We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity their dilation approach. We prove several results for operator-valued frames concerning duality, disjointeness, complementar...

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(·) ...

PREFACE 1. CONVERGENT. Does the set of cyclic operators have a non-empty interior?... 2. WEIGHTED. Is every part of a weighted shift similar to a weighted shift?. . . . 3. INVARIANT. If an intransitive operator has an inverse, is its inverse also intransitive? 4. TRIANGULAR. Is every normal operator the sum of a diagonal operator and a compact one? 5. DILATED. Is every subnormal Toeplitz operat...

in the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. in particular, it isshown that the cauchy-schwarz inequality holds. moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy hilbert space has a complementary subspace.finally, the notions of fuzzy bo...

We give dimension-free and data-dependent bounds for linear multi-task learning where a common linear operator is chosen to preprocess data for a vector of task specific linear-thresholding classifiers. The complexity penalty of multi-task learning is bounded by a simple expression involving the margins of the task-specific classifiers, the Hilbert-Schmidt norm of the selected preprocessor and ...

Lipschitzian aggregation operators with respect to the natural T indistinguishability operator ET and their powers, and with respect to the residuation → T with respect to a t-norm T and its powers are studied. A t-norm T is proved to be ET -Lipschitzian and → T -Lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an Archimedean t-norm T with additive generator t, t...

The operator norm of an n × n complex matrix A is its norm as a linear operator on the Euclidean space I C; i.e., ‖A‖ = sup{‖Ax‖ : x ∈ I C, ‖x‖ = 1}. The Frobenius norm of A is defined as ‖A‖2 = ( ∑ i,j |aij|) 1 2 = (trA∗A) 1 2 . Both these norms are used frequently in analysis of matrices. They can also be described in terms of the singular values of A − the square roots of the eigenvalues of ...

The paper presents some new estimates on the gain term of the Boltzmann collision operator. For Maxwellian molecules, it is shown that the L 1 {norm of the gain term can be bounded in terms of the L 1 and L 1 {norm of the density function f. In the case of more general collision kernels, like the hard{sphere interaction potential, the gain term is estimated pointwise by the L 1 {norm of the den...