<p>We investigate multipliers on the space of holomorphic functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H(\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Omega \subset \mathbb {C}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is an open set. For Runge domains, we characterize these multipliers as convolutions with analytic functionals. Additionally, we explore Cartesian product domains, providing a representation of multipliers through germs of holomorphic functions. Finally, we identify the appropriate topology for analytic functionals, establishing a topological isomorphism with multipliers by utilizing the topology of uniform convergence on bounded sets inherited from the space of endomorphisms on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H(\Omega )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>