of summing operator. J. Math. Anal. Appl., 484 (1), 1-16.
• Karakuş, M. & Başar, F. (2022a). On some classical properties of normed spaces via generalized vector valued almost convergence. Math. Slovaca, 72 (6), 1551-1566.
• Karakuş, M. & Başar, F. (2022b). Characterizations of Unconditionally Convergent and Weakly Unconditionally Cauchy Series via w_p^R-Summability, Orlicz-Pettis Type Theorems and Compact Summing Operator. Filomat, 36 (18), 6347-6358.
• Karakuş, M. & Başar, F. (2024). Vector valued closed subspaces and characterizations of normed spaces through σ-summability. Indian J. Math., 66 (1), 85-105.
• Lorentz, G. G. (1948). A contribution to the theory of divergent sequences. Acta Math., 80, 167-190. McArthur, On relationships amongst certain spaces of sequences in an arbitrary Banach space, Canad. J. Math. 8 (1956), 192-197.
• Mursaleen, M. (2014). Applied Summability Methods. London: Springer.
• Mursaleen, M. (1983). On some new invariant matrix methods of summability. Quart J. Math. Oxford, 34, 77-86.
• Mursaleen, M. & Edely, O. H. H. (2009). On the invariant mean and statistical convergence. Appl. Math. Lett., 22, 1700-1704.
• Pérez-Fernández, F. J., Benítez-Trujillo, F. & Aizpuru, A. (2000). Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series. Czechoslovak Math. J., 50 (125), 889-896.
• Raimi, R. A. (1963). Invariant means and invariant matrix methods of summability. Duke Math. J., 30, 81-94.
• Semenov, E.M. & Sukochev, F.A. (2010). Invariant Banach limits and applications. J. Funct. Anal., 259 (6), 1517-1541.
• Semenov, E.M., Sukochev, F.A. & Usachev, A.S. (2019). The main classes of invariant Banach limits. Izv. Math., 83 (1), 124-150.
• Swartz, C. (2009). Multiplier Convergent Series. Singapore: World Scientific Publishing.
• Swartz, C. (2014). Operator valued series and vector valued multiplier spaces. Casp. J. Math. Sci., 3 (2), 277-288.