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We provide and prove some new fundamental properties of the Erdélyi–Kober (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">EK</mi></semantics></math></inline-formula>) fractional operator, including monotonicity, boundedness, acting, and continuity in both Lebesgue spaces (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula>) and Orlicz spaces (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>φ</mi></msub></semantics></math></inline-formula>). We employ these properties with the concept of the measure of noncompactness (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">MNC</mi></semantics></math></inline-formula>) associated with the fixed-point hypothesis (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">FPT</mi></semantics></math></inline-formula>) in solving a quadratic integral equation of fractional order in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>p</mi></msub><mo>,</mo><mspace width="3.33333pt"></mspace><mi>p</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>φ</mi></msub></semantics></math></inline-formula>. Finally, we provide a few examples to support our findings. Our suppositions can be successfully applied to various fractional problems.