Commutator inequalities for Hilbert space operators

Linear Algebra and its Applications, 2009

We prove several singular value inequalities for commutators of Hilbert space operators. It is shown, among other inequalities, that if A, B, and X are operators on a complex separable Hilbert space such that A and B are positive, and X is compact, then the singular values of AX − XB are dominated by those of max(A , B)(X ⊕ X), where • is the usual operator norm.

Singular value inequalities for commutators of circulant operators are given and new singular value inequalities are established.

Journal of Functional Analysis, 2007

It is shown that if A, B, and X are operators on a complex separable Hilbert space such that A and B are compact and positive, then the singular values of the generalized commutator AX − XB are dominated by those of X (A ⊕ B), where. is the usual operator norm. Consequently, for every unitarily invariant norm |. |, we have |AX − XB | X |A ⊕ B |. It is also shown that if A and B are positive and X is compact, then |AX − XB | max A , B |X | for every unitarily invariant norm. Moreover, if X is positive, then the singular values of the commutator AX − XA are dominated by those of 1 2 A (X ⊕ X). Consequently, |AX − XA | 1 2 A |X ⊕ X | for every unitarily invariant norm. For the usual operator norm, these norm inequalities hold without the compactness conditions, and in this case the first two norm inequalities are the same. Our inequalities include and improve upon earlier inequalities proved in this context, and they seem natural enough and applicable to be widely useful.

Mathematische Zeitschrift, 2008

Let X, Y , and Z be operators on a Hilbert space such that X and Z are positive. It is shown that XY − Y Z max (X , Z) Y. Applications of this commutator inequality are given.

Linear Algebra and its Applications, 2012

A singular value inequality due to Bhatia and Kittaneh says that if A and B are compact operators on a complex separable Hilbert space such that A is self-adjoint, B 0, and ±A B, then s j (A) s j (B ⊕ B) for j = 1, 2, ... We give an equivalent inequality, which states that if A, B, and C are compact operators such that ⎡ ⎣ A B B * C ⎤ ⎦ 0, then s j (B) s j (A ⊕ C) for j = 1, 2, ... Moreover, we give a sharper inequality and we prove that this inequality is equivalent to three equivalent inequalities considered by Tao. In particular, we show that if A and B are compact operators such that A is self-adjoint, B 0, and ±A B, then 2s j (A) s j ((B + A) ⊕ (B − A)) for j = 1, 2, ... Some applications of these results will be given.

Publications of the Research Institute for Mathematical Sciences, 1988

Several inequalities for Hilbert space operators are extended. These include results of Furuta, Halmos, and Kato on the mixed Schwarz inequality, the generalized Reid inequality as proved by Halmos and a classical inequality in the theory of compact non-self-adjoint operators which is essentially due to Weyl. Some related inequalities are also discussed.

Linear Algebra and its Applications, 2010

AMS classification: 47A30 47A63 47B10 47B15 47B47 Keywords: Singular value Unitarily invariant norm Commutator Compact operator Positive operator Self-adjoint operator Normal operator Inequality

Journal of Mathematical Analysis and Applications, 2002

Two types of commutator inequalities for the Hilbert-Schmidt norm are established. The first type of these inequalities is related to a classical inequality of Clarkson, and the second type is related to the unitary approximation of positive and invertible operators.  2002 Elsevier Science (USA)

ANNALI DELL'UNIVERSITA' DI FERRARA, 2017

In this paper, we obtain some Berezin number inequalities based on the definition of Berezin symbol. Among other inequalities, we show that if A, B be positive definite operators in B(H), and A B is the geometric mean of them, then ber 2 (A B) ≤ ber A 2 + B 2 2 − 1 2 inf λ∈Ω ζ(k λ), where ζ(k λ) = (A − B)k λ ,k λ 2 , andk λ is the normalized reproducing kernel of the space H for λ belong to some set Ω. A(λ) = Ak λ (z),k λ (z) ,

Linear Algebra and its Applications, 2005