Abstract In this paper, the functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mi>B</m:mi> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>φ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> u\in B{V}_{\varphi }\left[0,1] which define compact and Fredholm multiplication operators <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msub> </m:math> {M}_{u} acting on the space of functions of bounded <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>φ</m:mi> </m:math> \varphi -variation are studied. All the functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mspace width="-0.08em" /> <m:mo>∈</m:mo> <m:mspace width="-0.08em" /> <m:mi>B</m:mi> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>φ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mspace width="-0.08em" /> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> u\hspace{-0.08em}\in \hspace{-0.08em}B{V}_{\varphi }\left[0,\hspace{-0.08em}1] which define multiplication operators <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msub> <m:mo>:</m:mo> <m:mi>B</m:mi> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>φ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mi>B</m:mi> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mi>φ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> {M}_{u}:B{V}_{\varphi }\left[0,1]\to B{V}_{\varphi }\left[0,1] with closed range are characterized.