Generalizations of Bohr inequality for Hilbert space operators

Abstract

Let B(H) be the space of all bounded linear operators on a complex separable Hilbert space H. Bohr inequality for Hilbert space operators asserts that for A, B ∈ B(H) and p, q > 1 real numbers such that 1/p + 1/q = 1, |A + B| 2 p| A| 2 + q|B| 2 with equality if and only if B = (p − 1) A. In this paper, a number of generalizations of Bohr inequality for operators in B(H) are established. Moreover, Bohr inequalities are extended to multiple operators and some related inequalities are obtained. The results in this paper generalize results known so far. The idea of transforming problems in operator theory to problems in matrix theory, which are easy to handle, is the key role.

Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces

“…The first result regarding operator version of Bohr inequality was established in [14]. For refinements, generalizations and other related results see [8], [9], [11] and [33].…”

Untitled

“…Proof : We use an induction on n. The case n = 1 is just the parallelogram law (14). Suppose that (15) is valid for n = k for some integer k 1.…”

An Order-Preserving Linear Map from Matrices to Banach *-Algebras and Applications

“…e context of matrices is given in [5]. Bohr's inequality and related results were generalized to operator algebras in [6][7][8][9][10][11][12][13][14][15].…”

“…Zhang [15] used operator identities and inequalities for approaching operator inequalities related to (1). Recently, the idea of matrix ordering for discussing operator absolutevalue inequalities appears in [8,9,11]. However, the results mentioned above has been proved in separate ways.…”