Aspect of invariant properties of Orlicz amalgam spaces with respect to Banach spaces

Abstract Let G be a locally compact group, $$\Phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Φ</mml:mi> </mml:math> be a Young function. In this paper, we study the amalgam space $$W(L^{\Phi },Y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>Φ</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> defined on G , where the local component space is the Orlicz space $$L^{\Phi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>Φ</mml:mi> </mml:msup> </mml:math> and the global component is a Banach space Y containing the characteristic function of any compact subset of G . We obtain an equivalent norm on $$W(L^{\Phi },Y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>Φ</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . By using the equivalent norm, we investigate right translation invariance and completeness of the amalgam space $$W(L^{\Phi },Y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>Φ</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with a non translation invariant space Y , in general. We also present an example of a non translation invariant space Y such that $$W(L^{\Phi },Y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>Φ</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is right translation invariant in contrast to the literature.