Classes of Operator-Smooth Functions. I. Operator-Lipschitz Functions

Abstract

In this paper we study the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded. Apart from the standard operator norm on B(H), we consider a rich variety of symmetric operator norms and spaces of operator-Lipschitz functions with respect to these norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of operator-Lipschitz functions.

An Elementary Approach to Elementary Operators on B(H)

“…if σ(A) ⊂ L. It was also proved in [12] that a kind of smoothness of spectrum is necessary for the validity of (4.14): if for some normal operator A the inequality (4.14) holds for all X, then given a sequence λ n ∈ σ(A) converging to λ ∈ σ(A), there is a limit lim(λ n − λ)/|λ n − λ| = f (λ). Potapov and Sukochev [15] using a powerful technique of Banach space geometry and harmonic analysis proved that (4.12) extends to all S p ideals, if A = A * .…”

“…Farforovskaya [10]). Kissin and Shulman [12] proved that the statement remains true for the operator norm and S 1 -norm, if one imposes a restriction on spectra of normal operators. Namely, for each C 2 -smooth Jordan line L there is constant c = c L such that A * X − XA * ≤ c AX − XA (4.14)…”

Classes of Operator-Smooth Functions. Iii. Stable Functions and Fuglede Ideals

“…By Corollary 5.4 of [17], g is commutator S 1 -and S ∞ -bounded. Mityagin [21] obtained (see also Theorem 3.B in [5]) that any ideal J φ is an interpolation space for the pair (S 1 , S ∞ ).…”

“…This is equivalent to the condition that the unit disc of C is J-Fuglede. If J is Fuglede, it follows from Proposition 4.5 of [17] that the spaces of J-Lipschitz and of commutator J-bounded functions coincide on each compact in C. It was proved in [ We now return to the subject of J-Lipschitz functions. Combining the results of Proposition 4.5 of [17], of Corollaries 3.6 and 5.4 of [17], and of Corollaries 2.3 and 2.5 yields the following corollary.…”

“…A summary of the main results for the most important class of Schatten ideals is given at the end of § 5. We refer the reader to [17] for a review of the history of the subject, the impetus behind our study, general notation and various technical results.…”