Parametrization for singular values inequalities of compact operators

Abstract

We establish novel weighted parametric frameworks for singular value bounds concerning compact linear operators on complex separable Hilbert spaces. Our investigation introduces systematic ­parameter-dependent methodologies generating infinite families of refined inequalities that extend fundamental operator-theoretic results. The central achievement demonstrates the inequality:

Let T, S ∈ B(H) be compact operators on a complex separable Hilbert space such that T is self-
adjoint, S ≥ 0, and ± T ≤ S. For any l Î(0,1], the following singular value inequality holds:

2 ( ) (( ) ( )) 2(1 ) ( ) s s l l ls j j T S T S T £ + Å- + - Å j S S

for j = 1, 2, .... .