<p>In this paper, we obtain more results on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(A\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation>-hyponormal operators. We show that the results for the supercyclicity of hyponormal operators described in [33] remain true for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(A\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation>-hyponormal operators. Inspired by the work [44], an operator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\( T \in \mathcal{L}_{A}(\mathcal{H})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <msub> <mi mathvariant="script">L</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is in the class <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((\sqrt[n]{\mathbf{A.H}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mroot> <mrow> <mi mathvariant="bold">A</mi> <mo>.</mo> <mi mathvariant="bold">H</mi> </mrow> <mi>n</mi> </mroot> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(T^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(A\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation>-hyponormal for some positive integer <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(n\geq2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that an operator <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(T \in (\sqrt[n]{\mathbf{A.H}})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mroot> <mrow> <mi mathvariant="bold">A</mi> <mo>.</mo> <mi mathvariant="bold">H</mi> </mrow> <mi>n</mi> </mroot> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is never supercyclic, moreover if it satisfies the translation invariant property then it is <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(A\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation>-hyponormal. In particular, if <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\( T \in \mathcal{L}_{A}(\mathcal{H})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <msub> <mi mathvariant="script">L</mi> <mi>A</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is an invertible operator then <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(T\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>T</mi> </math></EquationSource> </InlineEquation> and its inverse <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(T^{-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> have a common nontrivial invariant closed set. We also give a condition under which an <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(A\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>A</mi> </math></EquationSource> </InlineEquation> operators has a nontrivial hyperinvariant subspace.</p>