In this article, we present a Cauchy–Schwarz type submajorization inequality for elementary operators in noncommutative fully symmetric spaces \(E(\mathcal {M})\) , where \(\mathcal {M}\) is a von Neumann algebra. As an application, we show that \(\begin{aligned} \left\| \sum _{n=1}^{\infty } a_{n}^* xb_{n}\right\| _{E(\mathcal {M})} \le \left\| \left( \sum _{n=1}^{\infty } a_{n}^* a_{n}\right) ^{1 / 2} x\left( \sum _{n=1}^{\infty } b_{n}^{*} b_{n}\right) ^{1 / 2}\right\| _{E(\mathcal {M})} \end{aligned}\) for all \(x \in E(\mathcal {M})\) and \(\{a_n\}, \{b_n\}\in \ell _2(\mathcal {M})\) , and \(\begin{aligned} \left\| \int _{\Omega } f(t)^* xg(t) d \nu (t)\right\| _{E(\mathcal {M})}\leqslant \left\| \sqrt{\int _{\Omega } f(t)^{*} f(t) d \nu (t)}x\sqrt{\int _{\Omega } g(t)^{*} g(t) d \nu (t)}\right\| _{E(\mathcal {M})} \end{aligned}\) for all \(x \in E(\mathcal {M})\) and \(f, g\in L_2(\Omega , \mathcal {M})\) . Furthermore, by combining the above two inequalities with the properties of operator monotone functions and specific holomorphic functions, we derive several interesting inequalities.
