Further singular value and norm inequalities for matrices

Let A, B, and X be complex matrices of order n. We establish several novel singular value and norm inequalities, including: mu(k) (R(AX* B*)) <= root mu(k) (A vertical bar X vertical bar A* circle plus A vertical bar X vertical bar A*) parallel to B vertical bar X*vertical bar B*parallel to <= = parallel to X parallel to parallel to B parallel to mu(k) (A circle plus A), mu(k) (AB * + BA*) <= mu(k) ((A* A + B* B) circle plus (AA* + BB*)), and for positive semidefinite matrices A and B, mu(k) (AX - YB) <= max{parallel to A parallel to, parallel to B parallel to}(mu(k) (( X - Y) circle plus (X - Y))/2 + max{parallel to X parallel to, parallel to Y parallel to}). The first and second inequalities establish variants of well-known singular value inequalities, while the third result generalizes a commutator inequality for positive semidefinite matrices previously established by Kittaneh. Additionally, we extend several classical singular value inequalities to functional settings, broadening their scope and applicability.