Generalized variation and translation operator in some sequence spaces. (English) Zbl 0671.47019

Musielak, J.; Waszak, A.

Hokkaido Math. J. 17, No. 3, 345-353 (1988).

Let X be the space of all real sequences. We consider two subsets X(\(\psi)\) and X(\(\phi\),\(\psi)\) defined by means of the sequential modulus and \(\phi\)-variation of sequences. Let \(X_{\zeta}\) be a modular space defined by a pseudomodular \(\zeta\) in X. Then \(\bar c=e_ 1\oplus e\) where \(e_ 1=(1,0,0,...)\) and \(e=(1,1,1,...)\), \(X^{\sim}_{\zeta}=X_{\zeta}/\bar c\), \(X^{\sim}(\psi)=X(\psi)/\bar c\) and \(X^{\sim}(\phi,\psi)=X(\phi,\psi)/\bar c\). With \(2.4(+)\) condition on \(\psi\), one obtains:
(1) The quotient spaces \(X^{\sim}_{\zeta}\) and \(X^{\sim}(\psi)\) are Fréchet spaces, and they are \(\zeta^{\sim}\)-complete.
(2) The two-modular space \(X^{\sim}(\phi,\psi)\) is \(\gamma\)-complete. These results are applied to obtain an approximation theorem by means of the m-translation on an Orlicz sequence space.

Reviewer: Chang Shaochin

MSC:

47B37	Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46A04	Locally convex Fréchet spaces and (DF)-spaces
46A45	Sequence spaces (including Köthe sequence spaces)

Keywords:

two-modular structure; generalized variations; translation; operator; sequential modulus; \(\phi\)-variation; modular space; quotient spaces; Fréchet spaces; two-modular space; Orlicz sequence space