<p>This article investigates a second-order differential operator with Samarskii-Ionkin-type conditions. It proves that the operator is positive in <i>C</i>[0,&#xa0;1]. Moreover, it investigates the structure of the interpolation spaces generated by this operator. Furthermore, for each <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \eta \in (0, 2^{-1})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <msup> <mn>2</mn> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, it proves the topological equivalence of this interpolation space and the Hölder space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\overset{\circ }{C}}\,^{2\eta }[0, 1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>C</mi> <mo>∘</mo> </mover> <mmultiscripts> <mspace width="0.166667em" /> <mrow /> <mrow> <mn>2</mn> <mi>η</mi> </mrow> </mmultiscripts> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Hence, it establishes that this operator is positive in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\overset{\circ }{C}} \,^{2\eta }[0, 1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi>C</mi> <mo>∘</mo> </mover> <mmultiscripts> <mspace width="0.166667em" /> <mrow /> <mrow> <mn>2</mn> <mi>η</mi> </mrow> </mmultiscripts> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. It also applies theoretical outcomes to obtain noval coercivity estimates for solutions of some specific type parabolic equation.</p>