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For bounded linear operators defined on complex infinite-dimensional Banach space, H. Zariouh, in an article [1] introduced and studied the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>g</mi><mi>a</mi><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In this study, through techniques using the local spectral theory of operators, we discover the sufficient conditions that allow the transfer of the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>g</mi><mi>a</mi><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> from two tensor 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 their tensor product <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">S</mi></mrow></semantics></math></inline-formula>. The stability of the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>g</mi><mi>a</mi><mi>z</mi><mo>)</mo></mrow></semantics></math></inline-formula> in the tensor product under perturbations is also investigated. The theory is exemplified by considering suitable classes of operators such as shift operators, convolution operators, and <i>m</i>-invertible contractions.