Notes on super-operator norms induced by Schatten norms

Notes on super-operator norms induced by schatten norms

Let Φ be a super-operator, i.e., a linear mapping of the form Φ : L(F) → L(G) for finite dimensional Hilbert spaces F and G. This paper considers basic properties of the super-operator norms defined by ‖Φ‖q→p = sup{‖Φ(X)‖p/‖X‖q : X 6= 0}, induced by Schatten norms for 1 ≤ p, q ≤ ∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is prov...

Approximation properties of certain operator-induced norms on Hilbert spaces

We consider a class of operator-induced norms, acting as finite-dimensional surrogates to the L2 norm, and study their approximation properties over Hilbert subspaces of L2. The class includes, as a special case, the usual empirical norm encountered, for example, in the context of nonparametric regression in a reproducing kernel Hilbert space (RKHS). Our results have implications to the analysi...

Tight Bounds for Sketching the Operator Norm, Schatten Norms, and Subspace Embeddings

We consider the following oblivious sketching problem: given ∈ (0, 1/3) and n ≥ d/ 2, design a distribution D over Rk×nd and a function f : R × R → R, so that for any n× d matrix A, Pr S∼D [(1− )‖A‖op ≤ f(S(A), S) ≤ (1 + )‖A‖op] ≥ 2/3, where ‖A‖op = supx:‖x‖2=1 ‖Ax‖2 is the operator norm of A and S(A) denotes S ·A, interpreting A as a vector in R. We show a tight lower bound of k = Ω(d2/ 2) for...

Numerical Radius Norms on Operator Spaces

Abstract. We introduce a numerical radius operator space (X,Wn). The conditions to be a numerical radius operator space are weaker than the Ruan’s axiom for an operator space (X,On). Let w(·) be the numerical radius norm on B(H). It is shown that if X admits a norm Wn(·) on the matrix space Mn(X) which satisfies the conditions, then there is a complete isometry, in the sense of the norms Wn(·) ...

Notes on Vector and Matrix Norms