Unitarily Invariant Operator Norms

Abstract

1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius1of a Hilbert space operator T. Among the many interesting developments, we may mention:(a) C. Berger's proof of the “power inequality”2(b) R. Bouldin's result that3for any isometry V commuting with T;(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

Numerical radius inequalities for Hilbert space operators

“…These inequalities, among other related ones concerning the submultiplicativity of the numerical radius, can be found in [2], [4]- [7], and references therein.…”

“…It is known that 1.064 < c < 1.169 (see [3], [15], and [16]). Akin to this problem, it has been shown in [4] that if A, B ∈ B(H), then (14) w…”

“…For a host of numerical radius inequalities, and for diverse applications of these inequalities, we refer to [4]- [6], [10], [11], and references therein. In Section 2 of this paper, we establish a considerable improvement of inequalities (1).…”

Multivariable ρ-contractions

“…Here we used the properties w(A ⊗ I H ) = w(A) and w(A ⊗ B) ≤ w(A) B valid for any A ∈ L(X ), B ∈ L(H) (see, e.g., [14]). …”

Some vector inequalities for two operators in Hilbert spaces with applications

“…It is known that 1.064 < c < 1.169, see [3], [35] and [36]. In relation to this problem, it has been shown in [25] that:…”