Some singular value inequalities on commutators

In this study, singular value and norm inequalities for expressions of the form \(SXT+Y\) are established. It is shown that if \(S,T,X,Y \in \mathcal {B(H)}\) such that X, Y are compact operators, then

$$\begin{aligned} \sigma _{j}\left( SXT+Y\right) \le \left( \Vert S\Vert \Vert T\Vert + \Vert Y\Vert \right) \sigma _j( X\oplus I).\end{aligned}$$

Additionally, we explore several applications of this inequality, which provide a broader framework for analysis and yield more nuanced insights. For \(X, Y\in \mathcal {B(H)}\) one notable application is the following inequality,

$$\begin{aligned} \sigma _{j}\left( \mid X-Y\mid ^{2}-2 \left( \mid X \mid ^{2}+\mid Y \mid ^{2} \right) \right) \le \left( 1+\mid \mid Y\mid \mid \right) ^{2} \sigma _{j}( \mid X \mid ^{2}\oplus I). \end{aligned}$$

These results extend existing inequalities and offer new perspectives in operator theory.