$$\mu $$-Norm and Regularity

Abstract

In [22] we introduce the concept of a µ-norm for a bounded operator in a Hilbert space. The main motivation is the extension of the measure entropy to the case of quantum systems. In this paper we recall the basic results from [22] and present further results on the µ-norm. More precisely, we specify three classes of unitary operators for which the µ-norm generates a bistochastic operator. We plan to use the latter in the construction of quantum entropy.

Entropy of a Unitary Operator in $$\mathbb C^J$$

“…This construction was proposed in [4]. In [4], it was shown that, for regular (see [16]) unitary operators on L 2 (T n , µ), where T n is the torus and µ is the Lebesgue measure on T n (d µ = (1/(2π) n )d x 1 . .…”

“…In [4,15,16,17] the question was raised about defining a function h on the semigroup Iso(L 2 (X , µ)) of isometric operators such that the diagram End(X , µ) h ւ ց Koop…”

“…An entropy of a unitary operator h(U) was constructed in [4,16,17] by using the concept of the µ-norm.…”

“…, X K } be a partition of X , and let σ ∈ S n,K . In [16] the following definition of entropy h 1 (U) of an operator U ∈ U(H) was proposed. Consider…”

“…If the limit (1.4) exists, then h 1 (U) is called the entropy of the operator U ∈ U(H). It is known [16] that h 1 (U F ) = h(F ) for U F = Koop(F ) where F is arbitary authomorphism of (X , B, µ). Unfortunately it is unknown whether the limit (1.4) always exist.…”