| Summary: | In this article, <i>a</i>-Browder’s classical theorem is considered through the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>B</mi><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula>, and we show that if two operators are norm equivalent, then property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>B</mi><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> holds for one if and only if it holds for the other. The necessary conditions for the difference of the spectrum and essential approximate point spectrum of the tensor product of two operators coincide with the product of differences between the spectrum and the essential approximate point spectrum of its two factors are investigated. We also discuss the necessary conditions for the tensor product of two operators to verify the property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>B</mi><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula> and simultaneously give the equivalence between their various spectra with their Browder spectrum, likewise with the Drazin spectrum.
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