On Some Generalized <inline-formula> <graphic file="1029-242X-2011-485730-i1.gif"/></inline-formula>-Difference Riesz Sequence Spaces and Uniform Opial Property

• Main Authors: Ba&#351;ar&#305;r Metin, &#214;zt&#252;rk Mahpeyker
• Format: Article
• Language: English
• Published: SpringerOpen 2011-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2011/485730
• ISSN: 1025-5834; 1029-242X

<p>Abstract</p> <p>We define the new generalized difference Riesz sequence spaces <inline-formula> <graphic file="1029-242X-2011-485730-i2.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-485730-i3.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2011-485730-i4.gif"/></inline-formula> which consist of all the sequences whose <inline-formula> <graphic file="1029-242X-2011-485730-i5.gif"/></inline-formula>-transforms are in the Riesz sequence spaces <inline-formula> <graphic file="1029-242X-2011-485730-i6.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-485730-i7.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2011-485730-i8.gif"/></inline-formula>, respectively, introduced by Altay and Ba&#351;ar (2006). We examine some topological properties and compute the <inline-formula> <graphic file="1029-242X-2011-485730-i9.gif"/></inline-formula>-, <inline-formula> <graphic file="1029-242X-2011-485730-i10.gif"/></inline-formula>-, and <inline-formula> <graphic file="1029-242X-2011-485730-i11.gif"/></inline-formula>-duals of the spaces <inline-formula> <graphic file="1029-242X-2011-485730-i12.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-485730-i13.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2011-485730-i14.gif"/></inline-formula>. Finally, we determine the necessary and sufficient conditions on the matrix transformation from the spaces <inline-formula> <graphic file="1029-242X-2011-485730-i15.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2011-485730-i16.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2011-485730-i17.gif"/></inline-formula> to the spaces <inline-formula> <graphic file="1029-242X-2011-485730-i18.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2011-485730-i19.gif"/></inline-formula> and prove that sequence spaces <inline-formula> <graphic file="1029-242X-2011-485730-i20.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2011-485730-i21.gif"/></inline-formula> have the uniform Opial property for <inline-formula> <graphic file="1029-242X-2011-485730-i22.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2011-485730-i23.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2011-485730-i24.gif"/></inline-formula>.</p>