We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless, we prove that the intersection of these operators with the positive operators forms a proper cone in the real Banach space of hermitian operators.

We introduce and study the entanglement breaking rank of an entanglement breaking channel. We show that the entanglement breaking rank of the channel Z:Md→Md defined by Z(X)=1d+1(X+Tr(X)Id) is d2 if and only if there exists a symmetric informationally complete positive operator-valued measure in dimension d.

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Abstract. The class of absolutely norming operators on complex Hilbert spaces of arbitrary dimensions was introduced in [6] and a spectral characterization theorem for these operators was established in [11]. In this paper we extend the concept of absolutely norming operators to various symmetric norms. We establish a few spectral characterization theorems for operators on complex Hilbert spaces that are absolutely norming with respect to various symmetric norms. It is also shown that for many symmetric norms the absolutely norming operators have the same spectral characterization as proven earlier for the class of operators that are absolutely norming with respect to the usual operator norm. Finally, we prove the existence of a symmetric norm on the algebra B(H ) with respect to which even the identity operator does not attain its norm.Mathematics subject classification (2010): Primary 47B07, 47B10, 47L20, 47A10, 47A75, Secondary 47A05, 47L07, 47L25, 47B65, 46L05.Keywords and phrases: Symmetric norms, s. n. functions, s. n. ideals. R E F E R E N C E S[1] M. D. ACOSTA, Denseness of norm-attaining operators into strictly convex spaces,

We study the space of all d‐tuples of unitaries u=(u1,…,ud) using dilation theory and matrix ranges. Given two such d‐tuples u and v generating, respectively, C*‐algebras scriptA and scriptB, we seek the minimal dilation constant c=c(u,v) such that u≺cv, by which we mean that there exist faithful ∗‐representations π:A→B(H) and ρ:B→B(K), with H⊆K, such that for all i, π(ui) is equal to the compression PscriptHρfalse(cvifalse)false|scriptH of ρ(cvi) to scriptH. This gives rise to a metric prefixdnormalDfalse(u,vfalse)=logmaxfalse{c(u,v),c(v,u)false}on the set of equivalence classes of ∗‐isomorphic tuples of unitaries. We compare this metric to the metric dHR determined by prefixdHRfalse(u,vfalse)=inf∥u′−v′∥:u′,v′∈Bfalse(scriptHfalse)d,u′∼uandv′∼v,and we show the inequality prefixdHRfalse(u,vfalse)⩽KprefixdnormalD(u,v)1/2where 1/2 is optimal. When restricting attention to unitary tuples whose matrix range contains a δ‐neighborhood of the origin, then prefixdnormalDfalse(u,vfalse)⩽dδ−1prefixdHRfalse(u,vfalse), so these metrics are equivalent on the set of tuples whose matrix range contains some neighborhood of the origin. Moreover, these two metrics are equivalent to the Hausdorff distance between the matrix ranges of the tuples. For particular classes of unitary tuples, we find explicit bounds for the dilation constant. For example, if for a real antisymmetric d×d matrix Θ=(θk,ℓ), we let uΘ be the universal unitary tuple (u1,…,ud) satisfying uℓuk=eiθk,ℓukuℓ, then we find that cfalse(uΘ,unormalΘ′false)⩽e14false∥normalΘ−Θ′false∥. Combined with the above equivalence of metrics, this allows to recover the result of Haagerup–Rørdam (in the d=2 case) and Gao (in the d⩾2 case) that there exists a map Θ↦Ufalse(normalΘfalse)∈B(H)d such that Ufalse(normalΘfalse)∼unormalΘ and false∥U(Θ)−Ufalse(Θ′false)false∥⩽Kfalse∥normalΘ−normalΘ′false∥1/2. Of special interest are: the universal d‐tuple of noncommuting unitaries normalu, the d‐tuple of free Haar unitaries uf, and the universal d‐tuple of commuting unitaries u0. We find upper and lower bounds on the dilation constants among these three tuples, and in particular we obtain rather tight (and surprising) bounds 21−1d⩽cfalse(uf,u0false)⩽21−12d.From this, we recover Passer's upper bound for the universal unitaries cfalse(normalu,u0false)⩽2d. In the case d=3 we obtain the new lower bound c(u,u0)⩾1.858, which improves on the previously known lower bound cfalse(normalu,u0false)⩾3.