Notes on some inequalities for Hilbert space operators

Annals of Functional Analysis, 2015

The Cauchy-Schwarz (C-S) inequality is one of the most famous inequalities in mathematics. In this survey article, we first give a brief history of the inequality. Afterward, we present the C-S inequality for inner product spaces. Focusing on operator inequalities, we then review some significant recent developments of the C-S inequality and its reverses for Hilbert space operators and elements of Hilbert C *-modules. In particular, we pay special attention to an operator Wielandt inequality.

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to the arithmetic-geometric mean inequality. For the class of all normal operators, we shall present new general characterizations.

Canadian Mathematical Bulletin, 1999

Let Ai , Bi and Xi (i = 1, 2,…,n) be operators on a separable Hilbert space. It is shown that if f and g are nonnegative continuous functions on [0, ∞) which satisfy the relation f(t)g(t) = t for all t in [0, ∞), then for every r > 0 and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

Linear Algebra and its Applications, 1994

Porta, and Recht recently proved that (ISTS-' + S-'TSIj > 21jTI(. A generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic-geometric-mean inequality. Some related inequalities are also discussed. 1.

1968

Linear Algebra and its Applications, 2011

We prove some refinements of an inequality due to X. Zhan in an arbitrary complex Hilbert space by using some results on the Heinz inequality. We present several related inequalities as well as new variants of the Corach–Porta–Recht inequality. We also ...

Linear Algebra and its Applications, 2009

We prove singular value inequalities for positive operators. Some of these inequalities generalize recent results for commutators due to Bhatia-Kitttaneh, Kittaneh, and Wang-Du. Applications of our results are given.

Integral Equations and Operator Theory, 1992

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: II[qo(P)]-lT[tp(P)]ll < 12 max{llTII, IIp-1TPII} for any bounded operator T on H, where q~ is a continuous, concave, nonnegative, nondecreasing function on [0, IIPII]. This inequality is extended to the class of normal operators with dense range to obtain the inequality II[tp(N)]-lT[tp(N)]ll < 12c 2 max{llTII, IIN-ITNII} where tp is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with q~ by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form tp(N), where q0 is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space,

Linear Algebra and its Applications, 2004

Let B(H) be the C *-algebra of all bounded linear operators on a complex Hilbert space H, S be an invertible and selfadjoint operator in B(H) and let (I,. I) denote a norm ideal of B(H). In this note, we shall show the following inequality:

Journal of Mathematical Inequalities, 2016

|AX| r. |XB| r , for any real number r > 0 and every unitarily invariant norm. . In this article we derive several refinements of Cauchy-Schwarz norm inequality for operators. In particular, we show improvements for the results of Hiai and Zhan [Linear Algebra Appl. 341 (2002) 151-169]. Besides, new type inequalities close to Cauchy-Schwarz norm inequality will be introduced. for positive operators A, B and arbitrary X. Hiai and Zhan [4] proved that if A, B, X ∈ B(H) such that A, B are positive and r > 0, then the function f (v) = A