Abstract
We give singular value inequality to compact normal operators, which states that if is compact normal operator on a complex separable Hilbert space, where is the cartesian decomposition of , then Moreover, we give inequality which asserts that if is compact normal operator, then .Several inequalities will be proved.
Highlights
Let B H denote the space of all bounded linear operators on a complex separable Hilbert space H, and let positive operator T T *T 1 2 as s1 T s2 T and repeated according to multiplicity
The Jordan decomposition for self-adjoint operators asserts that every self-adjoint operator can be expressed as the difference of two positive operators
To prove the left hand side of the inequality, we will use the inequality which is well known for commuting self-adjoint operators and it asserts that
Summary
We will give singular value inequality to the normal operator A iA* , where A is normal: The right hand side of the inequalities is well known. Let A A1 iA2 be the Cartesian decomposition of the normal operator A , which implies that A1 A2 A2 A1 . The inequality A12 A22 A1 A2 , we get the right hand side of the theorem.
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