Amalgam Uzaylarındaki Bazı Çarpanlar ve Rölatif Tamlanış Üzerine

Year 2020, , 810 - 814, 30.11.2020

Cihan Unal

https://doi.org/10.35414/akufemubid.709302

Abstract

G, $\mu $ Haar ölçümüne sahip yerel tıkız değişmeli bir grup olsun. Bu çalışmada ilk olarak, $\left( L^{p},\ell ^{q}\right) (G)$ amalgam uzayı tanıtıldı ve bazı temel özellikleri verildi. Ayrıca $\left( L^{p},\ell ^{q}\right) (G)$ amalgam uzayının doğrusal bir A alt uzayı için bir $\overset{\sim }{A}$ rölatif tamlanış tanımlandı ve bu tamlanışın bazı özellikleri ele alındı. Son olarak; $Hom_{L^{1}(G)}\left( L^{1}(G),A\right) $ ile $\overset{\sim }{A}$ arasında cebirsel bir izomorfizma ve homeomorfizma olduğu ispatlandı.

Keywords

Amalgam uzayı, Banach modül, Rölatif tamlanış, İzomorfizma

On Some Multipliers and the Relative Completion in Amalgam Spaces

Abstract

Let G be a locally compact abelian group with Haar measure $\mu $. First of all, in this paper, the amalgam space $\left( L^{p},\ell ^{q}\right) (G)$ is introduced and some basic properties of amalgam space are given. Moreover, a relative completion $\overset{\sim }{A}$ for a linear subspace A of amalgam space $\left( L^{p},\ell ^{q}\right) (G)$ is defined, and is considered several properties of it. Finally, it is proved that there is an algebraic isomorphism and homeomorphism between $Hom_{L^{1}(G)}\left( L^{1}(G),A\right) $ and $\overset{\sim }{A}$ .

Keywords

Amalgam space, Banach module, Relative completion, Isomorphism

References

• Bertrandis, J. P., Darty C. and Dupuis, C., Unions et intersections d'espaces $L^{p}$ invariantes par translation ou convolution. \emph{Ann. Inst. Fourier Grenoble} 28 (1978), no. 2, 53-84.
• Busby, R. C. and Smith, H. A., Product-convolution operators and mixed-norm spaces. \emph{Trans. Amer. Math. Soc.} 263 (1981), no. 2, 309-341.
• Fournier, J. J. and Stewart, J., Amalgams of $L^{p}$ and $\ell^{q}$. \emph{Bull. Amer. Math. Soc.} 13 (1985), no. 1, 1-21.
• Doran, R. S. and Wichmann, J., Approximate Identities and Factorization in Banach Modules. Lecture Notes in Mathematics, Springer-Verlag, 1979.
• Duyar, C. and Gurkanli, A. T., Multipliers and relative completion in weighted Lorentz spaces. \emph{Acta Math. Sci.} 23B-4 (2003), 467-476.
• Duyar, C. and Gurkanli, A. T., Multipliers and the relative completion in $L_{w}^{p}(G)$. \emph{Turk. J. Math.} 31 (2007), 181-191.
• Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis v. I, II. Berlin-Heidelberg-New York, Springer-Verlag, 1979.
• Holland, F., Harmonic analysis on amalgams of $L^{p}$ and $\ell^{q}$. \emph{J. London Math. Soc.} 2 (1975), no. 10, 295-305.
• Quek, T. S. and Yap, L. H., Multipliers from $L^{1}(G)$ to a Lipschitz space. \emph{J. Math. Anal. Appl.} 69 (1979), 531-579 .
• Rieffel, H., Induced Banach representation of Banach algebras and locally compact groups. \emph{J. Funct. Anal.} 1 (1967), 443-491 .
• Torres de Squire, M. L., Amalgams of $L^{p}$ and $\ell ^{q}$. Ph.D. Thesis, McMaster University, 1984.
• Stewart, J., Fourier transforms of unbounded measures. \emph{Canad. J. Math.} 31 (1979), no. 6, 1281-1292.