A spectral characterization of absolutely norming operators on S. N. ideals

Abstract

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,

Universally symmetric norming operators are compact

“…Let Π denote the set of all strictly decreasing convergent sequence of positive numbers with their first term equal to 1 and positive limit, that is, Π = {π := (π n ) n∈N : π 1 = 1, lim n π n > 0, and π k > π k+1 for each k ∈ N}. (We have used Π, in [8], to denote the set of all nonincreasing sequences of positive numbers with their first term equal to 1, and hence, in accordance with that notation, we have Π ⊆ Π.) For each π ∈ Π, let Φ π denote the symmetrically norming function defined by Φ π (ξ 1 , ξ 2 , ...) = j π j ξ j and observe that Φ π is equivalent to the maximal s.n.…”

“…In [8], we used the theory of symmetrically normed ideals to extend the concept of "norming" and "absoutely norming" from the operator norm to arbitrary symmetric norms that are equivalent to the operator norm, and established a few spectral characterization theorems for operators in B(H) that are absolutely norming with respect to various symmetric norms. It was also shown that for a large family of 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.…”

“…The subsequent discussion in [8] involves positive operators of the form of a nonnegative scalar multiple of identity plus a positive compact plus a self-adjoint finite-rank. It is not clear, a priori, if the operators of this form are absolutely norming with respect to every symmetric norm on B(H).…”

“…[8, Proposition 1.3] There exists a symmetric norm • Φ * π on B(ℓ 2 (N)) such that I / ∈ N Φ * π (ℓ 2 (N)).…”

Idempotent, model, and Toeplitz operators attaining their norms

“…It is also worth to point out that compact operators are always norm attaining. While the norm attaining property at the Banach space level has been studied extensively (for instance, see [1,12,13]), the same for operators on Hilbert spaces has so far received far less attention (however, see [7,14,15,16]). On the other hand, in 1965 Brown and Douglas [6, Lemma 2], in answering a question of H. Helson [11, page 12], established a close connection between arithmetic of inner functions, Toeplitz operators, and norm attaining operators on Hilbert spaces.…”