| Summary: | In this paper, we analyze local spectral properties of operators <inline-formula><math display="inline"><semantics><mrow><mi>R</mi><mo>,</mo></mrow></semantics></math></inline-formula><i>S</i> and <inline-formula><math display="inline"><semantics><mrow><mi>R</mi><mi>S</mi></mrow></semantics></math></inline-formula> which satisfy the operator equations <inline-formula><math display="inline"><semantics><mrow><msup><mi>R</mi><mi>n</mi></msup><mi>S</mi><msup><mi>R</mi><mi>n</mi></msup><mo>=</mo><msup><mi>R</mi><mi>j</mi></msup></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msup><mi>S</mi><mi>n</mi></msup><mi>R</mi><msup><mi>S</mi><mi>n</mi></msup><mo>=</mo><msup><mi>S</mi><mi>j</mi></msup></mrow></semantics></math></inline-formula> for same integers <inline-formula><math display="inline"><semantics><mrow><mi>j</mi><mo>≥</mo><mi>n</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> We also continue to study the relationship between the local spectral properties of an operator <i>R</i> and the local spectral properties of <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mo>.</mo></mrow></semantics></math></inline-formula> Thus, we investigate the transmission of some local spectral properties from <i>R</i> to <i>S</i> and we illustrate our results with an example. The theory is exemplified in some cases.
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