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Given a complex Banach space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">X</mi></semantics></math></inline-formula>, we investigate the stable character of the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>V</mi><mi>E</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> for a bounded linear operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">T</mi><mo>:</mo><mi mathvariant="script">X</mi><mo>→</mo><mi mathvariant="script">X</mi></mrow></semantics></math></inline-formula>, under commuting perturbations that are Riesz, compact, algebraic and hereditarily polaroid. We also analyze sufficient conditions that allow the transfer of property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>V</mi><mi>E</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> from the tensorial factors <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">T</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> to its tensor product.