Subadditive inequalities for operators

Abstract

In this article, we present a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators. For example, under suitable assumptions on the spectrum of the positive operators A and B, we prove that 2 1´r pA`Bq r ď A r`Br for r ą 1 and r ă 0, and A r`Br ď 2 1´r pA`Bq r for r P r0, 1s .These results provide considerable generalization of earlier results by Aujla and Silva.Further, we present several extensions of the subadditivity idea initiated by Ando and Zhan then extended by Bourin and Uchiyama.A real-valued continuous function f on an interval J is said to be operator convex (resp.operator concave) if f pA∇ v Bq ď presp. ěq f pAq ∇ v f pBq for all v P r0, 1s and for all selfadjoint operators A, B P BpHq whose spectra are contained in J. A continuous function f on J is called operator monotone increasing (resp. decreasing), ifA ď B ñ f pAq ď presp. ěq f pBq .

Bounds in normed spaces using convex functions

“…More precisely, given a function f : [0, ∞) → R, and two positive operators A, B ∈ B(H), is there a relation between f (A + B) and f (A) + f (B)? We refer the reader to [2,3,9,16] for a discussion of this interest. In the next result, we use the gradient inequality of convex functions to show new sub and super-additive inequalities of this type.…”

Separated spectra and operator inequalities

“…Later, Bourin and Uchiyama [5] proved the validity of (3.5) for the concave f (not necessarily operator concave). We also refer the reader to [10] for further discussion in this direction. In the next result, we prove a subadditive inequality for convex functions, under the separated spectra condition.…”

“…). Again, a convex function (in the scalar sense) is not necessarily operator convex and the function f (t) = t 3 , t > 0 provides such an example [4, Finding conditions that make scalar increasing functions satisfy the operator monotony property or those properties that make scalar convex functions satisfy operator convexity have received a considerable attention in the literature as one can see in [9,10].…”

On The Operator Hermite--Hadamard Inequality

“…In [7], it is shown that convex functions satisfy (1.3) if some empty intersection conditions are imposed on the spectra of A, B. In this article, we present several forms of (1.3) using the Mond-Pečarić method for convex functions.…”